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HANDBOOK

DOUBLE STARS.

<-v6f'.

A

HANDBOOK

OF

DOUBLE STARS,

WITH A

CATALOGUE OF TWELVE HUNDRED DOUBLE STARS AND EXTENSIVE LISTS OF MEASURES.

With additional Notes bringing the Measures up to 1879,

FOR THE USE OF AMATEURS.

EDWD. CROSSLEY, F.R.A.S.; JOSEPH GLEDHILL, F.R.A.S., AND^^iMES Mt^'^I^SON, M.A., F.R.A.S.

"The subject has already proved so extensive, and still ptomises so rich a harvest to those who are inclined to be diligent in the pursuit, that I cannot help inviting every lover of astronomy to join with me in observations that must inevitably lead to new discoveries." — Sir Wm. Herschel.

*' Stellae fixac, quae in ccelo conspiciuntur, sunt aut soles simplices, qualis sol noster, aut systemata ex binis vel interdum pluribus solibus peculiari nexu physico inter se junccis composita. Stellanim siniplicium numerus est quidem major, at vero non nisi te'r'vel forusse bis tantum major quam systematum compositonim." — 2,

MACMILLAN & CO. 1879.

(E

0iAChS^

Hazell, Watson, and Viney, Printers, London and Aylesbury,

PREFACE.

This work has arisen out of our own wants as students of that branch of astronomy which deals with Double Stars, and it is on this account that we think it will be useful to others who are occupied in the same work. There does not exist any book which gives information sufficiently detailed to be of value to any one who seriously takes up this study. He must hunt through scores and hundreds of volumes if he wishes to get an accurate and complete list of the previous measures of any particular double star. These measures are scattered up arid down the astronomical periodicals of all nations. If he wishes to know with what instruments, with what apertures, and what micrometers these measures were taken, a fresh research awaits him. And if he proceeds to attempt an orbit, he will fail, unless he is a tolerably expert mathematician, from want of sufficient guidance and detail in the various mathematical papers and pamphlets that have been devoted to this subject.

This branch of astronomy is peculiarly suitable to ama- teurs. It does not require long previous training ; it does not demand unintermittent and severe work, nor the resources of a permanent observatory and staff. All it needs is a good telescope, a good eye, some patience, much conscientiousness, and — more than all — such an amount of guidance and co-

VI PREFACE.

operation as shall convince the amateur that his work is not useless, but that he is really contributing something, however small, to astronomical knowledge. And the construction of double-star orbits has always had a fascination for amateurs from the days of Admiral Smyth and 7 Virginis to the present time ; and it is perhaps the only branch of mathematical astronomy which is quite within the range of unprofessional mathematicians.

We venture to hope that this book will be of use in guiding amateurs in their work, — in pointing out what stars are of especial interest, what stars have had few or conflicting measures taken of them, at what times observations of certain stars are especially needful, and what stars have been so frequently and satisfactorily measured that for the present they need no attention. This sort of information has become a necessity owing to the extension of the subject and the number of observers. The Herschels, the elder Struve, and Madler, might with equal advantage measure every double star they saw ; but later observers must select their objects if they do not wish much of their work to be wasted. And so we find that Otto Struve, and Dawes, and Secchi, and others, have chosen stars that were certainly or probably of interest as subjects for their own work.

There has probably been no time in which so much work has been done in measuring double stars as during the last six or seven years. They have witnessed Burnham's lists of new double stars, which testify so highly to his telescope, his eye, his climate, and his industry ; Otto Struve's two im- portant volumes on his father's and his own double stars ; Dembowski's lists in the Astronomische Nachrichten ; Duner's valuable volume of observations made at Lund ; in America,

PREFACE. Vll

the work of Hall, Stone, etc. ; and in our own country, that of Knott and others.

The recalculation of orbits, also, is occupying much attention, both among foreign astronomers and at home ; and every year will enable this to be done with greater accuracy, and to be attempted for a greater number of stars.

This work, then, consists of four parts. The first part is historical, and descriptive of instruments and methods ; the second is mathematical ; the third part contains lists of measures of the most interesting double and multiple stars, with historical notes on those which are of special interest ; the fourth part is bibliographical.

In Part I., Chapter I. contains a historical introduction by Mr. Gledhill. Chapter II. is on the equatorial and the observatory, by Mr. Crossley ; Chapter III. is an account of the equatorials which have been used by double-star observers, by Mr. Gledhill ; Chapter IV. on micrometers, by Mr. Crossley; and Chapter V. on methods of observing, by Mr. Gledhill.

In Part II., Chapters I. and II. give a detailed account, with a fully worked example, of determining an orbit and an ephemeris by a purely graphical construction, founded on Herschel's and Thiele's methods, with some fresh extensions, by Mr. Wilson. Dr. Doberck, who has had very great experi- ence in double-star calculations, has contributed Chapter III., giving an example of the application of analysis to a double-star orbit already approximately known by graphical methods, and shows how greater accuracy may be obtained by it ; and Mr. Wilson gives Chapter IV. on the relative rectilinear motion of double stars ; Chapter V. on the effects of proper motion and parallactic motion ; and Chapter VI. on

Vlll PREFACE.

the mode of combining observations, and determining their weight.

Part III. contains a catalogue of double stars selected as of special interest, with a list of all accessible measures, and notes, etc., by Mr. Gledhill.

Finally, Part IV. contains the bibliography of the whole subject, and is due to Mr. Gledhill.

We may, perhaps, venture to say a word or two on the importance of this part of astronomy. It can scarcely fail to happen that accurate measures of double stars, especially when combined with a study of proper motion, will give in the future some sounder knowledge of the structure of the heavens. The calculation of double-star orbits, and the comparison of observed and calculated places, will bring out not only errors in the observations or of the computer, but the existence of forces that had been unsuspected. Resisting media and the laws of their condensation, unseen companions, and possibly new laws of force, may be discovered. And these investigations must throw light on the origin of these double and multiple systems, and thus indirectly on our own solar system.

Again ; if the difference of the linear velocities of the components of a binary system can be directly ascertained by the spectroscope, this fact, combined with a good know- ledge of the orbit and of the period of revolution, and of the apparent mean angular distance, will lead to a know- ledge of the parallax of the system, and therefore also to a knowledge of their mass.

At present we cannot see the significance of all that has been discovered : for example, the fact that the orbits hitherto computed are all elliptical, and very nearly all of

PREFACE. IX

large eccentricity, is too uniform to be an accident, and yet it is too isolated a fact to build theories on with safety. It does, however, seem to prove that these are genuine systems ab initio, and are not formed by the fortuitous approximation of single stars.

Nor, again, have we found the reason why the type of triple stars, such as /t Herculis, 7 Andromedae, f Cancri, fi Bootis — a bright primary and a faint binary companion — should be so common. When, further, we come to examine into the colours of binaries, we cannot yet see to what previous stage in their history is owing the absence of red stars in these systems, and the frequency of other colours which in their turn are rare in solitary stars. Spectroscopic observation will doubtless add some information on the point of fact, but will only remove the difficulty one stage further on. Again, the phenomenon of variable and temporary stars has always suggested the notion of a revolving dark companion. This may need further examination, and light may be thrown on the subject from tracing the gradual development of binary systems. In a word, the further study of binaries will help our successors to know what is the development-order of star systems and planetary systems.

The present work, therefore, is intended to facilitate the labours of future students of sidereal astronomy, by sup- plying the materials for the study of double stars in a convenient form, and as complete (so far as it is intended to go) as our utmost pains could make it.

The distribution of double stars has not been investigated, and it is perhaps at present premature to attempt it until more is known about them in both hemispheres; but there are already plain indications that it is not entirely fortuitous.

X PREFACE.

A knowledge of their distribution will scarcely fail to throw light on the great problem of the structure of the sidereal universe.

Similarly, it will be observed that we devote no chapters to the variability of colour or intensity in the components of double stars. We have been debarred from this branch of the subject by want of time, by the badness of our climate, and by the unsuitability of our instruments. It is to be hoped that this work will be taken up by some one else. Small telescopes, and especially small reflectors, are well suited to the examination of colour ; but if possible a careful spectroscopic examination of each star should be made. We have, however, provided in the bibliographical part of the book some references to the chief works and papers on this subject.

We therefore commend this study to amateurs. They may be encouraged by the thought that, with few exceptions, all the great workers in this branch of astronomy have been amateurs ; and be stimulated to exertion by the thought that observations made now will certainly be of value to their successors. The stars will not stand still. How can we be idle, and let slip the time for observations, which, if not made now, can never be made hereafter .'

Bermerside,

September, 1879.

CONTENTS. PART I.

HISTORICAL, AND DESCRIPTIVE OF INSTRUMENTS AND METHODS.

CHAPTER I. Historical Introduction — Early Observations — Discovery of Binary

Stars— The great Observers and their Work . . . . i

CHAPTER II. The Equatorial : its construction and adjustments — The Clock —

Observing Chairs — The Observatory 1 1

CHAPTER III. Some account of the Equatorials which have been used by Double- Star Observers 31

CHAPTER IV. The Micrometer — Construction — Methods of testing it and for

finding the value of one revolution of the screw . . . -So

CHAPTER V. Methods of observing Double Stars — Measurement of Angle and Distance — Special Methods — Dawes's Prism — Occasional Methods — A set of Measures — Specimen Forms of Registry — Weights — Contracted Apertures — Precautions and Hints . . 64

PART II.

ON THE CALCULATION OF THE ORBIT OF A BINARY STAR.

U/

CHAPTER I. Introduction — Statement of Problem — Method of Solution adopted — Preparing the Observations for Use — Reduction to a Selected Epoch— Drawing of the Interpolating Curve— Smoothing the Curve — Employment of Measures of Distance — Drawing the Apparent Ellipse— Determination of the real Ellipse ... 84

XU CONTENTS.

CHAPTER 11. PAGE

Example of an Orbit worked by a Graphical Method . . . io6

CHAPTER III. Dr. Doberck's Example of an Orbit worked by Analytical Methods 1 18

CHAPTER IV. On Relative Rectilinear Motion 134

CHAPTER V. On the Effect of Proper Motion and Parallax on the Observed

Position Angles and Distance of an Optically Double Star . 139

CHAPTER VI. On the Errors of Observation and the Combination of Observations 144

PART III.

THE CATALOGUE AND MEASURES.

Introductory Remarks 151

A Catalogue of Binary and other Double Stars deserving of

attention 152

Lists of Measures, with Historical Notes, etc 175

Supplementary List of Measures 405

Appendix 4x1

Additional Notes to Measures 417

Binary Stars classified 418

Note on Systematic Errors in the Measures of Angle and Distance

of Double Stars ^o

PART IV.

BIBLIOGRAPHY.

LIST A.

Some of the most important Works and Papers on Double Stars . 425

LIST B. Some Papers on the Micrometer 440

LIST C. Some Papers on the Colours of Double Stars 457

ADDITIONAL NOTES 459

LIST OF ILLUSTRATIONS.

Bennerside, Halifax. (Frontispiece.') Temple Observatory, Rugby The Bermerside Equatorial Observing Chairs . Bermerside Observatory Parallel-wire Micrometer .

PACE

I

ID

28,29

30

5«

LIST OF PLATES.

Map of the Pleiades

Interpolating Curves of Castor

Graphical Construction of Orbit of Castor

Interpolating Curve of 61 Cygni

A 4- B Looped Curve of fCancri — — — and C

Facing page

56 112 116 138

24S

DIAGRAMS.

Inverted field of telescope, showing how position aisles are reeistered .......

Curve on Millimetre paper Apparent Orbit of S. bo 2-73

2. 1 196 2. 1356 Z. 1424 Z. 1516

2. 1523 2. 1670 2. 1687 2. 1728

s. 1757

2. 1888 2. 1937 2. 1938 2. 1998 2. 2032 2. 2055 2. 2120 2. 2173 2. 2272 2. 2382 2. 2383 2. 2579 2. 3062

66

95 180

183

247 2S9

262 269 270 281 286 287 290

303 308 312 320 324 328

335 343 349 359 360

371 403

ERRATA.

Page 154. No 100. The Dec. should be 28° 55'; see also No. i(X> in the "Measures," p. 202. 154. No. 116. For the magnitudes, read 6, "]. 163. No. 524. The Dec. is 2° 15'. 163. Line I, read, £ Scorpii.

165. The Ref. No. 623 is given twice ; the second should be 624. 275. The formula given are Doberck's modified by Duner. Doberck's formulce are

P = 8i°'25 - o°-567 (t - 1850) + o°-oo57 (/ - 1850)'. A = 2"-47 + o"-oi3 (t - 1850).

373. W. and S.'s positions of J Cygni should be 339'i, 335"8. And in the diagram, 1875 should be at the other end of the curve.

406. Line 34. The date of De.'s measure (38°-9, o"'85) is i867*9.

407. Line 22. The measure of h. 4649 in l837"5 wis also by Hj.

The plate facing p. 248, illustrating the looped path of f Cancri f and C V

is taken from the Observatioru de foulkooa, vol. ix.

DOUBLE STARS.

PART I.

CHAPTER I.

HISTORICAL INTRODUCTION.

The history of double-star astronomy begins with the year 1779, a year for ever memorable as that in which the greatest of observers began the investigations which created a new department of observational astronomy.

The results of the occasional attention 01 astronomers to this class of observation prior to the time of Herschel were small indeed. Riccioli, about the middle of the seventeenth century, saw that ifUrsse Majoris was double, and Kirsch also noted the same fact in 1700. Huyghens saw 6 Orionis as a quadruple star in 1656; in 1664 Hooke first saw 7 Arietis as a double star and a Centauri appears to have been the fourth double star which yielded to the power of the telescope, as Feuill^e is said to have discovered it in 1709 at Lima. Bradley separated 7 Virginis in 17 18, and both Messier and Cassini watched the occultation of the components by the moon.* Castor was found to be a double star in 1719, 61 Cygni in 1753, ^ Cygni in 1755 ; then followed 7 Andro- medse, e Lyrae, 70 Ophiuchi, X Cancri, /3 Scorpii, f Ursae Majoris, etc. Pigott discovered three in I779.t Nor must

♦ See the Histoire de PAcademie Royale des Sciences, for the years 1678 1720, 1774. + Phil. Trans., vol. Ixxi.

I

2 DOUBLE STARS.

the numerous wide pairs detected by Christian Mayer pass unnoticed. This industrious observer, working at Mannheim with an eight-feet mural quadrant by Bird and a power of about 60 to 80, observed and catalogued a considerable number of stars with Comites.* A short extract from his book f will give a good idea of the character of the objects and his mode of observation : —

1777-	Stella cum comite. Comes Aldebaran Comes Electra Comes Algol	Gradus lucis.	Differentia Ascensionis rectae.	Differentia Declinationis.
Jan. I .. 13	8-9 Teles. 8	0° 2' 14" -2 008 0 2 49-5	0° 12' 29" 0 0 32 -5 0 9 9-5

At the end of the volume a table of the new pairs discovered by him (72 in number) is given ; among them are the follow- ing:—

	Mag.	Differentia in R.A.	Differentia Declinationis.	Dist.
7 Andromedte Castor j" Cancri	2, 6 1, 6 7> 8 5. 5 3. 7 6, 8 3. 7	sec. 0-9S 07 00 o-s 0-S3 0-2 206	3-8 77 6-3 40 3-0 19-9	15-2 ii-o 77 9-9 8-9
y Virginis
a Herculis
e Ly rse
fi CvPTii	36-6

In 1777, Maskelyne, in a letter to Mayer, says that he saw a Herculis double in August 1777, magnitudes 3 and 6, the preceding star being the fainter, and that the distance of the centres was 7". Mayer also wrote two other papers on this subject. J

To return. It was in 1779 that Sir William Herschel began to direct his wonderful energy to the observation of double

* Mayer says that Flamsteed first used the word comes for the smaller star of a pair.

t See his work, De 7iovis in caelo sidereo Phanomenis, etc., 1779.

X " Dc centum stellarum fixarum comitibus, eorumque insigni usu ad determinandum moium proprium fixarum; " and " De tniris fixarum comitumque mutaiionibus a me observatis a tempore eel. Flamsteedii."

HISTORICAL INTRODUCTION. 3

stars ; and his famous paper is so interesting, and so fully exhibits the state of this department at the time he wrote, that a short account of it may here with propriety be given.

The great historical problem of finding stellar parallax had presented itself to him, and with his usual ardour he set himself the task of grappling with all its difficulties. After noticing Galileo's method, and the previous attempts to carry it out by Hooke, Flamsteed, Molineux, and Bradley, and pointing out the cause of their failure, he proceeds to describe his own method, viz., to measure the position angle of two stars of unequal magnitudes at two opposite points of the earth's orbit. He states the essential conditions to be, (i) that the stars be near each other; (2) that their magnitudes be very unequal. He then criticises the attempt made by Dr. Long, and points out the causes of his want of success, viz., unsuitable double stars, and want of adequate optical power. (Dr. Long had chosen 7 Arietis, Castor, 7 Virginis, etc., and his magnifying power did not exceea 70.) His own method is then shown to be independent of refraction, nutation, precession, change of obliquity of the ecliptic, and aberration. The highest possible power is to be used ; and a figure showing a Lyrse under powers from 460 to 6450 is given. Having fully satisfied himself that the method was sound and practicable, the next step was the selection of suitable pairs of stars. And here his own noble words may fitly be quoted : —

" I resolved to examine every star in the heavens with the utmost attention, and a very high power, that I might collect such materials for this research as would enable me to fix my observations on those that would best answer my end. The subject has already proved so extensive, and still promises so rich a harvest to those who are inclined to be diligent in the pursuit, that I cannot help inviting every lover of astronomy to join with me in observations that must inevitably lead to new discoveries." — Phil. Trans., vol. Ixxii.

4 DOUBLE STARS.

It was in this spirit, and with this glowing enthusiasm, that Herschel began those sweeps and measures which have added so much to our knowledge of the sidereal universe.

A full description of his method of finding the position angle and distance apart of the components of a double star, statements respecting the accuracy of his estimations and micrometric measures, etc., are then given. Then comes the catalogue of his discoveries. The pairs given number 269, and they are arranged in six classes, according to dis- tance : Class I., close pairs requiring " indeed a very superior telescope, the utmost clearness of air," etc. II., those suitable for "very delicate measures of the micrometer." III., from S" to IS". IV., from 15" to 30". V., from 30" to i'. VI., from I' to 2'.*

Of these 269 objects, 227 were new, 9 were known before Mayer's time, and 33 were known to Mayer and other observers. A single extract will show the form and character of the information given respecting these stars : —

"16. 71 Coronas borealis, Fl. 2.

" Sept. 9. — Double. A little unequal. They are whitish stars. They seem in contact with 227, and though I can see them with this power, I should certainly not have discovered them with it ; with 400, less than \ diameter ; with 932, fairly separated, and the interval a little larger than with 460. I saw them also with 2010, but they are so close that this power is too much for them, at least when the altitude of the stars is not very considerable ; with 460 they are as fine a miniature of e Bootis as that is of a Geminorum. Position 59° 19' n following."t

In 1803 appeared Herschel's celebrated paper announcing the discovery of binary stars, and this was followed in 1822 by a list of 145 new double stars.

* Herschel's first measure of a double star is said to have been that of the trapezium in Orion. t Phil. Trans., 1782.

HISTORICAL INTRODUCTION. 5

During the first twenty years of this century, notwith- standing the splendour of the discoveries above described, double stars were but little observed. No doubt the prin- cipal cause was the want of instruments of suitable power and construction. In 1816 Sir John Herschel began to review the double stars discovered by his father, and was soon joined by Sir James South. For a list of his papers containing measures, etc., see List A, Part IV. For this distinguished observer, double-star measurement ever pos- sessed a charm ; and from time to time, all through his long life, catalogues, measures, etc., were contributed by him to the Memoirs of the Royal Astronomical Society. Valuable results were also obtained during Sir John's stay at the Cape of Good Hope; and just before his lamented death he was busy at work on a general catalogue of double stars.

Two years before the reviews began at Slough, Friedrich Georg Wilhelm Struve, in the distant and ill-furnished ob- servatory of Dorpat, was turning his attention in the same direction. Although an 8 feet transit by Dollond, and a 5 feet telescope by Troughton (power 126), were the only instru- ments at his command, he began to observe the positions, and occasionally to measure the position-angles and distances, of double stars. These results are to be found in the early volumes of the Dorpat observations. And in order to facilitate the study of this subject, he published in 1820 the places of double stars. In 1821 the fine Ertel Circle was received, and in 1824 the famous Fraunhofer refractor was added. Then began the great survey of the heavens between the pole and 15° of south declination, for the purpose of dis- covering new double stars, and the formation of a general catalogue of them. From 1824 to 1835 Struve and his assistants devoted themselves almost entirely to the execution of this noble scheme, and in 1837 appeared the results in the magnificent work entitled MensurcB Micrometrica Stellarum duplicium et muUiplicium. Nor did double stars

'/

O DOUBLE STARS.

lose their attractiveness at the observatory of Dorpat after the conclusion of this vast undertaking. In 1839 the splendid observatory at Poulkova was established, and in 1861, on the resignation of his father, the directorship was placed in the hands of Otto Struve. From year to year careful and syste- matic measures have been made up to the present time, and the latest publication of the distinguished son of the great Struve is a noble series in two volumes of measures of the most important double stars.

Here, too, must be mentioned the labours of Admiral Smyth. With an 8 feet equatorial, this excellent observer measured 680 stars between 1830 and 1843, and the results were published in 1844, under the title Cycle of Celestial Objects. In i860, the Speculum Hartwellianum, containing later measures, etc., was published.

Madler, observing with the Dorpat refractor, measured a large number of double stars between the years 1834 and 1845, and published the results in 1847, in an elaborate work entitled Untersuchungen iiber die Fixstern-systeme. In this fine work are given extensive lists of double stars having probable direct motion, probable retrograde motion, and certain motion ; chapters dealing with the orbits of the most important binaries ; very complete lists of measures ; a chapter on the combinations of double stars to form " higher systems," etc., etc.

Between 1830 and 1868 Dawes communicated many important lists of measures and papers on double stars to the Royal Astronomical Society. His great catalogue was, however, not published till 1867. This work is enriched by the addition of valuable introductions, notes, and lists of measures made by previous observers.

Valuable measures were made at Lord Wrottesley's obser- vatory between the years 1843 and i860.

Powell and Jacob, at Madras, made many useful measures, the former from 1853 to 1862, and the latter from 1853 to 1857.

HISTORICAL INTRODUCTION'. 7

The Baron Dembowski began his fine series of measures in the year 1852 at Naples. He proposed to measure all the Dorpat " lucidcz" within the reach of his instrument. This important undertaking he successfully accomplished between the years 1852 and 1858 ; and a more valuable contribution to this department has rarely been made. In 1862 he resumed the examination of those Dorpat stars which exhibited changes in angle or distance ; and the careful measurement of the great binaries has been con- tinued up to the present time. The last review also included the measurement of a large number of the double stars discovered at Poulkova.

Secchi, in the years 1856 to 1859, paid considerable at- tention to double stars, and in i860 appeared his Catalogo di 1 32 1* stelle Doppie misiirate col grande equator iale di Merz air osservatorio del Collegia Romano. Some years later he also published Serie seconda delle niisiire microm^triclte, fatte all' equator iale di Merz del Collegio Romano, dal 1863 al 1866 inclusive, stelle doppie e Nebulose dal P. A. Secchi.

In 1861, the late Rev. R. Main, Radcliffe Observer, began to observe a selected list of double stars. These observations have been published from year to year in the volumes issued by the observatory up to the present time. They have all been made with the Heliometer.

At Mr. Barclay's observatory the measurement of double stars has always held a prominent place in the work of the observers Mr. Romberg and Mr. Talmage.

Dun^r, at the Lund Observatory, issued a volume of double star measures in 1876. It contains his results from 1867 to 187s, and is a valuable addition to the works on double-star astronomy.

Mr. O. Stone and his assistants at the Cincinnati Observatory have for some time paid special attention to double stars, and several lists of measures have already been published. * The number is really 1 22 1.

8 DOUBLE STARS.

Mr. Burnham, of Chicago, has published no less than nine catalogues of double stars, his own discoveries, since 1871 : all these objects have also had their positions and distances either measured or estimated by this most industrious observer.

Dr. William Doberck, at Markree Observatory, has taken up this branch of astronomy with great spirit and success. For some of the results of his labours see List A.

Professor Pritchard, of the new Oxford University Obser- vatory, assisted by Messrs. Plummer and Jenkins, is making careful measures of the principal binaries, and is also engaged in a re-investigation of their orbits, by a method possessing some new features, and which seems to yield good results.

M. Camille Flammarion has devoted himself with great ardour to double-star investigations : his catalogue of im- portant objects, with lists of measures, will shortly be published.

This subject has always attracted the attention of patrons and wealthy amateurs, and the names of Lord Wrottesley, George Bishop, Esq., J. G. Barclay, Esq., Colonel Cooper, Edward Crossley, Esq., Isaac Fletcher, Esq., M.P., and G. Knott, Esq., must here be mentioned as deserving of special praise for the spirited manner in which they have established and supported observatories for the prosecution of this class of observation.

Lastly, compilers of useful catalogues of binary stars and the writers of handbooks must not be forgotten : among the former, Mr. A. Brothers, F.R.A.S., and among the latter the Rev. W. A. Darby, M.A., and, above all, the Rev. T. W. Webb, M.A., deserve especial mention.

Measures by the following observers and others have also been published: Auwers, Bessel, Bond, Brunnow, Challis, Dunlop, EUery, Encke, Engelmann, Ferrari, Fletcher, Galle, Gledhill, Hall, Hind, Hold en, Jacob, Kaiser, Knott, Lassell, Maclear, Miller, Mitchell, Morton, Newcomb, Nobile, Powell, Schiaparelli, Seabroke, Sporer, Waldo, Wilson.

THE BERMERSIDE EQUATORIAL.

CHAPTER II.

THE EQUATORIAL: ITS CONSTRUCTION AND ADJUSTMENTS.

In making a few remarks upon the instruments required by double-star observers, it is not our intention to give an ex- haustive description, but rather to confine ourselves to a few points which may serve as some guide to the amateur who wishes to provide himself with these instruments, or who, being already equipped, desires to set to work with confidence.

It is first of all necessary to be furnished with a good refractor or reflector, equatorially mounted, of sufficient aper- ture, and driven by clockwork. And we do not hesitate to say that we much prefer a refractor, as being more stable in its adjustments, less disturbed by atmospheric conditions, and more durable in its optical surfaces, — conditions which seem to us to do more than counterbalance any advantages arising from the smallness of the star discs, and the absence of colour obtained from good reflecting telescopes.

We will assume that an equatorially mounted refractor is chosen. This should be of not less than six inches aperture, in order to be generally useful. An aperture of eight or nine inches would be a liberal and handsome provision. Good work may be done on some stars with smaller apertures, but we are afraid they would cause disappointment by their limited power.

To obtain a good instrument, it is best to secure the ser- vices of a first-class maker, who has made large equatorials his speciality. Among English makers it is hardly necessary to mention such names as those of Messrs. Troughton and

12 DOUBLE STARS.

Simms of London, T. Cooke and Sons of York, and Mr. Howard Grubb of Dublin, whose well-known achievements speak for themselves.

We will now take up the different parts of the equatorial, beginning with the object glass. This requires the greatest possible amount of skill and patience in its construction, and great care should be exercised in its selection by the employ- ment of suitable tests.

After examining the lenses in their cell by transmitted light, to discover any flaws of serious magnitude (for minute sand- holes and bubbles are not serious), and then looking at its two outer surfaces by reflected light to see if the polish is uniform and good, replace the object glass in the tube and turn it upon some elevated object, as a church spire or chimney with a bright sky background. Focus carefully with a low power, and if the outlines are sharply defined and free from colour, the probability is that the glass is fairly achromatic. To render this test more severe, Stokes recommended that half the object glass should be covered by a semicircular piece of cardboard.

For the next test the instrument must be directed to the sky at night, and some patience and judgment will be needed in selecting a night suitable for the work. Examine the moon or any of the larger planets at an elevation of not less than 30° above the horizon, the higher the better; and if there be sharp- ness of outline, distinctness of detail, and absence of vibration, the night is one suitable for the purpose. Now turn to stars of different magnitudes, as near the zenith as may be, using a high power; and if clean round discs are obtained, free from wings and stray light, the result is so far satisfactory. Next examine the rings which surround the central small disc, when the eyepiece is moved a little within and without the focus. If the rings are circular, and each of uniform brightness all round, and sharply distinct from one another, the lens may be con- sidered well centered and corrected. If the glass should fail

THE EQUATORIAL. 1 3

under this test, it must be carefully adjusted by the centering screws. It is, of course, best to have this done by the maker before the instrument leaves his workshop.

The central portion of the glass may now be covered with a disc of paper whose diameter is two-thirds of that of the aperture. Focus sharply on a star; remove the disc, and cover up the outer portion of the object glass with a diaphragm whose aperture is also two-thirds of that of the glass. If the focus remains unaltered, the figure is good.

The tests for separating and illuminating power may next be applied. Close double stars and minute points of light will supply the means. This can only be effectually done on the finest nights.

For lists of test objects and valuable information on these and other cognate matters, the excellent little book by the Rev. T. W. Webb, Celestial Objects for Common Telescopes, should be consulted.

It is scarcely necessary to discuss at length the merits of the different forms of mounting of the equatorial. The German form of mounting is now almost universally adopted, and with modern excellence of manufacture it may be considered quite equal in steadiness to the old English form.

The essential points are rigidity, strength, durability, and accuracy, facility, and permanence of adjustment. The tube is often made of sheet brass ; sheet iron is lighter, cheaper, and more durable. The declination axis and polar axis should have plenty of bearing surface, and be of ample strength. The weight upon the polar axis should be relieved by friction rollers. The declination and hour circles should read by opposite verniers to 10" or 20" of arc, and i or 2 seconds of time respectively. The declination circle may be placed next to the telescope tube, so as to be read off conveniently by a reader from the eye-end, suitable illumination being provided. The hour circle should be moveable, and the telescope should have a clamp and slow

14 DOUBLE STARS.

motion in declination. The clockwork should be strong and powerful, a weak clock being one of the commonest defects of equatorials. Slow motion is also required in right ascension, and it is usually obtained by means of differential wheels in connection with the driving clock, an endless cord being brought to the eye-end. The tangent screw of the driving arc should be capable of perfect adjustment, and should not have to be removed from the arc for the purpose of releasing the telescope from the clockwork. This should be done by a clamp on the polar axis. The lamp for illuminating the micrometer is best placed at the end of the declination axis, away from the telescope, the axis being perforated for the light to pass through into the tube, whence it is reflected at right angles to the eye-end, either by a re- flector just outside the cone of rays, or by a tiny reflector say one-eighth of an inch in diameter, in the centre of the cone, and carried by an arm in such a manner that it can be moved to one side at pleasure. The first plan is perhaps the least objectionable ; the latter is, however, adopted by Mr. Grubb. In Messrs. Cooke's form of mounting, the whole instrument is carried upon a heavy central iron pillar, which takes up less space in the observatory than any other form, and does not interfere with the observing chair in any position of the instrument. The base of the pillar being turned true in the lathe, is also easily bedded in the foundation-stone.

The following principal adjustments should be provided for, viz., (i) the polar axis in altitude ; (2) the whole instru- ment in azimuth ; (3) the eye-end for coUimation ; (4) the verniers of both circles for index errors.

The declination axis is commonly set by the maker at right angles to the polar axis. When the bearing surfaces of this axis are not equidistant from the polar axis, or bear unequal weights, there may be a tendency to unequal wear, and therefore to change of inclination, unless the bearing surfaces are proportional to the weights they carry.

THE EQUATORIAL. 1 5

Before erecting the equatorial it will be well to see that the stand is carefully marked with a north and south point by the maker, and that a meridian line be drawn through the centre of the foundation-stone to the walls of the obser- vatory. After preparing and levelling the stone, it is now easy to set the instrument to the meridian line approxi- mately, or at least within the limits of the adjusting screws in azimuth.

We must now determine the following errors of the instru- ment, and make the necessary corrections : —

1. Error of altitude of the polar axis.

2. Index error of the declination circle.

3. Error of collimation, or deviation of perpendicularity of

telescope to the declination axis.

4. Error of azimuth, or deviation of the polar axis from the

plane of the meridian.

5. Index error of the hour circle.

6. Error of the declination axis from true perpendicularity

to the polar axis.

No. I and No. 2 are determined by the same set of obser- vations. Bring the telescope approximately into the plane of the meridian, say on the west side of the polar axis : put in the wire micrometer with a low power, and bring one of the moveable webs into the centre of the field of view approxi- mately. Make a star run along the web by means of the slow motion in declination: move the micrometer through 180°, and if the star will not now run along the web from side to side of the field, bring the web half-way towards the star by turning the micrometer screw, and then set the star on the web by the slow motion in declination. Again, turn the micrometer through 1 80°, and if the star now travels along the web, the latter passes through the centre of the field of view, or the centre of rotation of the position circle of the micrometer.

Now set the centered web on a bright star south of the zenith near the meridian whose position is given in the

1 6 DOUBLE STARS.

Nautical Almanac. Clamp in declination and read off the declination circle. Unclamp, swing the telescope over to the east side (being careful not to disturb the micrometer), set on the star again, clamp, and read off as before. It the star has north declination, the correction for refraction is subtracted from the readings ; if the star be south of the equator, add the refraction correction. If the star has north declination, and half the sum of the two readings corrected for refraction be greater than the true declination as given in the Almanac, the north pole of the instrument is too high ; if less, the pole is too low. If the star is south of the equator, and the result be too great, the pole is too low ; if too small, the pole is too high.

Half the difference between the two readings in either case is the index error of the declination circle. The following example will illustrate this : —

Jan. 28, 1878. Aldebaran was placed on the centered web.

Dec. Circle.

Telescope West, 16° 17' o") ,0 , „ ^t

16° 17' o"/ ™^^° "^ '7 o N.

Telescope East, 16° 16' 40") ,<, ,, „ ..

16° 16' o") ™^^" '° '" ^° N-

Sum

Half sum

Correction for refraction . . .

Observed declination True declination

Error of altitude of polar axis, too high 3"

The correction for refraction is obtained thus : —

Colat. of place 36° 18'

North declination of star 16° 16'

32° 33'	20"
16° 16'	40" 44"
16° 15' 16° 15'	56" 53"

Approximate altitude 52° 34'

Mean refraction, 44".

THE EQUATORIAL. I 7

The mean refraction is sufficiently correct for our purpose. A mean refraction table is to be found in all collections of mathematical tables, and in many astronomical handbooks.

3. If the polar axis is not far from the plane of the meri- dian, the error of collimation, — that is, the deviation of the telescope from perpendicularity to the declination axis, — can easily be determined as accurately as the hour circle will admit of. Thus : place the telescope on the west side, and near both the meridian and the equator. The micrometer having been undisturbed, turn it throug hgo" : the centered web now points to the pole. Set the telescope a little in advance of the nearest bright star, and note by the sidereal clock the time of transit across the web. Read off the hour circle : throw the telescope over to the east side, transit the same star, and read off as before. If the difference between the transit times be greater than that of the hour circle readings, the angle formed by the telescope and the declination axis is too great towards the eye-end, and the eye-end must be moved towards the declination axis. If the difference of the transits is less, the angle is too small, and the eyepiece must be moved away from the declination axis. Half the difference between the interval by the clock and that by the circle is the error.

The following example will exhibit the method of proceed- ing in this case : — Jan. 28, 1878. 8 Ononis. Dec, 0° 23' 28".

Clock. Circle.

M. 5. M, s.

Telescope West ... 20 26 25 32 „ East ... 23 45 28 40

3 19 38

Half the difference, 5*5 s. x cos. 23' 28" = error required.

As the clock interval is the greater, the eye-end must be moved towards the declination axis so as to diminish the angle between the telescope and the declination axis.

4. The error of azimuth is not so easily determined as the

2

1 8 DOUBLE STARS.

previous errors, on account of the difficulty in correcting for the effect of refraction. This can be done by calculation, as is fully explained in Loomis's Astronomy, Arts. 32, 145 ; but it can also be done quite effectively, and much more readily, by the following method. Centre the web of the micrometer, set the telescope to the true declination of a Greenwich star about six hours east or west of the meridian, and from 30° to 60° in altitude. Sweep to the star in right ascension with the finder, and if the star is some distance from the centre of the field, move the telescope in azimuth until it passes a little below the centre of the field. Now take a small clinometer, (which can be readily constructed with a piece of hard wood, a semicircular protractor, and a small plumb-line,) and place it on the telescope ; read off the altitude to the nearest degree. Rotate the micrometer until the fixed wires are approximately in the vertical plane. Find the mean refraction for the ob- served altitude from the Table of Refractions. Now bring the web that is not centered below the centered one by a distance equal to the angle of refraction. Set in azimuth so that the star will pass through the intersection of the lower web and the fixed wires of the micrometer. Repeat the operation on a star in the opposite quarter of the heavens ; and if this star also comes to the corresponding intersection the polar axis is in the plane of the meridian.

If the micrometer screw have 100 threads to the inch, and the focal length of the object-glass be measured from its centre, the angular value of one revolution of the screw will be known well enough for the above purpose. (See the chapter on the Micrometer.)

5- The index error of the hour circle can only be determined by an independent observation for time, unless the declination axis is provided with a striding level for the purpose of render- ing it horizontal, or truly east and west. In this latter case, all that is necessary after levelling is to set any division of the hour circle at the index point of the vernier which moves

THE EQUATORIAL. 1 9

with the telescope, then adjust the index point of the fixed vernier to the same division, and this will be the south reading. It is, however, still .more convenient, when it can be done, to set the fixed vernier east or west according as the Observatory is west or east, by the difference in time between the longitude of the Observatory and Green- wich : this will save the trouble of always having to add or subtract this quantity from the right ascension of a star when setting the telescope by the circles. If the declination axis is not provided with a level, which is seldom the case, as it is not indeed necessary, then sidereal time must be obtained from occultations of stars by the moon, from Greenwich time when telegraphed to the nearest post-office or railway station, by Dent's Dipleidoscope ; or, best of all, from a small transit instrument of about two inches' aper- ture; for such an instrument will give the time to the tenth of a second, and help to make the Observatory com- plete and independent.

The telescope can now be brought into the meridian by a star at the time of transit, and the fixed vernier set as before.

6. The error of the declination axis from true perpendicu- larity to the polar axis should be so small as to fall within the error of the setting of the instrument. It is not usual to provide an adjustment for this error, as such would tend to weaken the construction of the instrument. It should, however, be determined by the following method : —

Set the telescope on a star of not less than 40° north de- clination, and near the meridian ; transit, read ofiT the hour circle, and reverse the position of the telescope, as in the third adjustment. If there be no difference between the intervals, there is no error in the inclination of the declination axis to the polar axis : i.e., it is at right angles to it. If, however, the interval by the clock be greater than that on the hour circle, the declination axis towards the telescope is at too great an angle with the polar axis, — and vice versd. Half the

20 DOUBLE STARS.

difference of the intervals (expressed in arc) divided by the tangent of the star's declination gives the error of inclination required.

The whole of these six adjustments should be repeated several times, and also from time to time, as they are liable to change.

As the errors mutually affect each other, the second set of observations will be more accurate than the first, and should be made with greater care.

Having completed the adjustments of our equatorial, we are now ready to set the telescope upon any object in the heavens which we may wish to observe, whose right ascension and declination are given in our catalogues. First, set the telescope in declination, and then set the moveable hour circle to the right ascension of the object by the fixed vernier (with no correction for longitude if the fixed vernier is put to the Greenwich meridian, as above recommended). Now sweep the telescope in right ascension until the upper vernier comes to sidereal time by the clock, and the object will be in the field of view.

It will now be desirable to determine, approximately, the focal length of the object-glass, the angular value of the field of view with each eyepiece, and the magnifying powers of the eyepieces. The makers usually furnish the first and last of these, but it is well for the observer to ascertain these values for himself with some care.

Firstly : to find the focal length of the object-glass. This is not a very easy matter, owing to the difficulty of finding the optical centre of the glass. According to Troughton, " the measure.should commence from the interior part of the convex lens, at a distance from its exterior surface equal to one-fifth of the thickness of the double compound object- glass." (See Pearson, p. 19.) This point can of course be readily found by first ascertaining the thickness of the lens. A long, stout straight-edge, placed on the tube of the tele-

THE EQUATORIAL. 21

scope and made level, will enable the observer to find the

distance between the object end of the tube and the webs of

the micrometer adjusted to stellar focus. A plumb-line gives

the two points very quickly and accurately. If the telescope

be not a large one, the following method will give good

results : focus on a terrestrial object at a well-measured

distance, and mark the draw-tube ; then focus on the sun, and

again mark the tube ; then the formula

P _ P. (r - F) F

where F = the length of the solar focus required, F' the length of the conjugate focus obtained from the terrestrial object, and D the distance of the object. Of course, the distance between the two marks on the draw-tube should be measured very carefully by means of a finely divided rule and a pair of compasses. The distance between the telescope and the terrestrial mark must be measured from the object- glass.

Again ; the focal length may be accurately determined as follows : find the value in arc of say 50 revolutions of the micrometer screw. This will of course be readily done by separating the webs 50 revolutions, transiting a star near the equator (or, better, a star not far from the pole), and reducing the observed interval by multiplying it by the cosine of the star's declination, and by 15. Next, measure with great accuracy the linear value of the space between the webs,* then the proportion

2 tan ^ the arc : radius :: linear value : focal length will give the required quantity.

Secondly: to find the angular value of the field of view of the several eyepieces when in the telescope. This is easily done. Allow a star very near the equator to transit the field centrally, and convert the observed sidereal time into arc. If a chronometer or mean-time clock be used, the mean- * The practical optician can do this with very great accuracy.

2 2 DOUBLE STARS.

time interval must, of course, be converted into its equivalent sidereal interval, and then the arcual value found from the table. (See Loomis's Astronomy, p. 363.) Do this with each eyepiece. The angular value of negative eyepieces may also be found thus : as the field of view of a telescope depends partly on the focal length of the object-glass, and partly on the diameter of the diaphragm placed at its focus, the fol- lowing formula will give it : F is the focal length of the object-glass, and d the diameter of the diaphragm of the eyepiece, both in inches : —

d

F sin. l"

This is Delambre's formula.*

Thirdly : the magnifying powers of the eyepieces have to be found. One of the following methods may be chosen.

1. Measure the small illuminated circle seen in front of the eyepiece (which is the image of the object-glass), by means of the Dynameter. Then, the aperture of the object-glass is . to the diameter of its image at the focus seen through the eyepiece in the ratio of the focal length of the object-glass to that of the eyepiece. That is, the diameter of the object- glass divided by that of the small image gives the magni- fying power. The small image may, of course, be measured without the aid of the Dynameter, by means of a finely divided scale. Or the " Berthon Power-gauge " t may be used.

2. In the case of small telescopes the powers may be con- veniently found by means of a piece of white paper, say one inch long, on a black ground, fixed at a known distance from

* To take Pearson's example : let the focal length of the object-glass be 3'5 ft., and the diameter of the diaphragm of a negative eyepiece o'3

in. : then 4.2 x '000004848 = '000203616, and -r^~-i^^ = Hiz" = 24' 33"- t The Rev. T. W. Webb {Celestial Objects, p. 7) speaks highly of this little instrument, which he says may be purchased for Ts. td. of Mr. Tuck, watch-maker, Romsey.

THE EQUATORIAL. 23

the object-glass, a staff divided to inches being also placed near the paper. On looking through the telescope at the paper with one eye, and at the staff with the other at the same time, the number of inches on the latter covered by the paper will be seen, and the power at once found for that distance. From this terrestrial power, P', the stellar power P is obtained from the following formula, F being the stellar focal length and F' the terrestrial : —

p_ F X F

F' ■

3. The following method is convenient. Place a staff divided into feet and inches against a wall in a vertical posi- tion ; at a distance of three or four feet from the staff, hold the eyepiece to the eye, and, looking through it with one eye, and at the staff with the other eye, note how many feet and inches are contained in the diameter of the field of the eyepiece. For example, let the distance from the staff be 48 inches, and the observed diameter of the field 40 inches ; then the tangent of half the angle = ^ = o"4i6, and the angle is 45° 14', or 162840 seconds of arc. Now if the angular aperture of the telescope with this eyepiece be 33 sidereal seconds (found by transiting, centrally, a star very near the equator), or 495 seconds of arc, we have

n, ., . angular subtense 162840

Magiufying power = — 1— — = • -3i- = 329.

' ^ "^ angular diameter 495

4. Valz's method is useful for small telescopes. Turn the telescope towards any celestial object of known angular mag- nitude, say the sun, whose angular diameter is given in the Nautical Almanac, page II, of each month. Let the image be received on a screen kept at right angles to the tube, and having a line nicely divided into inches and tenths marked on it. Observe the horizontal diameter in inches and tenths of the image on the screen. Then if a be the sun's true diameter, A the angular diameter of the image on the screen'

24 DOUBLE STARS.

and D the distance between the middle of the eye-piece and the screen, then we have

tan i A = i-^,

and the magnifyine power = '15_iJ^ = _^^ _ .

° -^ » ^ tan 4 3 2 D tan i a

The measure of the image should be made when the sun is in the centre of the field of view.

The thickness of the webs of the micrometer may be found by bringing one up to a fixed web until the bright space between the two is estimated to be equal to the thickness of the web which is moved : read off" the divided head, and then carry the web into contact with the fixed web. Read off again. Repeat five or ten times. Take the mean value, and convert it into arc.

The following information, drawn up in a tabular form, may, for convenient reference, be pasted inside the box containing the eyepieces : focal lengths of telescope and finder ; angular value, in arc, of the field of view of each eyepiece of telescope and finder ; magnifying powers of the eyepieces ; value in arc of one revolution of the micrometer screen, and a table for taking out at sight the arcual value of revolutions and parts ; the thickness, in arc, of the webs of the micrometer.

For fuller information on these and other matters, the following works may be consulted : Loomis's Practical Astronomy, published by Harper and Brothers, New York. (This work is essential.) Webb's Celestial Objects for Common Telescopes. Pearson's Practical Astronomy. Chauvenet's Practical and Spherical Astronomy (London, Triibner and Co.); and Briinnow's Spherical Astronomy (Asher and Co., London). The Nautical Almanac for the current year, a collection of mathematical tables (such as Hutton's or Chambers's), and a good Star Atlas, are of course necessary.

25

The Clock.

A common well-made clock, if the pendulum be properly constructed and suspended, is all that is necessary for double- star observers. The piece supporting the pendulum should, of course, be very firm, and securely fastened to a good wall. The pendulum rod, 46 in. long, may be made of well- seasoned white deal soaked in melted paraffin, and % in. in diameter; the bob should be of lead, and cylindrical, its length (for a seconds pendulum) being, say, I4'3 in., diameter if in. with a hole a little more than f in. in diameter for the rod to pass through. The bob should be supported on the rod by means of a stout nut and screw, the latter having not more than thirty threads to the inch. A leaden bob of these dimensions would weigh about 13^ lb., which is found in practice to be a suitable weight. Such a clock, beating seconds audibly, would keep its rate unchanged for a few hours, and would meet all the requirements of double-star work. The rate would be obtained with the aid of a small transit instrument, or the equatorial itself, if well adjusted ; or the finder of the latter instrument might be used for this purpose. The rate should be small, and a losing rate, in order that the correction which becomes necessary from time to time may be made by putting the minute hand of the clock forward. If the clock be losing, say, ten or twenty seconds per day, the bob may be readily put near its true place by means of the nut under it, with the aid of the following formula : —

Change in one day = 43200 y seconds, where L is the breadth of one thread of the adjusting screw, and / is the length of the seconds pendulum ; from this the effect of one turn of the nut on the clock's rate is obtained. Or, to put it in a still simpler way : if n be the number of turns of the screw in i inch, then L = ^, /= 39-138; and the change in seconds for one turn of the screw = ;;^^fp = ^^-

26 DOUBLE STARS.

Assuming that the losing rate has been reduced to, say, two seconds per day, and that it is desired to make it about half a second, either of the following methods may be adopted : — (a) Place a small sliding metal collar on the rod, its weight being about xrrW^^ °^ *^^*- ^^ *^^ pendulum (bob and rod). At first this collar should be placed about 9 inches from the spring, and then gradually pushed downwards until the rate is what is desired. (3) Let the sliding collar take the form of a cup into which small shot may be put, and let it be _/£r«// to the rod at 19I inches from the spring. By trial the effect of one shot or of any number may be found, and the necessary change in the rate effected very readily.

The following extract from Baily's paper, in the Memoirs of the Royal Astronomical Society, vol. i., will be interesting in this relation.

Difference. Sec.

+ I '02 0-97

o'gi 0-85 079 074 0-68 0-63 0-56 0-51 0*46 0*40

o'34 0-28 0-23

O'll

+ o-o6

o'oo

— o-o6

O'll

0T7 0-23

Distance from axis in inches.	Variation in the rate per day.
I	+ I -08
2	2'IO
3 4 S 6 7	3-07 3-98 4-83 562 636
8 9	7-04 767
10	823
TI	874
12	9"20
13	9 60
14	9-94
15	I0'22
16	10-45
17	10*62
18	1073
19	1079
20	1079
21	1073
22	10-62
23	IO-45
24	+ 10-22

OBSERVING CHAIRS. 2^

If the pendulum is found to go slower in warm weather zxiA faster in cold, it is ww^^r-compensated, and more mercury should be put into the cylinder ; if faster in warm and slower in cold weather, mercury must be taken away, the quantity in each, case being found by trial.

Valuable information may be found in Baily's paper above referred to, in those by Bloxam (" Monthly Notices," vols. xiii. and xviii.), and in Denison's excellent " Clocks and Locks " (Adam and Charles Black, Edinburgh).

Observing Chairs.

As the work of the double-star observer is laborious, and often protracted, it is essential that he should be in a com- fortable position for his work.

Ordinary chairs and steps are quite insufficient for this purpose, though they often constitute the sole furniture of an observatory.

A special chair is required which will support the observer from head to foot, in any position of the telescope ; such is Dawes's chair (see Figs, i and 2). We have used it for several years, and should not like to be without it. It con- sists of a horizontal wooden frame on castors, 6 feet by 2 feet 4 inches, well braced to an upper frame, and inclined at an angle of 35° from top to bottom ; upon this upper frame is a sliding piece, carrying the seat which is nearly horizon- tal. The sliding piece is held at any point by a stout catch in a perforated iron plate on one side. The seat is 2 feet by I foot, and is padded ; the back is also padded, and it is so hinged to the seat that it can be raised to any position by means of a handle on the left-hand side, and then clamped to an arc on the right-hand side of the observer : this padded back is 2 feet by 2 feet 9 inches. It may thus

Fig. 2.

THE OBSERVATORY.

29

be raised and clamped at any angle without leaving the chair. Dawes used a rack for supporting the back, but the clamp is more convenient. An arm is also attached to the chair on the right-hand side ; this can be set at any angle by means of a notched arc, catch rod, and handle ; and it makes an excellent rest for the right arm. An iron hook on the left-hand side of the chair carries a reading lamp.

Fig. 3. (A Chair for occasional use.)

The Observatory.

The best form of Observatory is a square room with cylin- drical dome. The corners of the room are always useful, if not necessary, for tables, shelves, chairs, etc. ; and the cylindrical dome is manifestly more easily constructed than the spherical form. The shutters work horizontally, and are less liable to stick than curved shutters. Sufficient slope should be given to the roof to throw off a heavy fall of rain, and the top at least may be covered with thin sheet

30

DOUBLE STARS.

copper well painted. The conical form of roof is very effective, and also very cheap.

The Transit Instrument will require a small room, say 12 feet square, or rather less.

A Computing Room, on the north side of the Observatory, may be added, and this may be provided with a stove and chimney for heating the hot-water apparatus by means of which the observing rooms are kept dry in wet and cold

A'\f-

MR. EDWARD CROSSLLY S OlSLRVATORY, DERMLRSIDE

weather. The hot water must of course be turned oft some time before the work of observation begins.

Four windows, north, east, south, and west, are of great use in ventilating the Observatory, and in rapidly reducing the temperature inside as nearly as possible to that outside, so as to avoid currents of heated air, which are so detri- mental to optical definition.

31

CHAPTER III.

SOME ACCOUNT OF THE EQUATORIALS WHICH HAVE BEEN USED BY DOUBLE-STAR OBSERVERS.

AUWERS. (See KONIGSBERG.)

Barclay. (See Leyton.)

Bedford.

The mounting of the 8^ ft. equatorial was by Dollond, the Sisson form being used. The object-glass had a diameter of 5 '9 in., and was purchased in Paris by Sir James South. Tulley worked it. " It is considered by Captain Smyth to be the finest specimen of that eminent optician's skill, and will bear, with distinctness, a magni- fying power of 1200."

The declination and hour circles had a diameter of 3 ft. : the former read to 10". The negative powers were 22 to 1200, six of the highest being single convex lenses fitted in a polycratic wheel. The powers of the parallel-wire micrometer ranged from 62 to 850. The finder had an aperture of i'6 in.

The driving clock was invented by Mr. Sheepshanks, and had a steam-engine governor and absorbing wheel. It worked very well. — Monthly Notices, R. A. S., vol. i., and the Celestial Cycle.

Observer: Admiral Smyth.

Berlin.

The refractor at this Observatory is similar to the famous Dorpat telescope in all essential respects. Observers : Encke, Galle, Winnecke.

32 DOUBLE STARS.

BermerSIDE (Halifax).

Mr. Edward Crossley mounted his 9^ in. Cooke equatorial refractor in 1867. Its focal length is I48'5 in. The style of mounting is German. The diameter of the declination and hour circles are respectively 23J in. and I2| in., and they read to 10" and 2 sec.

The lamp, which gives a bright field to the micro- meter, swings at the end of the perforated declination axis.

The aperture, and amount and colour of the light for the bright field, are regulated from the eye-end by means of rods, and a rod and cords at the same end give the observer full control over the motion of the instru- ment in right ascension and declination.

The finder has an aperture of 2 J in., and a focal length of 2 ft. 4 in.

The negative eyepieces are ten in number: powers, 60 to 1000.

There are three micrometers, two filar and a double- image. The double-image and one of the filar micro- meters are by Simms, and the other filar by Cooke. The eyepieces for these instruments are, in all, seventeen in number, and the powers range from 100 to 1200. The new filar micrometer by Simms is divided on the face : diameter of circle 4§ in.

The driving clock is by Grubb of Dublin.

Observers : Crossley and Gledhill.

BESSEL. (See KONIGSBERG.)

Bond. (See Cambridge, U.S.)

Bonn.

The heliometer of this observatory has an aperture of 6 in.

The driving clock works "remarkably well," and its

EQUATORIALS. 33

construction is similar to that of the Poulkova refractor — Memoirs of R. A. S., vol. xx. BrunnOW. (See DLTNSINK.)

BuRNHAM. (See Chicago.)

Cambridge (Northumberland equatorial).

English mounting : the tube is square, and of deal. Object-glass by Cauchoix, ii| in. aperture, and 19^ ft. focal length; it was received in 1834. Hour circle S^ ft. in diameter, and reads to i sec. The circles were gradu- ated by Simms. — Main's An Account of tlie Observatories in and about London.

Declination axis, 5 ft. 8J in. long. Finder, 2f in. aperture, and 28J in. focal length. The declination is obtained by means of divided rods. For a full account, with elaborate drawings, see Airy's account of the instru- ment.— Account of the Northumberland Equatorial and Dome.

Observer: Challis.

Cambridge (U. S.)

This instrument is of the same style of mounting, size, and by the same maker, as the Poulkova refractor. Focal length 22 ft. 8 in., aperture i S in. " No colour except a purple tinge round very bright objects, such as the Moon and Venus." — Monthly Notices of R. A. S., vol. viii.

Observers : Bond and Waldo.

Cape of Good Hope.

Prior to 1847 the equatorial was a 46 in. by DoUond, aperture 3I in. There were four micrometers, viz., a spider-line position, an annular, and two rock-crystal. A flat-wire position micrometer was added subsequently. In 1849 the equatorial by Merz was mounted ; aperture nearly 7 in., focal length 8| ft. The tube is of wood, veneered with mahogany.

3

34 DOUBLE STARS.

The declination circle is 12^ in. in diameter, and reads to 10", and the hour circle has a diameter of 96 in., and reads to 4 sec. Tlie Huyghenian eyepieces have powers 86, 128, 200, 302. and 458. Those of the micrometer, 123, 161, 273, 347, and 464. The power of the double annular micrometer is 64. The divided circle of the position micrometer is 4 in. in diameter, is divided to 15', and reads to i' : the total range of the screw is 60 revo- lutions. One head only is divided.

Observer: Maclean

Challis. (See Cambridge.)

Chicago.

Mr. Burnham has made most of his discoveries with his 6 in. refractor by Alvan Clark. He has also used the fine 18^ in. Clark refractor of the Dearborn Obser- vatory, the 26 in. of the Washington Observatory, and the 94 inch of the Dartmouth College Observatory.

Cincinnati. (U. S.)

The object-glass was purchased in 1842 ; it was begun by Fraunhofer, and finished by Merz and Mahler. Dr. Lamont pronounced it " one of the best ever manufac- tured." Aperture 1 1 in., focal length 17 ft. Diameter of hour circle 16 in., of the declination circle 26 in. The powers range from 100 to 1400. The stand is of iron, and is filled with sand. The driving clock is by Clark and Sons, and is good. — Loomis's Recent Progress of Astronomy, and the Cincinnati Observations.

Observers : Mitchell, Stone, Howe, and Upton.

CROSSLEV. (See Bermerside.)

CUCKFIELD.

Mr. Knott's equatorial was mounted at Woodcroft, Cuckfield, and the measures lately published were made there between i860 and 1873. The object-glass has

EQUATORIALS. 35

a clear aperture of 7J in., a focal length of iiojin., and it was made by Messrs. Alvan Clark and Sons. The filar micrometer was made by Dollond ; diameter of position circle 3J in. ; it reads to tenths of a degree. The powers of the seven eyepieces range from 115 to SIS-

Dawes (Rev. W. R.)

In 1 83 1 this distinguished observer erected a S ft. achromatic at Ormskirk in Lancashire. It was by Dollond, and the mounting was like that of Smyth's refractor. The aperture was 3I in.; the circles 2 ft. in diameter; the powers used, 225, 285, and 625. — Memoirs of tJte R. A. S., vols. iv. and v.

The Newtonian reflector, the mirrors of which were presented to Dawes by Sir John Herschel, was mounted by Dollond, and applied to the polar axis of the 5 ft. telescope. Focal length about 7 ft., aperture 6j in. This instrument was used between 1834 and 1839, but not much. — Memoirs of the R. A . S., vol. xix.

In 1845 the Merz and Mahler equatorial was mounted at Cranbrook in Kent. The style of mounting was that of the great Dorpat refractor. The focal length was 8^ ft., and the clear aperture 6^ in. The object-glass was of first-rate quality. The hour circle read to 4 sec, and the declination circle to 10". Driving clock extremely steady and uniform. — Memoirs of the R. A. S., vol. xvi.

In 1859 the equatorial by Alvan Clark and Sons (now at the Temple Observatory, Rugby), was mounted at Haddenham (Hopefield Observatory), in Bucks. The glass was cast by Chance and Co. Aperture 8J in., focal length 1 10 in. The figure is excellent to the circum- ference, and the dispersion " but a little over-corrected."

The finder has an aperture of 2 in. The micrometer was a parallel-wire by Dollond. Driving clock : this is

36 DOUBLE STARS.

very good. Bond's spring governor renders the action very smooth. — Memoirs of the R. A. S., vol. xx.

Dawes's micrometer by Merz and Son was made in 1846. It was a parallel-wire, and was used with the 8 J ft. telescope. Powers 120,155,260,322,435, 572, and 690. His Amici micrometer was presented to him by Sir John Herschel : it was a double-image, and had but one power (1000 on the 20 ft. reflector). Dawes added three new eyepieces, which, on the 8 J ft. refractor, were 212, 360, and 508.

Dembowski (Baron).

This eminent double-star observer used an excellent dialyte by Plossl 5 ft. focal length and 5 in. aperture equatorially mounted, from 1852 to 1862. The power generally used was about 300. It was not provided with a driving clock.

In 1862 the refractor by Merz was erected. Its aperture is 'j\ in. The object-glass is a fine one, and the powers range from 100 to 720. The driving clock is moderately good. — Ast. Nachr., vols. xlii. and Ixii.

Doberck. (See Markree.)

DORPAT.

This noble instrument was erected in 1825. It was the work of Fraunhofer. The tube was of deal overlaid with mahogany, and the framework of the stand was of oak inlaid with mahogany and polished. The polar axis was 39 in. long. Aperture of the object-glass 9-6 in. ; focal length 14 ft. The hour circle, with a diameter of 13 in., was divided to minutes, and read to 2 sec; and the decli- nation circle, with a diam. of 19 in., was divided to 10 min and read to 10 sec. Powers 86, 133, 198, 254, 420, 532, 682, 848, 1 1 50, and 1500. The finder had an aperture of 2'4 in., and focal length of 30 in. The driving clock kept a star in the centre of the field when a power of

EQUATORIALS. 37

700 was used. — Memoirs of R. A. S., vols, ii. and xxxvi. Pearsons Astronomy.

Observers: X., O.S., and Ma. DUNER. (See Lund.)

DUNLOP.

Equatorial refractor, focal length 46 in. Micrometers, a parallel-wire and an Amici's double-image.

DUNSINK.

The object-glass is the work of Cauchoix : aperture 12 in. ; focal length 19 ft. The mounting was by Thomas Grubb.

Elchies.

The Elchies equatorial was mounted about 1850, by- Mr. J. W. Grant, at Elchies, in Scotland. The German form was adopted. One portion of the stand weighed 1 1 tons. Messrs. Ransome and May made the stand, and the object-glass was by Ross. The aperture was 1 1 in., and the focal length 16 ft. The axes were 5 ft. long, and 6 in. in diameter. The circles had a diameter of 30 in., and were i in. thick. The eyepieces were twenty-three in number. The parallel-wire micrometer had two eye- pieces, and one of the three finders had a focal length of 5 ft.

Encke. (See Berlin.)

Engelmann. (See Leipsic.)

Ferrari. (See Rome.)

Flammarion. (See Paris.)

Fletcher. (See Tarn Bank.)

Galle. (See Berlin.)

Gledhill. (See Bermerside.)

Greenwich.

In 1838 the Sheepshanks equatorial was mounted. Grubb of Dublin supplied the stand, which was of the

38 DOUBLE STARS.

German form. The object-glass was by Cauchoix : aper- ture 67 in. ; focal length, 8 ft. 2 in. Its definition was' found to be good, the principal defects being outstanding colour, and a diffusion of light from brilliant objects. Negative eyepieces, a wire micrometer, a comet eye- piece, and a double-image micrometer were provided. The driving clock was regulated by governor balls at the ends of a horizontal arm on a vertical spindle. When a certain velocity had been acquired, projections on the balls rubbed against a fixed horizontal ring.

The mounting of the great equatorial is in the English style, and was executed by Simms. Messrs. Ransome and Sims made the engineers' work. The object-glass, by Merz and Son, has an aperture of 12J in.,* and a focal length of 16 ft. 6 in., and it is a very fine one. The hour circle is 6 ft. in diameter, and the declination circle 5 ft. The driving clock is in the ground floor story, and the power is given by a flow of water acting through a turbine, the spindle of which passes up to the instrument. A Siemens' chronometric governor regulates the supply of water to the turbine. A Barker's mill drives the hour circle, and the regulation is obtained by a conical pendulum, Siemens' chronometric governor, and a spade dipping into a trough of water. — Greenwich Observations, 1864.

Hall. (See Washington.)

Hartnup. (See Liverpool.)

Herschel (Sir William).

The gigantic reflector was erected in 1787, at Slough. Two concentric circles of brickwork, 42 ft. and 21 ft. in diameter, battened from a breadth of 2 ft. 3 in. at the bottom, to i ft. 2 in. at the top, and capped with

* In the " Monthly Notices" the aperture is always given 12 j in. See vol. xxxvi.

EQUATORIALS. 39

paving-stones 12J in. wide and 3 in. thick, formed the foundation. A vertical beam 12^ in. wide was fastened in the centre, and around this the whole framework had its circular motion in azimuth.

The tube was of iron, 39 ft. 4 in. long, and 4 ft. 10 in. in diameter. The speculum was of tin and copper ; its weight 1050 lb., and diameter 4 ft. The power used seldom exceeded 200. — Pearsons Astronomy. See also Phil. Trans., vol. Ixxxv., for a full description.

Herschel (Sir John).

The 20 ft. reflector was constructed in 1820, by Sir William and his son. The mirrors were fine, diameter 18 in., and focal length 20 ft. With the whole aperture, powers 150 to 160 were ordinarily used, the eyepiece being a single lens of i^ in. focus. — Memoirs R. A. S., vol. ii.

The reflector used at the Cape by Sir John was the 20 ft. The three mirrors were all fine ; aperture i8j in. The 7 ft. refractor, aperture 5 in., was also used. — Cape Observations.

Hind. (See Regent's Park.)

Howe. (See Cincinnati.)

Jacob. (See Madras.)

Jenkins. (See Oxford University.)

Kaiser. (See Leyden.)

KONIGSBERG.

The famous heliometer of this Observatory is mounted like the great refractor of Poulkova. The focal length is 8 ft. 6 in., and the aperture 6| in., and a distance of 1° 52' can be measured. It was begun in 1824, by Fraun- hofer, and mounted in 1829. The position circle at the object-glass has four verniers, and reads to minutes. For ordinary use there are five eyepieces : powers, 45

40 DOUBLE STARS.

91, IIS, 179. 290. A circle micrometer of the Fraunhofer kind has a power of 65. The ring micrometer and net micrometer have powers of 66, 92, and 165. — As(. Nachr^ vol. viii.

ObseiTcrs : Bessel, Anwers, Peters, Luther, and Schliiter. I.ASSELL.

In 1 841 the Newtonian reflector, 9 in. aperture and 112 in. focal length, was erected at Starfield, near Liverpool. The declination circle was divided to 15', and read to 30". The hour circle was of the same size, and read to 2 sec. The diameter of the circles was about 2 ft.

In 1848, the 20 ft. equatorial was mounted. The tube was of sheet iron, y in. thick, and was 20 ft. long, and 25 in. diameter; its weight was 594 lb. The diameter of the speculum was 2 ft., and its weight 370 lb. The finder was a Newtonian reflector, aperture 4"2 in., focal length 42 in., power 27. — Memoirs of the R.A. 5"., vols, xii., xviii., and xxxvi.

The two 4 ft. specula were constructed and mounted in 1859 ^"'^ i860; their focal lengths were 441*8 and 448" I in. ; length of tube n ft. The mounting was equatorial, and the motion in right ascension was given by an assistant. Leipsic.

The mounting was by Pistor and Martins, and the optical part by Steinheil. Aperture, 8 Paris inches ; focal length 12 ft. ; powers, 72, 96, 144, 192, 288, 432, 576, and 720. Observer: Engelmann. Leyden.

The Leyden refractor is of Munich make. Aperture, 6 in. ; focal length, 8 ft. Observer: Kaiser.

equatorials. 4 1

Leyton.

The 10 in. equatorial refractor, focal length 12 ft., by Cooke, was erected at Leyton in i860, by J. Gurney Barclay, Esq. The mounting is in the German style. The polar axis is 4 ft. 2 in. long, and the declination axis 3 ft. 2 in. The declination circle is 2 ft. in diameter, and reads to 10" ; and the hour circle is 13 in. in diameter, and reads to 2 sec.

The finder has an aperture of 3 in., and a focal length of 3 ft.

The driving clock is regulated by a double conical pendulum.

Observers : Romberg and Talmage.

Liverpool.

This fine refractor was mounted in 1848. The mount- ing is a modified English form ; the optical parts were by Simms, and the engineer's work by Messrs. Maudslay and Field. The object-glass, which is a very fine one, was by Merz ; its aperture is 8^ in., and focal length 12 ft. The hour circle has a diameter of 4 ft., reads to 0"I sec, and has two microscopes. The declination circle has the same diameter, and reads to i"-o. There are six negative eyepieces (powers, 90 to I lOO), and the two micrometers (filar and double-image) have powers 150 to 600. The driving clock was made by Simms, and drives fairly. Observer: Hartnup.

Lund.

The instrument used by Dr. Dundr was mounted at the observatory of Lund in 1867. The tube and object- glass are by G. and S. Merz, of Munich. The rest of the mounting and the micrometer are by M. Emile Jiinger of Copenhagen. The style of mounting is modified German. The object-glass is a very fine one ; its aperture

42 DOUBLE STARS.

is 9-6 in., and the focal length 14 ft. The diameter of the declination circle is 21 '2 in., and reads to 2" ; and the hour circle, with a diameter of I9'6 in., reads to 0"2 sec, and, by- microscopes, to 002 sec. The micrometer is a filar. The driving clock is a good one, the regulator being the invention of Professor Holten of Copenhagen.

Maclear. (See Cape of Good Hope.)

Madler. (See Dorpat.)

Madras.

The 4 in. equatorial was made by Simms, in the German style ; focal length 6y2 in. The circles were for finding only, and read to minutes of space and seconds of time.

The micrometer was a parallel wire ; powers used 170 and 280. The spurious discs of stars were " sharp and round, but rather large." — Memoirs of the R. A. S., vols. XXV. and xxxii.

The Lerebours and S^cretan equatorial had an aper- ture of 63 in., and a focal length of 89 in. A second object-glass was furnished by them in 1852, which proved good, but not perfect. — Memoirs of the R. A. 5., vol. xvii.

Observers : Jacob and Powell.

Main. (See Oxford.) Markree Observatory.

This equatorial was mounted in 1834, at Collooney, County Sligo, by the late Mr. E. J. Cooper. The German style was adopted, and the cast-iron stand was placed on limestone blocks.

The object-glass was the work of Cauchoix. It is not a very good one. Aperture 13^ in. ; focal length 25 J ft. The diameter of the declination circle is i ft. 9 in. ; it is divided to J°. The diameter of the hour circle is 30 in. ; it is divided to minutes. The micrometer is of Munich make, and very good :

EQUATORIALS. 43

powers, lOO, 200, 300, 400, 500, 600, and 800. The position circle is 4 J in. in diameter, and reads to i'. The driving clock is a rough machine. — See Astr. Nachr., No. 2187. Observer : Doberck.

Milan.

The mounting is in the German style : both mounting and object-glass are the work of Merz and Mahler. The object-glass is a good one ; its aperture is 9*5 in., and focal length 10 ft. 7'9 in. The diameter of the hour circle is II in., that of the declination circle 157 in. The negative eyepieces furnish the following powers: 6"], 95, 155, 223, 322, 468. The filar micrometer was made by Merz: the powers are 87, 144, 210, 322, 417, 500, and 690 ; those generally used for double-star measurements are 322 to 690.

The driving clock, by Merz, is not a good one ; it has a conical pendulum. — Ast. Nachr., vol. Ixxxix.

Observer : Schiaparelli.

Miller. (See Whitehaven.)

Mitchell. (See Nantucket.)

Mitchell. (See Cincinnati.)

Morton. (See Wrottesley.)

Nantucket (U. S.)

Miss Mitchell's telescope was a 5 in. refractor by Alvan Clark.

Naples.

Aperture 5 J in. : focal length J% ft. : powers used 268 and 362.

Observer : Nobile.

Newcomb. (See Washington.) Nobile. (See Naples.)

44 DOUBLE STARS.

Oxford (Radclifife Observatory).

The mounting of the Oxford heliometer was designed and executed by Messrs. Repsold, and differs from the ordinary German equatorial. Aperture 7'S in. ; focal length 10*5 ft. The polar axis is 42^ in. long; diameter at upper pivot 4f in., and 3'85 in. at the lower. It is of steel, and the pivots turn in collars of bell-metal. It is perforated 2'i in. throughout.

The declination axis is 43 '4 in. long, 5 in. diameter in centre, 4-3 in. at the pivots. It is of steel, and perforated throughout, the bore being ig in. The tube is of ham- mered brass; diameter at object-end 13 in., at the eye-end 9"2 in. The position circle is 227 in. in diameter. The hour circle is at the north end of the polar axis, has a diameter of 33"8 in., is graduated to i min., and reads to 0'2 sec. The declination circle has a diameter of 34'3 in., is graduated to 4', and reads to i". The driving clock is governed by centrifugal balls, and the instrument is moved by a weight of about 30 lb. — Radcliffe Obs., vol. xi.

Observer: Main.

Oxford (University).

The equatorial refractor is by Grubb ; aperture 12J in.; focal length 176 in. The declination circle has a diameter of 30 in. There are two filar micrometers, and a double- image. The driving clock is not faultless. — Monthly Notices, vol. xxxvi.

Observers : Plummer and Jenkins.

Paris.

The instrument used by Flammarion is one of the equatorials of the Paris Observatory. The object-glass is by Lerebours, and has a diameter of about 15 in., and a focal length of 29 ft. It is not a very good one, and a diaphragm is therefore generally used. The hour circle has a diameter of 25 in., and reads to i'. The

EQUATORIALS. 45

declination circle is divided to 5', and has a diameter of about S ft. The parallel-line micrometer is by Brunner, and the powers generally used are 300 and 400. The driving clock is also by Brunner, and has a Foucault regulator.

Plummer. (See Oxford University.)

POULKOVA.

A very fine instrument was mounted at this Obser- vatory by Merz and Mahler. The weight of the instru- ment is 7000 lb. ; the clear aperture 15 in., and the focal length 225 ft. The driving clock is regulated by the friction of centrifugal balls against the interior of a conical box. There are 6 negative eyepieces, powers 152 to 1218; 21 positive eyepieces, powers up to 2000.

Observer : O.S.

Powell. (See Madras.)

Regent's Park.

In 1836 G. Bishop, Esq., erected an observatory in Regent's Park, London. The equatorial was by Dollond, and the mounting English in form. The tube was of brass, and painted. The aperture of the object-glass was 7 in., and its focal length io| ft. The hour and declination circles were of brass, and 3 ft. in diameter, the former being divided to minutes and read off to seconds, and the latter divided to 10' and read off to 10".

The eyepieces gave the following powers : 45, 70, 108, 200, 320, 460, 700, and 800, and a polycratic wheel carried six of them.

The prismatic crystal micrometer was by Dollond, powers 185, 350, and 520; the parallel- wire was also by Dollond, powers 63, 105, 185, 320, 420, 600; also 190 and 300.

The driving clock was by Dollond : it was driven by a powerful spring, and regulated by two fans, and was

46 DOUBLE STARS.

found to work "extremely well." — Bishop's Astr. Obs., 1852. Observers : Dawes and Hind,

Romberg. (See Leyton.)

Rome.

This fine instrument is mounted like the great Dorpat refractor.

Aperture 96 in. ; focal length 14-2 ft.

Driving clock, very good. " The rate of the regulating part of this instrument is controlled by the friction of two small brass balls against the sides of a conical box." — Monthly Notices of the R. A. S., vol. xvi.

Observers : Secchi and Ferrari.

Rugby. (See Dawes.)

Observers : Wilson, Seabroke, and A. Percy Smith.

ScHiAPARELLi. (See Milan.)

Seabroke. (See Rugby.)

Secchi. (See Rome.)

Smith. (See Rugby.)

Smyth. (See Bedford.)

South.

The 5 ft. equatorial was erected in 1797 in London. " The whole scheme of its fabric was cast by the late Captain Huddart, many years a worthy Fellow of this Society. All the tinned iron-work was made under the direction and inspection of the same able engineer." The brass-work was made by J. and E. Troughton, and the whole instrument was completed in 1797. The excellent object-glass of 3I in. aperture was by P. and J. Dollond. The powers used were 68, 116, 133, 240, 303, 381. That most used was 133, the others being double eyepieces. In some few cases a single lens, power 578,

EQUATORIALS. 47

was used. The diameter of the declination circle was 4 ft. — Phil. Trans., 1824, Part iii.

The 7 ft. equatorial had an aperture of 5 inches, and was, at the time it was made, the chef-d'ceuvre of Tulley. " In distinctness under high magnifying powers, it is probably excelled by no refractor existing." The ordi- nary observing power was 179 ; occasionally, 105 and 273 were used. — Phil. Trans., 1824, Part iii.

The 20 ft. refractor was mounted in 1829, at Ken- sington. The glass was by Cauchoix, and had a clear aperture of iif in. — Monthly Notices, vol. i.

Stone. (See Cincinnati.)

Struve and Otto Struve. (See PouLKOVA.)

Talmage. (See Leyton.)

Tarn Bank.

Mr. Fletcher's equatorial was erected at Tarn Bank in i860. The optical part was by Cooke, and the stand w£is made under the direction of Mr. Fletcher. The Sisson polar axis was used in the mounting. The object- glass has a diameter of 9^ in., and a focal length of 12 J ft. The declination circle has a diameter of 42 in., and reads to i" ; the hour circle is of the same size. The driving clock had 22^ lb. as a driving weight, and worked very well.

Mr. Fletcher's small equatorial, by Cooke, was mounted in the German style ; aperture, 4' 14 in.; focal length, 6 ft. This mounting was that used by Dollond, with a long polar axis. This axis was of mahogany, 9 ft. long, 9 in. square in the middle, and 7 in. square at the ends. The hour circle was 20 in. in diameter, read to 2 sfec, and was loose on the polar axis. The declination circle had a diameter of 20 in. also, and read to 10". Powers, 50, 100, 160, 230, 300, 420, and 600, with the parallel-wire micro- meter. The power generally used for double-star work

48 DOUBLE STARS.

was 300. The driving clock was a very elegant instrument and worked very well. The governor was like that used in steam engines. — Monthly Notices of R. A. S., vols, x., XX., XXV. ; Memoirs of the R. A. S., vol. xxii.

Upton. (See Cincinnati.)

Waldo. (See Cambridge. U.S.)

Washington. (The Great Refractor.)

This magnificent instrument has an aperture of 26 in. and a focal length of 390 in. The glass was by Chance, and Messrs. Alvan Clark and Sons were the makers of this noble lens. It was finished in 1872. The mounting is in the German style. The negative eyepieces are four in number, powers 155 to 1360. The positive eyepieces are sixteen in number, powers 173 to 1802. The tube is of steel, Jj- in. in thickness near the ends and \ in the middle. Length 32 ft. ; diameter of the middle one- third about 31 in. The object-glass is com- posed of an equi-convex front lens of crown-glass and a nearly plano-concave flint lens : thickness of the objective at the centre about 2-87 in. The glasses are free from all hurtful rings and striae, and are of nearly perfect figure. There are three micrometers, two filar and one double-image. There are two finders, apertures 2 in. and 5 in. The driving clock was invented by Professor Newcomb : with careful attention to the oiling, etc., it works satisfactorily. — Instruments and Publications of the United States Naval Observatory, Washington, 1845-76.

The smaller instrument was made by Merz and Mahler. Aperture 9*6 in., focal length 14 ft. 3 in. The object- glass was under-corrected for colour, and in 1862 it was refigured by Messrs. Clark and Sons : the focal length was increased about one inch, and the glass corrected for defective achromatism ; the definition also was im- proved. The flint disc is not perfect. Hour circle 15 in..

EQUATORIALS.

49

and declination circle 2i in. diameter. Finder 2-6 in. aperture, and 32 in. focal length. Micrometer, a re- peating filar, by Fraunhofer. The driving clock is regulated by a Fraunhofer centrifugal pendulum, but it is scarcely powerful enough. There are eight eyepieces, powers 90 to 899. — Washington Observations, 1865.

Observers : Newcomb, Hall, and Holden. Whitehaven.

In 1850 Mr. J. F. Miller, of Whitehaven, began his double-star measurements. The instrument was a very good equatorial refractor by Cooke, the mounting in the German style, and of the same size as Mr. Fletcher's instrument. The micrometer was by Simms, and proved to be a very good one. Diameter of position circle 5 in. ; power generally used 300. The clock-work, too, was good. — Memoirs of the R. A. S., vol. xxii. ; Astr. Nachr., vol. xxxiii.

Wrottesley.

English mounting : polar axis of four mahogany planks 14 ft. 3 in. long and 10 in. square in the middle ; pivots of bell-metal. Focal length 10 ft. 9 in.; aperture 7f in. ; flint glass by Guinand ; crown by DoUond. Mounted at Wrottesley, Staffordshire, in 1843. Decli- nation and hour circles each 3 ft. in diameter : verniers read to 10" and i sec.

Parallel-wire micrometer : position circle 4 in. diameter, reads to 6' ; powers used 450 and 320, and 600 and 820, occasionally. Driving clock not satisfactory. — Memoirs R. A. S., vols, xxiii. and xxix.

Observer : Morton.

50 DOUBLE STARS.

CHAPTER IV.

THE MICROMETER.

The parallel-wire micrometer is par excellence the instrument of the double-star observer. Though used for many other purposes, it is specially adapted to his work, and has not been superseded by any other form of micrometer.*

It consists of the following parts : first, a stout brass tube or adapter fitting into the eyepiece end of the telescope, and carrying at its outer end a position circle divided from 0° to 360° in the direction contrary to the figures on a watch dial, and read off by two opposite verniers to tenths or twentieths of a degree ; it is also provided with clamp and slow motion. The moveable vernier plate has attached to it the micrometer box, which is generally 5 to 6 inches long, \\ to 2 inches wide, and \ inch deep. The micrometer screws enter the box at each end, their divided and milled heads being outside. The screws, of a hundred threads to the inch, enter their respective frames, which fit nicely within the box, and move parallel to one another like two tuning- forks, one just small enough to work within the other. Across these frames, in the centre of the field, are stretched the fine webs at right angles to the direction of the screws. To prevent slack, the two frames are pushed towards one

* There are many other forms of micrometer, the most important being Airy's and Amici's, both double-image micrometers. The former consists of a positive eyepiece containing four lenses, the third from the eye being concave and divided into two halves, and each half carried by its own screw. Amici's double-image micrometer consists of two prisms, and has been used by Dawes and Doberck. It is considered the best of the kind.

THE MICROMETER.

51

another by spiral springs, thus bringing the inner heads of the screws against the ends of the box. These heads are often made square with the shaft of the screw; but they are much better made spherical, so as to fit into conical bearings at the ends of the box. A flat comb plate is placed over the moveable frames across the open centre, with a fine-toothed comb cut so as to form a chord to the circle of the field of view at right angles to the moveable webs. This comb plate carries two stout parallel wires (called position wires), about 12" apart, across the centre of the field, and at right angles to the moveable webs and parallel to the comb. The eyepieces are attached

Fig. 3. (Parallel-wire Micrometer.)*

outside the box to a sliding-piece, moved by a screw for centering over the webs in the direction of their motion. The webs, position wires, and comb should be clearly defined with a high power at the same time. The eyepieces should as much as possible slide into the same adapter, to save screwing and unscrewing.

* One reading lens is removed to show the slow-motion clamp.

52 DOUBLE STARS.

It is usual to insert in the stout brass tube or adapter, close to the position circle, a thin ivory ring with openings all round through the adapter, to admit light for the pur- pose of giving dark ground illumination to the webs. English makers usually furnish both screws with heads divided into a hundred parts, and figured o, lo, 20, etc., so as to give an increasing reading when the webs are moved towards the heads or against the spiral springs. Obser- vations are always taken by setting the screw in this direction, as it is found in practice to give the best results. German makers divide only one of the heads, and simply use the other screw for setting in different parts of the field. It is desirable that both screws should have easy play through not less than fifty revolutions. A divided head to one of the screws is quite suflScient, and for distinction we will call this the micrometer screw, and the other the setting screw.*

We have now to determine the value of the revolutions of the micrometer-screw in seconds of arc, and for this purpose we can make the setting screw and its moveable frame an eflScient auxiliary. Let the comb be divided in such a manner that every fifth notch is a longer one, and each tenth notch numbered by small holes — one, two, three, etc., counting from the notch nearest to the setting screw as Zero. Let the following webs be placed on the moveable frame of the setting-screw: No. I, at Zero; No. 2, at 1775 revolutions ; No. 3, at i8'2S ; No. 4, at ig'o; No. 5, at 20'o; No. 6, at 25-0 (in the centre) ; and No. 7, at so'o. On the micrometer screw but one fine web is needed, and it is placed in the centre of its moveable frame : let us call this web No. 8.

We are now in a position to step the micrometer screw throughout its whole length with great ease and accuracy,

* These are marked A and B, respectively, in Figure 3, and are held simply by friction, so as to admit of being set to any reading.

THE MICROMETER. 53

viz., at every five revolutions by webs No. 5 and No. 6 ; at every single revolution by webs No. 4 and No. 5 ; at every half revolution by webs No. 2 and No. 3 ; and also at every quarter revolution by webs No. 3 and No. 4.

It will probably suffice to test only the ten central revolu- tions for parts of a revolution. Use a high power and good illumination. The operation may be thus described. Bring No. 5 to Zero and No. 8 beyond Zero : the latter must now be brought carefully just into contact with No. 5, first on one side and then on the other, the head being read off to tenths of a division each time. No. 8 must now be brought up to No. 6 in precisely the same way, and this will complete the first step of five divisions. No. 5 must now be brought to five revolutions, and No. 8 set as before, first on No. S and then on No. 6 ; and this will be the second step : carry on this process throughout the fifty revolutions. Repeat this several times, and the mean readings of each step will give the comparative value of each five revolutions with great accuracy. Each group of five revolutions must now be tested in precisely the same way for each single revolution, by means of webs No. 4 and No. 5 ; and each of the ten central revo- lutions for parts of a revolution with webs Nos. 2, 3, and Nos. 3, 4. It is, of course, impossible for webs Nos. i to 7 to be placed absolutely at the distances named ; but the exact distance will be determined by the observations and the proper allowances made in the computations.

Having thus obtained by the most accurate as well as the most convenient method the comparative value of the different parts of the screw, it now only remains to convert these values into seconds of arc. This is done by transits of a slow moving star from web No. i to web No. 7, the distance being fifty revolutions of the screw. The best stars for this purpose are a, j8, and h Ursae Minoris, whose places are given in the Nautical Almanac.

If the telescope used has, say, 6 in. aperture and 9 ft. focal

54 DOUBLE STARS.

length, the value of the fifty revolutions will be 954'93 ± seconds of arc. This, at the equator, is equal to 63662 seconds of time, or i'' = o"o66 seconds of time: but if we multiply 0066 by the secants of the declinations of P, S, and o Ursae Minoris respectively, we get 0'25i8, vi2J, and 2-859 seconds of time. Now as it is difficult to take a single transit with greater accuracy than 0*25 sec, the advantage of a slow star is at once apparent. If, for in- stance, the transit of S Ursse Minoris be taken to 0'5 sec. by a single observation, the value of the screw will be obtained with an accuracy of i in 2000; but as one obser- vation cannot be relied on, a large number of transits of different stars should be taken, and in this way an accuracy of I in 5000, or even of i in 10,000, can be secured.

It is usual to express the value of the screw in seconds of arc for one revolution ; and if an auxiliary table be constructed giving the value of parts of a revolution, any measured dis- tance can be readily converted into arc.

The effect of change of temperature on the screw is so small that it may be entirely neglected. The effect of refraction, however, cannot be so disregarded when the above transits are observed out of the meridian; and the following is a simple and convenient mode of dealing with this, since it enables the observer to transit, when away from the meri- dian, and to correct his results at once if the altitude be not less than about 20°. Find the altitude of the object to the nearest degree or half-degree by the clinometer. Observe the transit as above and read off the position circle ; then bring the micrometer box into a vertical position by means of the plumb-line of the clinometer. Read off the position circle, and the difference between the readings will give the angle with the vertical, or the parallactic angle. The full effect of mean refraction on the position of the star, sup- posing the transit to be in a vertical plane, must now be multiplied by the cosine of the angle with the vertical, and

THE MICROMETER. 55

this will give the correction for refraction in seconds of arc. It is always subtractive in the case of transits. The interval of transit must now be multiplied by the cosine of the declination to reduce it to the equatorial value, and then converted into seconds of arc. The correction for refraction must now be added. This method is also apphcable to correct the measures of low wide double stars : in this case the correction is always additive.

The correction for curvature of path must be applied in observations of a and S Ursae Minoris, but for /3 it is in- sensible. Convert the observed interval into arc. Then twice the sine of half the arc thus obtained, divided by the arc expressed in terms of the radius, will give the factor by which the observed interval must be multiplied to reduce it to the true value. Dembowski preferred /3 to S as requiring no correction for curvature, and taking less time to observe, and so lessening the chance of instrumental disturbance during transit.

The micrometer screw may also be tested by two terrestrial marks, and the angular value determined if the distance of the marks from the object-glass be ascertained ; but the definition so near the surface of the earth will rarely be found good enough for this kind of observation.

A powerful theodolite may also be used for this purpose, the two telescopes being turned towards each other, and the angular distance of the webs read off on the horizontal circle.

If the micrometer will include the sun's disc, its value may be obtained from the sun's diameter. In this case the horizontal diameter should be measured. If the vertical diameter be taken, the sun should have a considerable altitude, and the correction for refraction must, of course, be applied. The sun's semi-diameter for noon of each day will be found in the Nautical Almanac on page II of each month.

56 DOUBLE STARS.

Some observers make use of the pairs of stars in the Pleiades whose places were determined by Bessel with the greatest care. The following pairs, consisting as they do of small stars of nearly the same magnitude, will be found very useful for this purpose ; and to aid in their ready identifica- tion a rough map is also given.

k (Asterope) 1

8 .

9 •

f (Atlas) h (Pleione'

31 •

32 .

35 •

36 •

Mag.

7-8

7-8

8-9

8-9

4-5

5-6

8

8

9

9

R. A. {1880).

54 54 55 55 55

2I'I9

28-33 24 '66 58-88 22-16 40-46 53-56 8-23 19-06 3040

Dec. (1880).

24 10

9 23 49

48

23 41 46

24 I o

23 52 51

46-84 1201

15-83 57-27

11-53 11-68

45-54 52-12

41-55 5-57

From the formula r = y (J 8)^ + {A af cos^ mean 8 we find the following distances for the four pairs k 1 ; 8, 9 ; 31. 32; 35.36:—

kl 8,9 3'. 32 35. 36

Diff. of R. A. (A a).

127-14 34-22

74-67 131-48

Diff. of Dec. (A {).

94-83 18-56 53-42 95-98

149-92

36-39 86-64

153-86

In order that the observer may be able to check the pre- ceding results and also to select other pairs for special pur- poses, the following extract from Bessel's work {Astronomische Untersiichungen, Erster Band) is given : —

'+^ ^

I

S

fN

.^

.^

%

.r

^

»«^

THE MICROMETER.

57

K 5

p p

0 b

1 I

nn	r^ \n
U	O	<J
O 1	0 1	o 1

^ Tj- Tt" Tf- ;^ -^ Tj- ^-^ Th Tt- 7I- -^ rj- _Tj- -^i* ^-^ ■* Th

"ooobbbbbbbbbbbbbd M 11 1 I I I I I I M I 1 1 I

M M -I i-t^Tj-OM-^. moo 00 00 r^ I- u-) ui ui 5 t^ r^ r^ !>. r«« i>.'^o 'OvO'Ovo loij^i-nu^^rt*

O LTiOr^^^'-oo wvo -^i-i 11 o O N Lor^ ^ rosp T}-p^fOO^p^►-| O •-< On '^vO r-. u^ i— in

(N >0 "O >-< vb r^'O o->(S 0"O O b V"" wO^

o ro -^ ro 't^h r^

N N M N N

p p

b b

++

o

11 r^ t^ N « O

pop bob

+++

O O ^^"-1 O O^ONfO u^oo ^O t^ ON ON r->. r>. r^ r^ r-^ t^ t^ r-^vo ^ r^vo o vo vo vo vo ^o o

"bbbbbbbbbbbbbbbbb

H- + + + + + + + -h-

a^ONlOO N "-00 '- t^vO ■- N »M I- 00 O *n 1- mo O a^ONO^a^^^TJ-0^'-« ThrOM OnOO

00 00 ON pNoo 00 ON pN ^ y^Np oo oo on on 00 On i~>. ^^oo 00 t^cJo 00 rn '(J- in

	u		"aT '-^
			■ 1^1 • . ■ • PO cj ri u i-« N »-0^^>
	0 • 4> * •	d
	^^ 00	OVOO "	N^wg- mmmm
0	1 -^^ 2	0.0	1 crv^^ 0 1 1 1
<	;^^<	?<	N N N <*

The following table from S-'s Mensiira Micrometrica will give a good idea of the accuracy of the work done with the parallel wire micrometer \e is the probable error of a single distance, and /of a single measured angle].

58

DOUBLE STARS.

A. Table of the probable Errors of single measures of 2.'s lucida:, i.e., those whose companions are not below the 8th magnitude.

Class.	Mean Distance.	No. of Stats,	No. of Measures.	e	/
I.	070	44	176	0-074	2 309
11.	1-48	III	447	■086	I 52-4
III.	308	128	563	•099	I 8-2
IV.	5-62	119	469	•116	0 489
V.	979	SI	222	•127	0 302
VI.	'3-94	46	199	■127	0 239
VII.	1938	48	184	■145	0 183
VIII.	2819	48	178	•156	0 149

B. Table of the probable Errors of single measures of S.'s reliqua, i.e., those whose companions are between the 8th and nth magnitudes.

		Mean	No. of	No. of		/
		Distance.	Stars.	Measures.
	I.	0-75	28	94	0-087	0 / 2 270
	II.	1-54	186	642	•109	2 1-9
	III.	293	383	1299	•122	I 29-S
	IV.	582	426	1428	■156	I 7-1
	V.	10-00	278	783	•184	0 47-1
	VI.	1388	161	455	•201	0 38-7
	VII. & VIII.	22-6o	383	1064	•207	0 270
C. Table of the probable Errors of single measures of Stars, the
companions of which are below the i ith magnitude.
	Class.	Mean Distance.	No. of Stars.	No. of Measures.	e	/
	II. & III.	2''s9	14	49	0176	0 / 2 27-8
	IV.	592	17	55	•221	2 2-1
	V.	10-46	22	59	•362	1 20-7
	VI.	14-19	II	37	•376	0 59-6
	VII. & VIII.	21-93	12	35	■371	0 55-6

Dr. Dundr, of Lund, gives the following results for the value of his micrometer : they were obtained from transits of Polaris : —

1867.

Sept. II ... II ...	• • i7'3i3 •303
19 ... 20 ...	•313 •308
21 ... 24 ... 26 ...	•309 •326 .. -336

1868. Oct.

3 ••■	.. 17-322
12 ...	•301
21 ...	■309
22 ...	•300
25 ...	•311
26 ...	•315

Mean, r = i7"'3i3 ± o"-oo2.

THE MICROMETER.

59

The Baron Dembowski made a very elaborate investigation of his micrometer in 1873. He used star transits, terrestrial marks, and auxiliary webs or types, as he calls them, in the micrometer. The following extracts exhibit some of his results : —

Litres means that all the transits taken on any given day are observed with the telescope in the same position with respect to the meridian, E. or W., the time of observation being any whatever within three hours of the meridian passage of the star.

Conditionnh means transits observed with the instrument alternately E. and W. of the meridian, at the same culmination, the same number of observations being made on each side.

The values of the entire scale, and the probable errors are as follows : —

	Sets. 10		Inter- vals.		Probable error.
/S Ursae Minoris	libres	84	50 rev. = 1054-484	11 rio-iyo	T centig.+28-4
	7	»»	60	•874	•198	+ 04
	lO	conditionnes	80	•384	■150	+30-4
	7	*i	84	■836	•292	- 08
	10	»»	80	•486	•209	+ 176
S Ursse Minoris	14		28	•942	•311	+21-2
By Gauss' method	18	double sets	...	•375	780	+127

And by the method of least squares he deduces the following results : —

Value of the 50 rev. = io54"-578 — (T — i9°-72). 0-01420. Probable error of the coefficient of (T — i9°72) = 0-00295.

Hence it is inferred that the absolute value of the entire scale is known within the limits ± o"-o6.

The next table enables us to see the result of his examina- tion of each S rev. of the scale, four different methods being used : —

6o

DOUBLE STARS.

Methods used.

Polaris : 13 transits (iibres) .. 5 U. Minoris : 14 „ (condit.) .. Terrestrial marks, 14 measures Types, 15 measures

Mean

Value of I rev

The results from Polaris which are underlined in the tables are excluded from the means.

Rev. o to 5.

105 247 r. =

■382 r. =

■397 r- =

'16 •10 •06

105 -342 2 1 "-068

Rev. 5 to 10.

io5"-662 r. = o"2i

105-453 r. = -zi •444 r. = '08 ■372 r. = -07

lo5"-423 2l"-o8s

Rev. ID to 15.	Rev. 15 to 20.	Rev. 20 to 25.	Rev. 25 to 30.
io5"-573 !■• = o"-29	'°5"'475 >■• = o"'28 •468 r. = -18 ■404 r. = -lo ■423 r. = -07	i05"-358 r. = o"-29 •297 r. = -26 ■381 r. = -09 •388 r. = -07	ios"-498 r. = o"-35
i05"-382 r. = -15 •401 r. -= -12 ■390 r. = -03	•536 r. = -18 •380 r. = -08 •457 r- = '04
ios"-39i 2i"o78	105' '442 2i"-o88	105 -356 2i"-o7i	i05"-468 2i"-094

Rev. 30 to 35.	Rev. 35 to 40.	Rev. 40 to 45.	Rev. 45 to 50.
io5"-459 r. = o"-34	io.;"'6o7 r. = o"-34	io5"-826 r. = o"-22
•348 r. = -29 •436 r. .= -11 •438 r. = -06	■53b r. = o"-27 ■484 r. = '09 •499 r. = -05	i05"-690 r. = -32 •670 r. = '08 ■655 r. = -06	io5"-655 r. = o"-2o •632 r. = -07 ■590 r = -05
I05"'420 2 1 "084	io5"-5o6 2l''-I0I	i05"-672 2i'-i34	i05"-626 2l"-I25

These results present, on the whole, an increasing value from o to 50 revolutions ; a minimum value appears at 20 to 25, and the maximum is reached at 40 to 45. The probable error of one measure does not exceed o"'07.

Then the value of each of the ten central revolutions (20 to 30) is given, by two different methods : —

Method.

Rev. 20 to 21.

Rev. 21 to 22.

Terrestrial mark : 50 measures Types: 13 measures

Mean

2 1 ".078 r. = o"-o5 •063 r. = 'Oi

21" '070

21 070 r. = 0-05 ■079 r. = 'OI

2l"-074

THE MICROMETER.

6l

Rev. 22 to 23,	Rev. 23 to 24.	Rev. 24 to 25.	Rev. 25 to 26.
2 1 "-079 r. = o"o5 ■076 r. = 'OI	2 1 "066 r. = o"'05 ■082 r. = -Ol	2i"-o85 r. = o"-o5 •c8i r. = -oi	2i"'o83 r. = o"'05 •088 r. = -OI
2i"-o77	2l"-o74	2i"o83	2i"-o85
-
Rev. 26 to 27.	Rev. 27 to 28.	Rev. 28 to 29.	Rev. 29 to 30.
2l"-o8o r. =-05 ■090 r. = '02	2i"o86r. = o"-04 •095 r.'= .02	2i"o99 r. = o"-o5 •100 r. = -02	2i"-ii7 r. = 0" 05 ■092 r. = -02
2i"-o85	2 1 "090	2 1 "-099	2l"-I04

Here, as in the preceding results, the mean values increase on the whole from 20 to 30; and De. finds that the pro- bable error of one measure does not exceed o"'05.

R^sum^ of the mean values of each quarter of the ten central revolutions in the seven different series, and the pro- bable error of one measure : —

Series.	1st Quarter.	2ndQ	jarter.	3rd Quarter.	4th Quarter.
I.	5"-oos	0"l20	5"-i75	o"-o93	5"-5i3	o"-o88	5" -390 o"-o89
II.	•301	•107	■^0*	•059	■733	•099	•336 -088
III.	•III	•110	■185	•085	•356	•036	•432 -070
IV.	■143	•079	■195	•049	•277	•047	■469 -045
V.	•195	•064	•164	•090	■298	•076	•426 ^108
VI.	•281	•097	•139	•043	•270	•063	•394 '062
VII.	■325	■008	•194	■024	•199	■004	■366 -026

Series.	4th Quarter.	3rd Quarter.	2nd Quarter.	1st Quarter.
I.	4"^907 o"^i44	S"-i49	o"-i58	5"-263 o"-i4i	5"-764 o"^i8o
II.	5 -lis ^126	•197	•116	■644 ^104	•127 •OSS
III.	4 -920 -076	■351	•089	■266 •oSi	■547 -165
IV.	•876 ^049	•308	•030	■332 '069	•567 -069
V.	S ^086 -114	•144	•059	•544 ^060	•299 -174
VI.	•309 -121	•268	•093	•315 -048	•192 '146
VII.	■276 -022	•241	•026	•562 -029	•004 -024

The objects used in obtaining the series I. to VII. were as follows: — For I., II., double distances of 5 Lyrae; for III., IV., v., double distances of two terrestrial discs ; for VI. double distances of /i Draconis ; and for VII. the distance between two auxiliary webs in the micrometer.

62

DOUBLE STARS.

Taking the mean of the values for each quarter of a revo- lution obtained by the positive and negative movements of the screw, the following results for each series are found : —

	Mean of the values for each Quarter.
I.	S"-384	S -219	S"-33i	5"- 148
II.	•214	•429	•215	•225
III.	■329	•225	■353	•176
IV.	■355	■263	•292	•172
V.	•247	■359	•221	•256
VI.	•236	•227	•269	•351
VII.	•164	■378	•220	•321

The means of these series for each quarter are S"'276, S"-300, 5"-272, s"-233.

Difference between the mean measured value of a Quarter of a Revolution and the mean value ^"'zji.

I.

II.

III.

IV.

V.

VI.

VII.

+ o'-ii3

- '057 + -058 + -084

— -024

- '035

— '017

o '052 •158 •046 •008 ■088 •044 ■107

+ o"'o6o — -056 •082 ■021 •050 •002 •051

+ +

0"I23

•046

•095 ■099 •015 •080 •050

In making the seven series of measures, the micrometer was removed from the telescope after each series.

Remarking on the whole investigation, De. is led to the following conclusions : —

1. The values of the four quarters of a revolution are not equal inter se.

2. Greater inequalities still are found between the + and — readings.

3. These inequalities do not depend on any defect in the division of the head.

The micrometer used at Bermerside Observatory (see the illustration, p. 51), was made by Mr. Simms last spring. It is a beautiful instrument, and a very careful examination of the screw by Dembowski's method (see p. 59) has shown that it may be regarded as perfect, at least for the purpose of double-star measurement.

THE MICROMETER. 63

From upwards of 200 transits of stars the value of i rev. was found to be I3"'8372, with a probable error ±o"*oo4.

The screw (marked A in the illustration) which is the one used in measuring double stars, was tested with the following satisfactory results : —

1. From ten careful settings of the micrometer web close to one of the fixed webs, it was found that the probable error of the mean was ±o""CX33, and the probable error of one setting ±o"-oi4.

2. Careful stepping of the screw by 5 revolutions at a time showed the following differences from the mean value of eight sets of determinations : +o"'Oi4, +o"'003, — o"'006, O'O, — o"-oo8, +o"-oo4, +o"-oo4, -o"-ooi, +o"-oo5, — o"-oo5.

3. The ten central revolutions were then stepped singly, and the differences from the mean result were : — o""Oi4, +o"'00i, +o"'ooi, — o"*oo5, — o"'oo4, —d'-ooj, — o"-oos, +o"-o2i, — o"-oi2, +o"-oo7.

4. Each half revolution of the ten central ones was then measured five times, and the greatest difference from the mean result was +o""04.

5. Lastly, each quarter of the six central revolutions was stepped four times ; the greatest difference between the mean of the whole and the means of the several quarters did not exceed o"'02.

These results therefore show that there is no appreciable change of value in the different parts of the screw, and that there is no sensible eccentricity in its mounting.

The webs used for double-star work. No. 6 and No. 8, were measured, and the thicknesses found to be o"-4i5 and o"-372.

64 DOUBLE STARS.

CHAPTER V.

METHODS OF OBSERVING, ETC.

It is here proposed to give a somewhat full account of the methods of observing the positions and distances of double stars. The subject will be treated under the following heads : —

1. Methods of observing angles and distances.

(a) The methods adopted by Sir Wm. and Sir John

Herschel. (d) The methods used by Dawes and Dembowski

in the measurement of angles : Dawes' prism. (c) Special methods for very close stars. (d) Methods which may be occasionally used.

2. The number of measures of angle and distance required

to form a se(, or complete observation, with an example.

3. Specimens of Forms of Registry.

4. Weights.

5. On contracted apertures.

6. Best time for observing : weather, etc.

7. Precautions to be used while observing.

(i) Methods of Observing.

{a) The method Sir William Herschel adopted will be best given in his own words : " The distances of the stars are given several different ways. Those that are estimated by the diameter can hardly be liable to an error of so much as

METHODS OF OBSERVING. 65

one quarter of a second ; but here must be remembered what I have before remarked on the comparative appearance of the diameters of stars in different instruments. Those that are measured by the micrometer, I fear, may be liable to an error of almost a whole second ; and if not measured with the utmost care, to near 2". This is, however, to be understood only of single measures ; for the distance of many of them that have been measured very often in the course of two years' observations can hardly differ so much as half a second from truth, when a proper mean of all the measures is taken. As I always make the wires of my micrometer outward tan- gents to the apparent diameter of the stars, all the measures must be understood to include both their diameters ; so that we are to deduct the two semi-diameters of the stars if we would have the distance of their centres. What I have said concerns only the wire micrometers, for my last new micro- meter is of such a construction that it immediately gives the distance of the centres ; and its measures, as far as in a few months I have been able to find out, may be relied on to about one-tenth of a second, when a mean of three observa- tions is taken. When I have added inaccurate, we may expect an error of 3" or 4". Exactly estimated may be taken to be true to about one-eighth part of the whole distance : but only estimated, or about, etc., is in some respect quite un- determined ; for it is hardly to be conceived how little we are able to judge of distances when, by constantly changing the powers of the instrument, we are, as it were, left without any guide at all. I should not forget to add that the measure of stars, when one is extremely small, must claim a greater indulgence than the rest, on account of the difficulty of seeing the wires when the field of view cannot be sufficiently en- lightened.

" The angle of position of the stars I have only given with regard to the parallel of declination, to be reduced to that with the ecliptic as occasion may require. The measures

5

66

DOUBLE STARS.

always suppose the large star to be the standard, and the situation of the small one is described accordingly. Thus, in Fig. 4, A B represents the apparent diurnal motion of a star in the direction of the parallel of declination A B ; and the small star is said to be south preceding at m n, north pre-

SoutR.

mth

-East

IToith.

Fig. 4.

ceding at op, south following at qr, and north following at St. The measure of these angles, I believe, may be relied on

W a?o° 0-

to 2°, or at most 3° except when mentioned inaccurate, where an error amounting to 5° may possibly take place. In mere

METHODS OF OBSERVING. 67

estimations of the angle without any wires at all, an error may amount to at least 10°, when the stars are near each other."*

The foregoing diagram will make this method of registering the position angles quite clear. The innermost circle repre- sents the inverted field of view ; the four quadrants are indi- cated by nf, sf, sp, np, and the angle is given by the position circle: e.g., in the case supposed in the figure the position would be entered as 45° nf. The outer circles exhibit the method first suggested by Sir John Herschel, and now in universal use. In this the quadrants are dispensed with, the zero of the position circle is at the north point, and the circle is read all round to 360° in the direction N.E.S.W. ; hence, according to this method, the above angle would be registered as 45° simply.

For distances, the methods used by Sir John Herschel and the later observers are identical.

{b) To measure accurately the position angle of a double star would seem at first sight to be a sufficiently simple process. Experience, however, has shown that in many cases it is most difficult. A glance at the measures of some double stars by different practised and eminent observers at the same epoch is quite enough to exhibit this fact in a striking way ; and a comparison of the individual measures of the same star on the same night by one and the same observer and instrument, abundantly confirms it. Some of the disturbing causes are obvious enough ; but even when the stars do not differ greatly in magnitude or brightness, and when the sky is clear and the air still, these discrepant measures still present themselves. And in the case of close and unequal pairs, " the eye, often at the very first glimpse, acquires a prejudiced bias." (Hj.) " When such stars are between the wires, the eye may un- consciously be directed to the edge of one wire rather than of

* Subsequent and more accurate measures show that Sir William's measures were liable to much greater errors than he here anticipates.

68

DOUBLE STARS.

the other ; " there is a tendency to place one of the double wires nearly in the direction of a tangent to the discs of moderately unequal stars." (Hj,.) Further, we are told that there is a tendency in the eye to " accommodate its judgment to the position of the wires " before they are brought up to correct parallelism with the line joining the centres of the star discs.

Nor is this all. Not only have we to get rid of widely discrepant results, we must also be on our guard against accordant measures. This latter difficulty is often a very considerable one. However, as we are here rather concerned with the methods by which these tendencies are to be de- stroyed or counteracted, we proceed to describe those used by the most successful observers of double stars : —

1. By repeated small movements of the wires in the same

direction till the eye is quite satisfied.

2. By bringing up the wires alternately from opposite sides

of the true direction. If three or more measures be made both ways, the mean result will probably be near the truth.

3. By a succession of small movements of the wires, tlie eye

being removed from tJie telescope for a moment after each ' alteration.

Whichever method be adopted, it will always be well to rest the eye a little, and to carry the webs some degrees away from the last position obtained, after each reading*

When the stars are so faint that only very little arti-

* " It will occasionally happen that, after taking two or three very co- incident angles, on recommencing after some slight interruption, a sudden difference of two or three degrees will occur, and a new set of angles, agree- ing well inter se, but differing from the former, will be obtained. In such a case it is most probable that the one or other result has been affected by some bias of the kind above alluded to ; and, as it is highly necessary to ascer- tain which it is, the following method of trying such rival measures against each other will often be found ser\'iceable. Suppose the two measures at issue were 63° and 65 ", each being a mean of three or four pretty coincident

METHODS OF OBSERVING. 69

ficial light can be used, it is still possible to obtain useful angles by employing the method of oblique vision. The illumination is gradually increased until the webs are just well seen ; and the eye is then directed, not to the star, but to another part of the field. " In this way, a faint star in the neighbourhood of a large one will often become very conspicuous." (H,.)

Before concluding these remarks on the measurement of position angles, some account of Dawes's prism should be given. This distinguished observer, soon after he began to measure double stars in 1830, discovered a tendency in his own eye to " obtain a different result in position when the line joining the centres of the stars was nearly parallel to the line joining the centres of the eyes, from that which was obtained when these lines were nearly perpendicular to each other ; and a still more decided difference was found to prevail when those lines formed a very oblique angle." He entirely overcame the difficulty by simply fixing a small prism to the eyepiece between it and the eye. By this means any double star can be placed in any desired position with respect to the horizon ; and it was the uniform practice of this great observer to confine himself entirely to the vertical and horizontal positions. Dembowski and Struve always observed with the head vertical. 0.5"., also, after accumulating a vast mass of measures, became aware of an error resulting from obliquity of position, and under- took a laborious series of measures of artificial double stars,

measures. As it is probable that one is decidedly right, and the other decidedly wrong, and as their difference is 2°, let the micrometer be set to 61" and 67°, one or the other of these being necessarily 4° in error, will be violently offensive, while the other will be affected only by an error which experience has already shown may be borne without detection in the particular star in question. Thus the false results will be made evi- dent ; and, in assigning weights to the measures, this must be taken into consideration as materially diminishing the influence due to it." — Sir John Herschel, in Memoirs of the Royal Astronomical Society, vol. v.

^0 DOUBLE STARS.

partly for the purpose of ascertaining the amount and law of this error ; and in his measures lately published both the observed and corrected angles and distances are given. The objections to the prism on the score of loss of light and impaired definition were regarded by Dawes, after nearly forty years' use of it, as quite unfounded. It is obvious, too, that the comfort of the observer, and therefore, to some extent, the accuracy of the measures, will be considerably increased by this simple apparatus.

Of the extreme difficulty which attends attempts made to obtain accurate measures of distance of close and unequal double stars, nothing need here be said. So keenly was this felt by the late Sir John Herschel, that he devised a method of obtaining the elements of the orbit of certain double stars from the measured angles alone, the measured distances being used collectively for finding the value in seconds of space of the scale used in the construction. Extreme care, much practice, a good sky, patient repetition on different nights, the destruction of bias by removing the eye from the instrument for a few moments, and carrying the web far away from the last setting after each measure, — these and such like precautions naturally suggest themselves to the observer.

{c) In the case of close pairs, the following suggestions, if carefully carried out, will often be found of use : —

1. Place one star centrally over a web, and note the change

of form which the disc undergoes, e.g., if it becomes elliptical in shape, place the other web so as to produce the same effect on the other star.

2. When the distance is less than one second, the two

following methods will frequently give valuable re- sults. Place the inner edges of the webs at a distance apart as nearly as possible equal to that which separates the two stars, using a high power; bring the stars close up

METHODS OF OBSERVING. 7 1

to the webs, and compare the two spaces ; correct, if neces- sary, and then read off the divided head of the micro- meter. Repeat this from six to ten times ; then, the reading when the webs are just in contact, together with the read- ings given by the above settings, will furnish the means for deducing the distance of the stars with considerable accuracy.

A better method, however, is that of first placing the threads a certain known distance apart, say i", bringing the discs between them, and trying to estimate and express in numbers the ratio between the distance of the discs apart and the distance of i". Make several or many esti- mations ; then, the distance between the threads being known, the true distance of the discs is readily deduced from the ratios. These two methods were used by Struve. Baron Dembowski takes one measure by estimation, then one with the webs, and places great confidence in the mean of the two.

This will be a suitable place for a few words on the Barlow lens. It was frequently used by Dawes, and he thus sums up its advantages : —

1. The diameter of the micrometer threads subtends only

about half the angle.

2. The moveable parallel threads are both as nearly in focus,

with double the magnifying power.

3. The value of the micrometer divisions with the lens

is only about half of its amount without it : hence a proportionably fine motion in the measurement of distance.

4. With any given power the threads are distinct to a

much greater distance from the centre of the

field. {d) The method of oblique transits described by Sir John Herschel may be noticed (see the " Cape Observations," p. 247). " If / be the polar distance of a double star; 6 its

72 DOUBLE STARS.

measured angle of position ; a the angle of position of an oblique wire across which both stars are allowed to transit by their diurnal motion ; A the interval of their transits across it in seconds of time, — then will the distance of the stars from each other be given by the formula

15 A sin/ cos a sin (a — 6)

Convenient values of a are 100°, or 110°, or (on the other side of the vertical) 260° or 250°. The inclination of the oblique wire ought to be towards the opposite side of the meridian to that of the line joining the two stars. In situa- tions not remote from the pole, a high degree of precision is attainable by this method."

Lastly, it is sometimes convenient, especially when the distance is very great, to measure differences of declination, and then to compute the distances of the components from them and the angles of position.

2. Number of Measures.

As regards the number of measures of position and dis- tance which should be taken of a star on the same night, the practice of eminent observers differs. However, it is quite certain that at least three of the angle, and three double measures of the distance, should be taken. Six of angle and twelve of distance (six double measures) would be much better. On the other hand, it is better to measure the same object on two different nights, than to make a large number of measures on one night only. Of course the importance of the star and the quality of the night will also affect the number of measures taken. Sir John Herschel usually made ten of angle and ten of distance : Dembowski, four of angle

METHODS OF OBSERVING, 73

and four double measures of distance : Wrottesley, ten of position and ten of distance. Dr. Doberck four of angle and one double distance.

The making of a complete observation of the position and distance of a double star may be thus described. After lighting the lamp which illuminates the field, and turning on the red or blue glass,* the micrometer is pushed into the tube, adjusted to distinct vision of webs and star, and the position circle set to zero. With eye on the star, the micro- meter is then turned bodily until the star runs along one of the distance webs (which has been placed near the middle of the comb), from side to side of the field. The thick position webs, now coinciding with the meridian, are then moved until the stars lie between them. Then, if Dawes's practice be followed, let the webs be brought up to true parallelism with the imaginary line joining the centre of the stars by a succession of small changes, the eye being removed from the telescope for a moment after each change. Read off the circle, and repeat from four to ten times. If the method of Dem- bowski be preferred, the webs will be brought up alter- nately from opposite sides of the true direction, the same number of measures being made in each direction. The webs should be moved away some distance each time, so that the eye may be freed from any bias.

If the circle be not set to zero at the outset, the necessary correction must of course be applied to each reading when the set is complete.

It is well to examine the zero reading of the circle after the measurement of each star, to avoid errors from accidental derangement. To take an example, let the star run along

* "The colour I employ is that afforded by a brown-red glass of the Claude-Lorraine kind, which throws a strong sunshine glow over a landscape, almost verging to orange. A fuller red is even yet superior for distinct definition of wires." — Herschel, in Memoirs of R. A. S., vol. V. Dembowski and Doberck prefer Cinnabar red glass.

74 DOUBLE STARS.

the equatorial wire at 91° 30' by the position circle: then will — i°"S be zero correction. If five readings be now taken, the operation of reduction will be as follows : —

1127 \ 110-5

II4'2

112-3 111-8 /

The sum = 561-5

Mean = 112-3 Correction — 1-5

II0-8

For the distance : — Let us suppose that the companion is to the right of the principal star and the micrometer set to measure position. Fix one web, and place it on the centre of the principal star : now move the free web to the right until it bisects the companion star and read off the head. Carry the free web to the left of the fixed one, and bring the companion to the left until it is bisected by the latter; place the free web on the principal star ; and again read oflT. Repeat this double measure, bringing the web up in t/te same direction as before, i.e., from left to right, from four to ten times, reading off the divided head each time.

To take an example as before : As the divided head is held on the axis of the screw by friction only, it may be set approximately to zero, when the moveable and fixed webs are superposed. Suppose this to be done, then the following observation will illustrate the method : —

i2i7S'700g: — e' Lyrae.

d d d d

18-5 100— Si'o = 19-0 18-5 -81-5 = 18-5

iS'o — 8r'o = 19-0

18-7 -82-0 = i8-o

d The sum of these eight readings is 148-2, and the mean is d 18-52.

It still remains to convert these parts into seconds of arc. This is most readily done by means of a short table from which the values can be taken out at once.

METHODS OF OBSERVING.

75

Such a table may be thus constructed, supposing 13 "227 = I'. The first column gives the divisions and the others the tenths.

Div.	"o	'1	'2	■3	■4	■s	■6	■7	•8	■9
0	•0	•007	•015	•022	•030	II	II •04s	II •052	1' •060	•067
I	•07s	•083	•090	■097	■los	•112	■120	•128	■13s	•144
2	i -'SI	•158	•166	•173	•181	•189	•196	■204	•211	•219

etc.

etc.

etc.

etc.

Here 2-3 = o"-i73 at once from the table.

3. Forms of Registry.

The importance of having ready a supply of forms for the entry of measures need not be here insisted on. Annexe are copies of those used by Sir John Herschel, Smyth, and Wilson.

Sir John HerscheVs Form. Registry of the Micrometric Measures of Double Stars.

Number for Reference.	N.P.D.	Declination.	Right Ascension.
No.
Instrument used.	Date.	Star's Name.
	18 = 18	(Dec. of year.)
Diagram.	Quadrant.	Magnitudes. Colours.
Face to	Micrometer reads

76

DOUBLE STARS.

Form of Registry of the Micrometric Measures of Double Stars — continued.

Position.

Power.

W

Mean Z =

from » in direction nfsp

Distance.

I

Remarks.

Power. >,

Rev. Pts. Dec.

Mean

+

Div. by 2 Parts =

Seconds =

N.B. — When only positive readings are taken, a zero must be used, and the division by 2 omitted.

Sky Wind Steadiness Definition of Star Dist. from Merid.

General ] Pos. Judgment >

ofObs. ) Dist.

Observer.

Zeros of Position and Distance.

Star runs along the thread at .'. Zero for position Z =

Threads close at

Mean .', Zero for distance Z

To be used only in case opposite readings are not taken.

+

Determination of Place.

Clock (or clock + 24h). —

Hour Circle :

+ if East, - if West ; 1 if read on to 24 h. always \

Instrumental correction True R. A.

Declination Circle, )

+ North, — South I

Instrumental correction

True Declination

METHODS OF OBSERVING.

77

Admiral Smyth's Form.

MiCROMETRic Measures of Double Stars, at Bedford, with the SJ Feet

Refractor.

Star's Name.	Right Ascension, 1830.	Declination, 1830.
Diagram.	Quadrant.	Magnitudes. Colours.
		A = B = C = D =
Position.	Distance.	Remarks.
Power.	w	Power.	n "(3 c 1 2 a c E 1 1 + H m	+ + + + +	Rev. Pti. Dec.	w
•	-
Mean Z =		Mean]		Sky Wind Steadiness
		Div. b] Parts Second	f 2			Definition of Star Face to
from « in direction nfsp	=	Dist. from Merid. General ( Pos. Judgment < of Obs. 1 Dist.
Zero of Position.	Date.
Star runs along the equa- ) torial wire at \ .'. Zero for position Z =	O I	18

78 DOUBLE STARS.

The Rugby Form. TEMPLE OBSERVATORY.

No. 187

DOUBLE STARS.

R. A. Decl.

Magnitudes.

Position. /^ \ Distance

Zero. Readings.

Direct. Indirect. J Diff.

Position = Distance =

4. Weights.

Several practised observers have accustomed themselves to assign weights to every position and distance. Sir John Herschel, for example, gives the following account of his mode of doing this. " Although it is impracticable to estimate correctly in numbers the goodness of a measure, yet such is the powerful influence of atmospheric circumstances on this very delicate class of observations, as to render it imperatively necessary either to observe only on those rare nights when that cause of error does not exist, or to jnultiply observations on inferior nights, and reject, freely, all which exhibit great deviations, or which do not give satisfaction at the time. If this be not done, the greatest confusion will arise. The assignment of a weight to each measure, accord- ing to the best judgment the observer can form, offers a middle course, free from the objectionable point of arbitrary

METHODS OF OBSERVING. 79

rejection, and admitting a multiplication of observations on different nights, which is, indeed, quite indispensable for coming at the truth in all the more difficult cases. The scale I have adopted is from i to 10; i applying to the worst possible measurement in the most unfavourable cir- cumstances, and 10 to the most perfect which can be had in the most favourable." In casting up the mean of a set of measures, if the weights were pretty equal the arithmetical mean was adopted : if the weights differed much, the mean was found by the rule for finding the centre of gravity of a number of weights. — Sir John Herschel, in Memoirs of tJie R. A. S., vol. V.

It will be understood that the assignment of the weight must precede the reading of the circle or divided head. Dembowski began to use Sir John Herschel's method in 1854.

Dawes followed Sir John Herschel's plan after 1831. He observes : " Scarcely any liberty has been taken in the rejection of observations considered tolerably satisfactory at the time. Occasionally the micrometer has been set to a suspected read- ing, and a re-examination instituted. If not found decidedly bad, it has been suffered to remain ; if otherwise, another completely detached observation has been taken. If this last differed widely from the suspected one, and nearly coincided with the rest, it has been taken in its stead ; if not, both the suspected measure and that taken to prove it have formed part of the set."

Wrottesley computed the probable errors and weights by the usual formula prior to 1857. After that year a more elaborate method was adopted : see Proceedings of R. S., vol. X.

Secchi assigned weights (i to 5) according to the agreement among the individual measures of the set ; 5 was the highest and I denoted an approximate result.

For a fuller treatment of this subject, see page 144.

So DOUBLE STARS.

5. Contracted Apertures.

Sir John Herschel was probably the first obser\-er who made constant use of these contrivances. In 183 1 he has the follow- ing remark in the notes to his measures : " The action of a telescope is often surprisingly improved by stopping out the central rays, by a round disc from a fifth to a sixth of the diameter of the object-glass, which should be well sheltered." *

In 1834 Dawes wrote: "The use of a central disc on the object-glass having been suggested to me by Sir John Her- schel, for the purpose of diminishing the images of the stars, I have frequently employed one from an inch to an inch and eight-tenths in diameter. The effect is decidedly good on the stars themselves, if not too faint to bear the loss of light. The separating-^ower of the telescope is increased ; but the concentric rings accompanying bright stars are multiplied, and rendered more luminous, and are also thrown further from the disc. Hence small stars may often be obscured or distorted by the ring passing through them." In the introduction to his last great catalogue, this eminent observer again takes up the use of apertures. His long experience enabled him to speak with much confidence, and the following is a summary of the contents of the chapter. He seldom used the central round disc before the object-glass, because it increased the number and brightness of the rings, and caused the rings to hide faint companions of bright stars and elongate the discs of nearly equal stars ; — a perforated whole aperture was used with great advan- tage, and the " perforated cardboard used for making the Berlin-wool work is very suitable for bright stars." For fainter stars, a piece of cardboard covering the whole object-glass and pierced with holes in concentric circles may be used. These contrivances reduce the size of the discs and the bright- ness of the rings. The concentric prismatic rings produced are so distant as not usually to interfere with companion

* " Sheltered : '' i.e., provided with a dew-cap of ample length, blackened inside.

METHODS OF OBSERVING.

stars. Angular apertures were used by Sir John Herschel, especially the inscribed triangle for destroying the rings round bright stars ; but the rays often obliterate or distort the small companion star. Dawes recommends the inscribed hexagon. In order to destroy the tendency of discs to become triangular, " especially when the wind is in the east or south-east," he recommends " cutting off three equidistant segments from the whole aperture of the object-glass, the base of each of which is the chord of 60°. Then, the chords being placed so as to coincide in position with the angles of the telescopic inverted image, those angles will be reduced by the larger circular aperture between the segments, and a fairly round image will be substituted for the triangular one."

, " A smaller aperture may sometimes show a very delicate and close companion to a bright star, when a larger aperture fails to do it."

The following table, from Dawes, may be of use in enabling the observer to form a correct estimate of the separating power of his object-glass : —

	Aperture	Least	Aperture	Least	Aperture	Least
	in	separable	in	separable	in	separable
	inches.	distance.	inches.	distance.	inches.	distance.
	I'O	/se	40	i'-'h	8-5	0-536
	1-6	2-85	4-5	i-oi	90	0-507
	2-0	2-28	50	0-91	9-5	0-480
	225	2-03	SS	083	100	0-456
	2'S	1-82	60	076	I2-0	0-380
	275	1-66	6-5	070	150	0-304
	30	1-52	70	0-65	200	0-228
	3-5	1-30	7-5	o-6i	25-0	0-182
	3-8	1-20	80	0-57	300	0-152

6. Best Time for Observing, etc. The state of the atmosphere during double-star observation should always be described in the note-book. Secchi indi- cates the state of the sky by means of the initial letter of the words signifying very fine, good, middling, and bad. He considers the night very fine when distances under i can be

6

82 DOUBLE STARS.

readily measured, the discs being sharp and dear ; good, when distances from i" to 2" can be dealt with, the discs being less sharp than in the preceding; middling, when the discs are badly defined and unsteady ; bad, when discs 3" apart cannot be clearly separated. Some observers express these condi- tions by numbers. From the experience of Dawes and Struve it would seem also to be worth while noting the direction of the wind. Both these practised observers frequently found that easterly winds were associated with triangular discs.

As regards the best time for observing, perhaps not much can be said, so much depends on local circumstances. Sir John Herschel, in the south of England, found that " the best time for astronomical observation, and especially for these measurements, is between midnight and sunrise. In the long nights of winter, it is true, distinct vision often comes on an hour or two before midnight, and in all seasons occasionally, of course, much earlier." He then notes the unsteadiness of the discs as morning twilight comes on, and uses the following descriptive terms in his notes : " twirling,'' " moulding," " con- vulsed," " twitchings," "wrinkled," "burred," "glimmering." " The rarest of all states of the atmosphere is that in which the rings are destroyed and the stars are seen perfectly round and tranquil."

In conclusion. Sir John's experience with respect to the action of dew and the use of the dew-cap is worthy of note. " The least dew on the object-glass must be most carefully avoided, as it produces a singular contortion in the stars, which I have usually termed wrinkling ; the discs are much diminished, the rings multiplied and rendered narrower, and are kept in constant motion ; and a material change of the apparent angle of position is often produced by the displace- ment of their centres." The remedy he found to be a tube of tinned iron about 20 in. long, bright without and blackened within, and fixed on the object end of the telescope. (This was for his 7 ft. refractor.)

methods of observing. 83

7. Precautions. The following precautions and hints may be of use to amateur observers. They are drawn from the experience of such observers as S., H., Da., De., and Se. : —

1. At the outset it must be remarked that the observatory

(doors, windows, slit, and ventilators) should be thrown open at least an hour before observation begins, in order to reduce the temperature of the room to that of the external air.

2. If the definition be bad and the motion great, it is useless

to attempt the measurement of double stars. In short, if a power of at least 300 cannot be used, the results cannot generally be of any value.

3. Very bright stars should be measured in daylight or

twilight.

4. The observations should be made near the meridian if

possible.

5. The observer should be in an easy position, — the prism

effectually secures this ; and the driving clock ought to go smoothly.

6. The bright-field should be used almost exclusively —

red and blue colours are most in use.

7. Use the highest powers possible, and always the same

powers.

8. A moderate number of measures of an object on each of

two nights is better than a large number on one night.

9. Use printed forms.

10. Enter date, hour, weather, and distance from meridian,

before observation begins.

11. Notes on definition, general impression as to the

value of each measure or each set, etc., cannot well be too copious.

12. In all doubtful cases make a sketch, and add full de-

scription.

PART II.

CHAPTER I.

ON THE CALCULATION OF THE ORBIT OF A BINARY

STAR.

INTRODUCTION. In his Lettres Cosinologiqiies, first published in 1761, the astronomer Lambert has the following remarkable words : " By observing the groups in which the stars are very much condensed, we may, perhaps, be enabled to ascertain whether there are not fixed stars which revolve in sufficiently short periods of time around their common centre of gravity." At the time these words were written, not more than from forty to fifty double stars were known to astronomers. In 1784 (see Phil. Trans., vol. Ixxiv., p. 477), about four years after Sir William Herschel began his famous discoveries of double stars, Michell wrote : " It is not improbable that a few years will inform us that amid the great number of double stars, triple stars, etc., observed by Herschel, there are some which form veritable systems of bodies revolving about one another." And again, in an earlier paper, {Phil. Trans., 176"/, vol. lii.,) Michell writes : "If, however, it should hereafter be found that any of the stars have others revolving about them, for no satellites shining by a borrowed light could possibly be visible, we should then have the means of discovering the proportion between the light of the sun and the light of those stars, relatively to their respective quantities of matter." Maupertuis, Cassini, and no doubt other thoughtful astronomers in the early part of the eighteenth century, speculated on the existence of siderial systems, but

ox THE ORBIT OF A BINARY STAR. 85

none with such clearness as did Christian Mayer of Mannheim. This diligent observer studied the proper motions of many- bright stars by means of the small Comites he discovered near them, and speculated on binary systems, elliptical orbits, the origin of new stars, variables, a central sun (?), etc.* The actual discovery, however, of pairs of stars physically connected and in orbital motion was reserved for Sir William Herschel. In the year 1779 he began to sweep the northern heavens in search of double stars, and in his first catalogue, presented to the Royal Society in 1782, gave descriptions and measures of 269 of these objects. About twenty-five years after the conclusion of these sweeps for double stars, he carefully remeasured the angles and dis- tances. The observed changes in angle and distance formed the subject of his great paper, "Accounts of the changes that have happened during the last twenty-five years, in the relative position of double stars ; with an investigation oj tJu cause to which they are owing!' {Phil. Trans., 1803, Part ii.) In this paper he showed that "many of them are not merely double in appearance, but must be allowed to be real binary combinations of two stars, intimately held together by the bond of mutual attraction." And Castor is the star whose changes he first submits to examination. Indeed this splendid object seems to have commanded much of his attention for years before the publication of his famous discovery of binary stars ; for Sir John Herschel says of this star that its " unequivocal angular motion seems to have first impressed on my father's mind a full conviction of the reality of his long-cherished views on the subject of binary stars." — Memoirs of R. A. S., vol. v., p. 196. Here too it is worth while noting that in 1798 Dr. Hornsby, reflecting on the well-marked proper motion of Castor, and the fact that the distance of the components had not changed for twenty years, drew the inference that both stars were moving with * See his Gruendliche Vertheidigung, etc., 1777.

86 DOUBLE STARS.

the same velocity and in the same direction, but quite failed to see that these facts supplied unequivocal evidence of physical connexion.*

In this way Sir William Herschel detected about fifty binaries. Since his time the list has been largely extended, and the researches of Struve, Madler, and others brought the number up to about six hundred.

In the paper above referred to, rough guesses at the periods

of revolution of some of the binaries were made by Herschel ;

e.g., he assigned a period of about 342 years to Castor.

It was reserved, however, for his distinguished son, Sir

John, to grapple successfully with the interesting problem

of finding by a graphical method the orbit which one

star describes relatively to the other. If S represent the

principal star, to which the motion of the companion is

> referred, and if at successive

. epochs the positions of the latter

have been observed to be as in

4 . ^* the figure, Si, Sj, S3, S4, Sj

*/ ' it is plain that, assuming that the

Fig. 6. observations are sufficiently nume-

rous and accurate, a curve can be drawn through them which will represent the orbit. The positions thus marked down will not always form part of an ellipse ; they may lie in a straight line. For instance, the charted positions of the com- panions of Vega, 5 1263, and 5" 1516, appear to be well represented by straight lines ; while 7 Virginis, Castor, ^ Ursae Majoris, certainly move in elliptic orbits. It is possible, too, that the path may be some other curve, the knowledge of which will in its turn throw light on the forces and con- ditions which obtain in these sidereal systems.

To describe a method by which the elements of the orbit of a binary star may be obtained without the aid of the higher de- partments of mathematics, is the object of the present section. * See Grant's History of Physical Astronomy, p. 559.

on the orbit of a binary star. 87

Statement of the Problem. From the observations of angle and distance at given epochs, to draw tJte apparent orbit which one star describes relatively to the other, and thence to determine the elements of the true orbit, and to construct an ephemeris.

The first part of the problem consists, then, in a careful study of the observations to determine their relative value, and in so arranging them as to obtain the apparent orbit. A little explanation will here be necessary. The orbit, or portion of it seen by us, is the apparent orbit : it is the projection on the background of the heavens of the true orbit, i.e., the pro- jection of the true orbit on a plane at right angles to the line of sight. Suppose, for example, the plane of the true orbit to be at right angles to the line of sight, then will the revolving star be seen to describe an elliptic path round the primary star in the focus, and the true and apparent orbits will co- incide. If the plane of the orbit pass through the earth, and present its edge to the observer, the revolving star will appear to recede from, approach, occult, or be occulted by, and again recede from, the star in the focus of the ellipse. The plane of the orbit, again, may be but a little inclined to the line of sight, and then the companion will appear to pass a little below and above the principal star. In one word, the plane may have any inclination to the visual ray, and the projection will present corresponding phenomena. Hence, a circular or elliptic orbit, if its plane were oblique to the line of sight, would be projected into an ellipse ; if the plane passed through the earth, the projection would be a straight line ; and an elliptic orbit might be so situated as to have a circle for its projection.

The history of binary stars already furnishes us with illus- trations on this point. Take the star ? Herculis, discovered to be double by Herschel in July 1782. On looking at this object in October 1795, it was still seen double. Soon after the companion disappeared. During 1821, 1822, 1823, and

88

DOUBLE STARS.

1825, the utmost endeavours of Herschel and Struve failed to elongate it. Encke caught it double in 1826. Of this phe- nomenon Herschel says: " My observations of this star furnish us with a phenomenon which is new in astronomy ; it is the occultation of one star by another." Here then is an example of the orbital plane being in the line of sight. The period of this binary is about thirty-five years.

Once more : 7 Virginis is already a famous binary. It was Known as a double star in the seventeenth century. Herschel found the distance 5"7 in 1780; in 183 1 it was 2"o. In 1836 Herschel wrote: "7 Virginis, at this time, is to all appear- ance a single star." About 1837 it again separated, and the distance is now nearly 5".

42 Comae is a fine example of a binary, the plane of whose orbit coincides with the visual ray.*

Perhaps the accompanying figure will help to render this

quite clear. Let C C be the di- rection of the line of sight, A B G the real ellipse whose focus is S and centre C. Then will its pro- jection on a plane at right angles to the line of sight be the ellipse A' B' G'. And it will be observed that S', the projection of S, does not coincide with the focus F. The principal star, therefore, will not in general occupy the focus of the apparent ellipse, but will be displaced into some other position.

In many binary stars the observations do not yet extend over a sufficiently long period to enable us to compute any

• Examples.— In 2 186, 1967, 2737 (AB) the plane of the apparent orbit coincides very nearly with the visual ray. The apparent orbit is nearly circular in S 1037, 1126, and X Ophiuchi ; the orbit is extremely elongated in 2 1516 (AC), 1909, and 2822. In 2 1348, either the position of A in the apparent orbit is very eccentric, or the plane of the orbit is ffreatly inclined to the visual ray.

Fig. 7.

ON THE ORBIT OF A BINARY STAR.

89

satisfactory orbit. In some, the portion of the orbit traversed since observations virere commenced does not include any of the critical points ; while in yet other cases complete revolu- tions have been made since the date of the discovery of the stars. Of this last class, f Ursas Majoris, period about sixty- one years, 17 Coronas Borealis, period about forty-two years, and f Herculis, period about thirty-four years, may be given as examples.*

The next part of the problem consists in determining the real from the apparent orbit, and the position occupied in the apparent orbit by the principal star. And when all this has been done satisfactorily we are in a position to put our orbit to the test by the construction of an ephemeris, i.e., a series of computed positions and distances for the epochs of past and future observations. And if the computed quantities fairly agree with the measures made in past years, we must then proceed to compute positions and distances for future years at intervals of from a quarter of a year in the case of stars having rapid motion, to five or ten years in cases where the period extends over centuries.

That it is quite possible, however, for an ephemeris to represent all past observations in a satisfactory way, and yet to fail completely when it comes to be compared with future measures, will be evident on a little reflection. The subjoined table, however, will bring out the fact very clearly : —

Position.

Epoch.		LiWlJ.	Observer.	No. of Nights.
Computed.	Observed.
1848-0	239-0	249-16	Dawes.	7
1850-0	234-4		>j
1852-0	227-3	246-39		I
1854-0	212-6	24621	*f	7
1855-0	195 s
1856-0	164-4	245-44	Dembowski.	7

The computed places are from an ephemeris for Castor con-

* In 02 208 and 298 it is probable that we shall soon be in a position to attempt the computation of the elements of the orbits.

go

DOUBLE STARS.

structed by Sir John Herschel from an orbit which he published in 1832. The observations used by him extended from 1719 to 1831. The observed places are put by the side for comparison.

The small number of observations at the disposal of the computer, and the very small portion of the orbit dealt with, must, of course, be here remembered. Yet this orbit repre- sented the previous measures very fairly indeed.

Even when a star has been measured by skilful observers during more than an entire revolution, it is not always an easy matter to obtain elements which will furnish materials for a good ephemeris. Take ^ Ursae Majoris as an example. Its duplicity was discovered by Sir William Herschel in 1780; the companion was then not far from its apastron ; the peri- astron was reached in 18 16, and again in 1876, and hence its period is about sixty years. Now in 1872 Dr. Ball gave a set of elements, and an ephemeris furnishing positions up to 187875. The subjoined table will show how far the pre- dicted positions agree with recent measures.

Epoch.

1872-50 187275 187300

1873-25 1873-50

18737s 1874-00 1874-25 1874-50

187475 1875-00 1875-25

i875'5o 187575 1876-25 1877-25 1878-00 1878-25 1878-50

Position.

Computed. Measured.

22-4

17-5 12-5

7-3 2-2

357-2 352-I 347-0 342-2 337-6 333-2 329-0

325-1 321-4 3147 303-2 2961 293-9 291-8

19-39 358-91

333-63

317-56

304-8 294-9

85-5

in 1872-32, by Dembowski. in 1873-33, >.

in 1874-35, ,.

in 1875-27, „

in 1876-30, in 1877-26, ,

»» "

in 1878-45, by Wilson.

The agreement here is of course not satisfactory.

ON THE ORBIT OF A BINARY STAR. gi

Methods of Solution adopted.

The first part of the problem— that is, the determination of the most probable apparent orbit — may be best solved by the methods given by Sir John Herschel (' Memoirs of the Royal Astronomical Society,' vols. v. and xviii.), with some slight additions. We shall give a brief explanation of it, but the method will be best understood by working through an example.

To pass from the apparent to the real orbit is a geometrical problem of considerable difficulty. Fine analytical solutions of it have been given by Savary (' Connaissance des Temps pour I'an 1830 at 1832'), Sir John Herschel ('Memoirs of the Royal Astronomical Society,' vol. v.), Encke (' Ueber die Berechnung der Bahnen der Doppelsterne, Berliner Astr. Jahrbuch fiir 1832'), Villarceau ('Methode pour calculer les orbites relatives des 6toiles doubles.' 'Connaissance des Temps pour I'an 1852 et 1877'), and Klinkerfues ('Ueber eine neue Methode die Bahnen der Doppelsterne zu berechnen.' Gottingen, 1855). Purely geometrical solutions have been given by Thiele ('Ast. Nachrichten/ No. 1227, vol. lii.), and by the writer ('Monthly Notices,' vol. xxxiii., p. 375). Of these, Thiele's is by far the most elegant, and it is the one we shall here adopt. The construction of an ephemeris, and the comparison of the observed with the calculated places, is essential for the completion of the problem. This will be effected in the present paper by a graphical method.

It must, however, be understood that the graphical method is only introductory, and that subsequent analytical methods are necessary in order to correct the elements, and attain the highest degree of accuracy that the observations permit of.

To Prepare the Observations for Use. Some of the earlier measures of position and distance have

92 DOUBLE STARS.

to be deduced from the differences of right ascension and declination observed by Bradley, Piazzi, Lalande, and others. The process is as follows : —

Let S, S' be the two stars, whose right ascension and declination are a, S, and a + Aa, h -\- Ah respectively, S being the principal star.

Let Aa, J8 be expressed in seconds of

arc : then if P is the pole, P S Q, P S' Q'

Fig- 8. declination circles meeting the equator

in Q, Q', and S H is an arc of a small circle parallel to Q Q',

QQ' = Ja, S'H = JS.

Let P S S' = ^, S S' = p, the position and distance required to be calculated from Aa, JS observed.

Then tan e = tan S S' H = |g = ^^^-^ (i.),

and p = J8 sec. 0, (ii.),

from which Q and p may be obtained.

It must be observed that since 6 is always measured in the direction n,f, s,p, if Aa is positive, that value of 6 between o° and i8o°, which satisfies (i.) must be chosen ; and if Aa is negative, the value between i8o° and 360°. p is always positive.

Further, if Aa, AB are observed with assigned limits of error, it is advisable to ascertain what are the corresponding limits of error in 6 and p, by substituting in succession those values of Aa, AB which give the greatest and least values to 6 and p.

Throughout the whole of the working of this problem it is advisable to have angles expressed in degrees and decimals of a degree.

Reduction to a Selected Epoch.

In all cases before observations made at different times can be combined, the effect of precession on the angle of position

ON THE ORBIT OF A BINARY STAR. 93

must be eliminated. For it must be remembered that angles of position are measured from the great circle which passes through the star and the pole, and that in consequence of precession the pole is constantly shifting its place, having a slow retrograde motion round the pole of the ecliptic. Hence the position that this circle occupies a^ some selected epoch must be taken as the zero of position, and all observations must be referred to it.

The subjoined figure will show how the effect of this motion of the pole on the angle of position of any star can be com- puted.

Let E be the pole of the ecliptic ; P, P' positions of the pole at an interval of a year; T the intersection of equator and ecliptic, from which the right ascension is reckoned ; S S' a double star in right ascension a° and declination 8°. Draw the circles P S, P' S ; then P S F is the A 9 required.

Since T E, T P are quadrants, T P E is a right angle, and therefore P' lies on T P. Draw Y'p perpendicular to S P.

Fig. 9.

riG. 9.

Then P P is known from the constant of precession to be for the year 1850, and very approximately for any other year, 2o"-os64= o°-ooSS7i2;

94 DOUBLE STARS.

and P'/ = P P sin P' P/> = P F sin a;

also F'p = Ad cos h, as in the last section. .-. J 0 cos S = P' P sin a ; and A6 = o°-oo5S7i2 sin a sec S. The exact formula is {20"os64 — o"-000097(/ — 1850)} sin a sec 8.

It appears further that the effect of precession is to increase the angle of position in the case chosen. Hence, in order to bring up to a certain date old observations of position taken t years before that date, we must add to those angles of posi- tion the quantity o°'oo5S7i2 sin a sec S x /. It is plain that this will be + for values of a from 0° to 180°, or from o"* to 12'' and — for values from 180° to 360°, or from I2'» to 24^

Example. — Dawes in the year 183 r34 observed the angle of position of 77 Coronas Borealis in right ascension IS*' 18" 14^ and declination 30° 43' 31", to be SO°-46. Reduce this to the epoch 1880.

Converting the right ascension into degrees, it becomes 229° 35' 30". Hence AO = o°-oosS7i2 sin 229° 35' 30" sec 30° 43' 31" X 48-66 = — o°2292 ; and the corrected angle is therefore So°-23.

Drawing of the Interpolating Curve.

When a table has been thus constructed, giving, for some selected epoch, the angles of position and distances at a number of dates, the next problem is how to use this mass of materials. It will be at once obvious that the observations are not very harmonious, but that there are serious discre- pancies not only between different observers, but between the same observer and himself. And if the points were simply charted out according to the observed positions and distances, they would not lie on a curve, but on a broad irregular band.

Sir John Herschel was the first to suggest {Mem. R. A. S., vol. V.) a graphical method of obtaining the positions at any

ON THE ORBIT OF A BINARY STAR.

95

selected epochs with a high degree of accuracy; a method which necessarily gives no weight to exceptionally bad obser- vations, and makes use of all the good observations, both before and after any epoch, to determine the angle at that epoch.

Take a sheet of paper ruled in fine squares, — that called millimetre paper * is the best, — and let the divisions running horizontally, suppose, represent angles, each division standing for a tenth of a degree, and the divisions running vertically represent years, each division standing for a tenth of a year.

On this convention a dot on the chart represents a single observation.

The subjoined chart therefore represents the following table of observations,

t.	6.
1870-23	210-05
1870-38	209-95
1870-40	210-30
1870-25	212-38
1871-08	211-10
1871-34	212-08
i87i-4i	212-08
1872-10	214-62
1872-19	214-44
1872-20	213-21

mm.

zio' in sis' sjf 2u'

Fig. 10.

and the curve drawn among them cannot be very far from the truth, and is influenced by all the observations except the two outlying ones, which are obviously bad.

By this means we can obtain more accurate estimates of

what the angle would be at any assigned date, or what is

more used, of the date at which the angle would have an

assigned value, than we can from the observations directly.

For example, from the diagram we see that in 1870-00 the

♦ Millimetre paper may be got at Messrs. Williams and Norgate's.

96

DOUBLE STARS.

angle would have been 2095, and that the angle was 213 at the time 187 171.

All the measures, therefore, of position of the star must be charted, and the 'interpolating curve,' as Herschel calls it, must be drawn among them. This is a matter of the highest importance. The curve must be smooth and flowing. It may have points of contrary flexure, but it can have no abrupt changes of curvature.*

When the curve has been drawn, note the time indi- cated by it at which the angle had in succession a series of values, proceeding by some common difierence, say of 2°, or of 5°, and construct a first table of interpolated angles and dates.

Let the subjoined table be a specimen :

e°.	/.
70	: 1837-34
75	183990
80	1842-12
8s	! i844"25
90	1846-08
95	1847-74

Smoothing the Curve.

The next process is to ' smooth ' the curve by an arith- metical examination of this table.

Let A( represent the result of subtracting any one number in column 2 from the number below it, and let the series of numbers so obtained be arranged in a column to the right of the column of t.

Similarly, let A't be the differences between the numbers in the column of A-i, and be placed in a column to the right ; and A^( be the diff'erences of A-t.

* A point of contrary flexure indicates a point where the line drawn to the principal star is normal to the apparent orbit of the star.

ON THE ORBIT OF A BINARY STAR. 97

The table will then be as follows :

6°.	/.
70	1837-34
75	1839-90
80	1842-12
85	1844-25
90	1846-08
95	1847-74

A/.	AV.	A^/.
2-56
2-22	- '34	+ ■25
2-13	- -09	— •21
1-83	- -30	+ ■1%
1-66	- -17

It is plain from this that the numbers are not quite right, that is, that the curve has not been drawn quite smoothly, or that some of the values of t have not been quite correctly estimated. For if they were, then the differences in each column ought to proceed regularly, and not show irregular and abrupt changes, as this series does, in the second and third differences.

It is necessary, therefore, to make slight changes in the second column such as will bring the difference columns into more perfect adjustment. To do this is not very easy, and requires patience. The following considerations may help in the process. The column of A^t is on the whole +, and therefore the column of A^t ought to have its terms, which are negative, continually decreasing in absolute magnitude. The -09 is therefore, too small, and the -30 too large. These can be changed in the right direction by increasing the 2-22 or diminishing the 2-13, and these in their turn make changes in the first column.

After successive attempts, we obtain the following result :

70

75 80

85 90

95

/. 1837-35 1839-87 1842-16 1844-23 1846-09 1847-74

M. A''/. 2-52

2-29 2-07 1-86 f6s

- -23

— -22

— -21

AV.

-01 •01 ■00

gS DOUBLE STARS.

By comparing this with the previous table, it will be seen that none of the dates have been altered more than '04 of a year, which would be represented on the chart by an almost imperceptible space.

The values of t so obtained may therefore be regarded as a still closer approximation to the truth than those obtained directly from the graphical process, and d fortiori than those obtained by direct observation. All small errors arising from imperfect drawing of the curve, or wrong estimation of the decimals, have been got rid of. But it must not be forgotten that these values are still hable to be affected by serious errors of judgment in drawing the curve, or by errors of single observations when the curve depends on single observations. The curve may be smooth, and yet not the right curve. Errors of this kind cannot be detected at the present stage of the problem, but will be revealed later on.

Employment of Measures of Distance.

In a precisely similar manner all the measures of distance should be charted on millimetre paper, and interpolated distances obtained, at equal intervals of time, and the distance curve ' smoothed.' The errors in observation of distance often bear a large ratio to the distance itself, and the inter- polated distances are far more trustworthy than any individual measures.

If now a series of corresponding values of r, 6 is found, and charted, these points will give a general indication of the nature of the curve. They will, for example, indicate whether the orbit is likely to be rectilinear or elliptical, and whether a sufficient portion of it has been described to make it worth while to attempt the computation. But in many cases it will be found that the points so obtained do not lie tolerably well on a curve, and that there will be liability to large error in attempting to draw a curve among them. This arises from the almost unavoidable error in the measurement of

ON THE ORBIT OF A BINARY STAR. 99

distances. Sir John Herschel, therefore, devised a method by which the relative distances could be obtained from the measurements of position alone, and this we now proceed to describe.

Determination of distance from the interpolating curve for angles of position.

If A C E is an ellipse, S C p

the focus, it follows from Kepler's second law that equal areas are described in equal times, that the rate of change of angular position is much more rapid in some part of the orbit than in others. I'^i^- n.

Let A S B, C S D, E S F, be equal areas ; then they would be described in the same time, and hence the change of posi- tion angle in that time would be A S B in one part of the orbit, C S D in another, and E S F in a third. And con- versely, if the change of angle is greater at one part of the orbit than at another, it follows that the distance must be less, and less to such an amount as to make the areas described in equal times equal.

If ;- be the distance at any time, A 6 the small angle described in the time A t, it follows that \ r'^ A6 \s the area described in that time; and therefore that the limit of r^ — must be constant at all parts of the orbit ; and therefore that r' varies as limit of ^. •

But from the table given above (p. 96) A 6 \s constant, and A t can be got by subtraction, and the limit of -^ may be got either from the formula

A^ _ j_ fiu_ _ _Af , A^^ \

or, very approximately, by taking half the sum of the differ- ences of the times that precede and follow the date selected.

lOO DOUBLE STARS.

For example, referring to the previous table, A 0 = 5°, and when d — 80°, by the first formula

^2 0C

I /2-07 , -21 , -ooN _

yl-T + 1- + TJ - 435, ■436;

and by the second formula

^2 ^ if2-29 + 2-07) _

when the angle is 70°, ^^ = L{^^ + '^ + 'f) = -527, and

therefore the values of r at 80° and 70° are as v/436 : v/527, or as 2088 : 2295.

In this manner relative values of r are obtained for all the values of 6 in the previous table, at intervals of 5" or iO°, and these will be in general more accurate than those obtained from direct measurement, as they depend on measures of position alone.

In order to compare with seconds of arc the unknown unit in terms of which these values of r are expressed, it will be necessary to take the whole series of values of r obtained in seconds at suitable points from its own interpolating curve, and the whole series obtained in the unknown scale from the formula above given, and compare the sums of the two series. Thus will be obtained the relative value of the two units to a high degree of approximation. Take the following values as an illustration : —

t, r. r^ on scale.

1830 4'5o 12500

1835 462 127-60

1840 475 I3i'40

Sums ... 1387 384-00

Here I3"87 are equal to 384-0 scale divisions, and therefore I scale division corresponds to o'' 036 12.

It will further be worth while to reduce to seconds each of the values of r, and chart them along with the interpolating curve which furnished the direct values of r, in order to see how far the calculated and observed values agree. A dis-

AN ORBIT WORKED BY A GRAPHICAL METHOD. Id

crepancy, systematically recurring between them, may lead, as in Otto Struve's recent investigation of the orbit of the dis- tant companion of f Cancri, to some novel and remarkable conclusions. (See Observations de Poulkova, vol. ix., and the Comptes Rendiis de I'Acad^mie de Paris, vol. Ixxix., p. 1463.)

To Draw the Apparent Ellipse. It may now be assumed that we have the values of r for a series of values of Q differing by 5°. Let these be converted into X and y by the formulae x=r cos B, y= r sin 6, and the points charted on the millimetre paper. They will be found to lie on a curve ; and if a sufficient portion of the orbit has been described, the curve will be sensibly an ellipse. And here it may be observed that these points furnish the best possible test of the skill with which our final interpolating curve has been drawn; for if any poitit or points lie out of the curve we must at once redraw that part of tlie interpolating curve. Assuming that the correction has been made, the ellipse passing through the points may now be found either by the graphical or analytical methods. If the former be adopted, an ellipsograph, or a piece of string and two drawing pins, with a little patience, will suffice for this purpose. The line once drawn in pencil should be carefully inked in with a fine pen. This is the apparent ellipse. No care must here be spared in drawing the best possible ellipse, and drawing a fine line. With a pair of compasses we may now at once measure off the maximum and minimum apparent distances, and obtain directly the angles at which they occur. The larger star A occupies the projected focus of the real ellipse.

Determination of the Real Ellipse : Thiele's Method.

We must next proceed to the method of determining the

real ellipse from the apparent one, and in doing this we shall

follow Thiele's method, and give a geometrical proof of the

elegant theorem he employs.

102 DOUBLE STARS.

The problem is this : — Given an ellipse and a point in it which is not the focus, it is required to firld the position and magnitude of the ellipse whose projection is the given ellipse, and the projection of its focus the given point.

The determination of the position and magnitude of the ellipse requires the determination of five elements, viz.,

(i) The angle a that the line of intersection of the two planes, or line of nodes, makes with a fixed line.

(2) i the angle of inclination of the planes.

(3) ^ the eccentricity of the ellipse.

(4) a the semi-axis major of the ellipse.

(5) \ the angle between the line of nodes and the line of

apsides, or the line to periastre. The solution depends on the following geometrical property of the ellipse : —

Let P S Q be any focal chord of an ellipse ; M X N the

corresponding directrix ; P M, Q N

perpendiculars to the directrix ;

P K, Q H perpendiculars to the

axis major ; S L the semi latus

Fig. 12. rectum, and L R perpendicular to

the directrix. Then, by similar triangles, HS:SK::SQ:SP,

and by the property of the ellipse S Q : S P : : Q N : P M ;

therefore QN:PM::HS:SK

::LR-QN:PM-LR; that is, Q N, L R, P M are in harmonic progression ; but Q N, L R, P M are respectively proportional to S Q, S L, S P j therefore the harmonic mean of S Q and S P is constant.

And if along the chord P S Q a point Y be taken, so that S Y is the harmonic mean between S P, S Q, the locus of Y would be a circle of which S would be the centre, and S L the radius.

If now the ellipse and this harmonic circle (as it may be called) be projected on a plane inclined to their own, the circle will be projected into an ellipse, the direction of whose

AN ORBIT WORKED BY A GRAPIHCAL METHOD. i03

major axis gives the line of intersection of the two planes, and the ratio of whose semi-axes is the cosine of the inclina- tion of the planes.

Conversely, if the harmonic ellipse be drawn, by taking, arithmetically or graphically, the harmonic means between the segments of a number of chords through the projected focus in the apparent ellipse, it follows that its major axis is equal to the latus rectum of the true ellipse; that its major axis is in the direction of the line of nodes; and that the ratio of its minor to its major axis is the cosine of the angle of inclination of the plane of the real ellipse to the plane of the apparent ellipse.

Further, if C is the centre (Fig. 13), S C A' is the projection of the major axis ; and §| = .?, the eccentricity of the real ellipse, this ratio being unaltered by projection. Hence we find in succession a, i.e. the angle which the line of nodes makes with the axis of ;r, the meridian through the star ; i, the inclination, from the condition cos i = ||, Sa and Si being the major and minor axes of the harmonic ellipse : e = ^-

J L S r- ) AC '

and ^ = — -^ = _^, L being the semi latus rectum.

Finally, X, i.e. the angle the line to the periastron makes with the line of nodes, is found as follows : —

Let X' be the angle X S C, V - /2 the angle A' S a in the annexed figure where A' is the projected periastron, and therefore known. \ is the angle A S a which is required.

Draw A' N, A N perpen- dicular to S a.

Then tan X = 1^ = M,

X "11 = sec i tan (\ - a), ^^^ ^^

and therefore X is known.

To construct the ephemeris graphically, it is necessary to divide the ellipse into equal sectorial areas by radii drawn from the focus. This may be accomplished as follows : —

104

DOUBLE STARS.

Let A PA' be an ellipse (Fig. 14), P any point in it, S the focus, C the centre ; A Q A' the auxiliary circle, Q P N an ordinate through P.

Let e be the eccentricity, a, b the semi-axes of the ellipse,

T the periodic time for the whole orbit. Then if t be the

time taken in describing the area ASP from perihelion to

the point P,

' ~ ' SCQ.

t_ T

ASP _ ASQ

IT ab TT a^

ACQ-

and therefore if u is the circular measure of A C Q,

t \ua* — ^aeasinu u — ^ sin u

In order, therefore, to divide the area by focal radii into equal

intervals, values must be given to « — ^ sin u in arithmetical progression.

Let A B A' be the semicircle described on the major axis of the ellipse as diameter, S the focus of the ellipse.

Divide the arc B A' into any number of equal parts, say of 10° each. Draw the tangent at B, and mark off along it from B parts equal to the arcs of 10°, 20° . . . 90°.

Through the points of division of the arc draw lines parallel

AN ORBIT "WORKED BY A GRAPHICAL METHOD. IO5

to C X, and through the points of division of the tangent at B draw lines parallel to C B, thus determining a number of points Vi Va . ., and through these points draw a curve B V X. We will call this the ephemeris curve.

If now P is any point on the ellipse, Q the corresponding point on the auxiliary circle, V the corresponding point on the ephemeris curve, Q V being parallel to C X, C X' equal to CX,ACQ = u. Then if V N is parallel to C B, X' N = « u. Join S B, draw V H parallel to S B, and join C Q. Since y-N ~ c5 ~ ^> ^""^ CQ ~ ^^" ^' therefore H N = ae sin u, and therefore X' H = a(u — e sin u).

Hence 2-^7$ = f-. and therefore the position P in the orbit can be at once found corresponding to any time /, and con- versely the time / can be found corresponding to any position P in the orbitj by simply drawing parallel lines.*

Lastly, this method can be adapted to the further problem of dividing an ellipse into equal areas by lines from any point which is not the focus. To do this, instead of the auxiliary circle, an auxiliary ellipse must be taken, which will be similar and similarly situated to Thiele's harmonic ellipse.

The working of this will be readily understood from the example annexed.

* This problem can also be approximately solved with equal accuracy by mechanical means. The latest and best method is that given by Professor Bruhns in the Vierteljahrschrift der Astronomischen Gesellschaft 1875, Heft. 4. For an improved form of this apparatus, also by Pro- fessor Bruhns, see Heft. 4, 1877. Dr. Doberck, however, prefers to use the tables he has published in the Ast- Nachrichten.

I06 DOUBLE STARS.

CHAPTER II.

EXAMPLE OF AN ORBIT WORKED BY A GRAPHICAL METHOD.

For this method we shall select Castor, as a double star of great historical interest, and sufficiently brilliant and widely • separated to be within the reach of all telescopes that are likely to be used by amateurs. The orbit has been frequently computed before, both by graphical and analytical methods, and a comparison of the results arrived at is very instructive as showing the difficulty and uncertainty in problems of this nature, when the portion of the orbit described bears a small ratio to the whole.

Table I. gives in chronological order the observations arranged as follows. In column i, headed t, is the date of the observation ; in column 2, the observed angle, headed & ; in column 3, the angle corrected for precession up to the year 1880, headed 6\ in column 4, the number of nights of observation,