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 01' THE 
 
 UNIVERSITY OF CALIFORNIA 
 
 Received 
 Accessions No. _ .^^ 
 
 Shelf No. 
 
LECTURES 
 
 ON 
 
 PRACTICAL ASTRONOMY 
 
THE AUTHOR desires to express Ms thankful acknowledg- 
 ments to the SYNDICS OF THE UNIVERSITY PRESS for their 
 liberality in granting him the whole expense of the Press-work, 
 Paper, and Illustrations of this Volume. 
 
LECTUBES 
 
 ON 
 
 PKACTICAL ASTKONOMY 
 
 AND 
 
 ASTRONOMICAL INSTRUMENTS 
 
 BY THE 
 
 KEY. JAMES CHALLIS, M.A., F.B.S., F.R.A.S. 
 
 PLUMIAN PROFESSOR OF ASTEONOMT AND EXPERIMENTAL PHILOSOPHY IN THE 
 UNIVERSITY OF CAMBRIDGE, AND FELLOW OF TRINITY COLLEGE. 
 
 _J D 
 
 UNIVERSITY OF 
 CALIFORNIA. 
 
 CAMBRIDGE: 
 
 DEIGHTON, BELL AND CO. 
 
 LONDON: GEOEGE BELL AND SONS. 
 
 1879 
 
 [All Eights reserved.] 
 
PRINTED BY 0. J. CLAY, M.A. 
 AT THE UNIVERSITY PRESS. 
 
PBEFACE. 
 
 I BEGAN in the year 1843 a Course of Lectures on 
 Practical Astronomy and Astronomical Instruments, 
 having been at that time seven years Director of the 
 Cambridge Observatory. Lectures on these subjects 
 had already been given by Dr Peacock as Lowndean 
 Professor, which he discontinued on the understanding 
 that I would undertake to carry them on. The circum- 
 stances that an Astronomical Observatory so well ap- 
 pointed as that of Cambridge was near at hand, and was 
 provided with various instruments of first-rate quality, 
 appeared to me to give facilities for lecturing on Prac- 
 tical Astronomy which ought to be taken advantage of ; 
 and accordingly I commenced lecturing in the above- 
 named year, having previously produced a 'Syllabus' of 
 the subjects of the lectures, and procured a considerable 
 number of wooden models, which, together with some 
 apparatus that had been collected by Professor Peacock, 
 I made use of in conducting and illustrating the Lectures. 
 I had also the advantage of having at command several 
 portable instruments pertaining to the Observatory. 
 At the end of the Syllabus a list of formulse applicable 
 to the reduction of astronomical observations was in- 
 troduced, accompanied by brief demonstrations. Also 
 
VI PREFACE. 
 
 in giving the lectures orally, I adopted the plan of 
 exhibiting in ink-writing on large sheets of white 
 paper descriptions of the instruments, and investiga- 
 tions of formula, with the requisite illustrations by 
 Figures, all being of such size as to be readily seen 
 by the students from their seats. I had recourse to 
 this method of lecturing partly because I thought it 
 would serve to convey adequate information with 
 little expenditure of the students' time, and partly 
 because I was unable to meet with a text-book on 
 this department of Astronomy which I could regard 
 as sufficiently accurate and complete. 
 
 The contents of the Syllabus, together with the 
 explanations inscribed on the above mentioned papers, 
 have formed the ground- work of the present publi- 
 cation. In fact, as far as regards the divisions of 
 the subjects treated of under the head of each instru- 
 ment, and the order of their treatment, the Syllabus 
 has been closely followed ; but in the course of writing 
 the Treatise, the composition of the Lectures, as 
 originally conceived, has been in various respects 
 modified and added to. The subjects which make 
 up the additional matter are the following : A 
 method of correcting the errors of a transit-instrument 
 for deviation of the pivots from the cylindrical form ; 
 a detailed description of the construction and appli- 
 cations of the collimating eye-piece ; the chronographi- 
 cal method of registering transit- observations ; a discus- 
 sion respecting personal equation in taking eye-and-ear 
 transits; an experimental investigation of the effect of 
 the flexure of a Mural Circle on the mean of its Micro- 
 scope-readings ; a description of the construction of the 
 new Transit-Circle of the Cambridge Observatory and 
 
PREFACE. Vll 
 
 the mode of using it in taking observations (inserted 
 with the consent of Professor Adams) ; the method 
 employed at Greenwich for making observations of 
 the Moon with an Altazimuth Instrument (not pre- 
 viously introduced, as far as I am aware, in an 
 Elementary Treatise) ; and the process of taking 
 observations with the Greenwich Reflex Zenith Tube. 
 The treatment of these subjects has given to the 
 work an extension much beyond what I at first 
 contemplated, but at the" same time its usefulness 
 as an astronomical manual may be considered to be 
 much increased by being thus made to exhibit with 
 a great degree of completeness the actual state of 
 observational astronomy. 
 
 Although the instruments of the Cambridge 
 Observatory, and processes of observation I adopted 
 in the use of them, have been more especially de- 
 scribed, and the Treatise consequently partakes some- 
 what of a local and personal character, I may venture, 
 I think, to say that as having been written after 
 twenty-five years of continuous labors in astronomi- 
 cal observations and calculations, and containing what 
 may have occurred to me in the course of that experi- 
 ence as contributory to the advancement or improve- 
 ment of practical astronomy, it will be found of some 
 general utility as respects the work carried on in an 
 Astronomical Observatory. 
 
 A particular account has been given of the in- 
 vestigation of formulae for the calculation of the 
 mean places and annual variations of the fixed stars, 
 and care has been taken to derive the numerical 
 coefficients from the most accurate data procurable. 
 The values assigned to the coefficients were calculated 
 
Vlll PREFACE. 
 
 for the year 1879, and generally may be used without 
 important error for epochs less than ten years before 
 and after that date. 
 
 I take this opportunity for saying a few words 
 respecting Practical Astronomy considered as a subject 
 included in the course of the Mathematical Studies 
 of the University of Cambridge. It seems to me 
 that the tendency of our mathematical instruction 
 and. examinations has been of late years to promote 
 the acquisition of a knowledge of formal relations 
 of symbols and the power of readily producing them, 
 apart from a distinct exercise of the reasoning 
 faculty. Now as far as regards Practical Astronomy 
 and Astronomical Instruments, it may be asserted 
 that it would be impossible for the student to make 
 himself acquainted with this department of science 
 without understanding the reasons of the mutual 
 relations of the different parts 0f the instruments, 
 and the processes of observation by which the intend- 
 ed purposes are effected. He must be able to see 
 how the many ingenious mechanical contrivances which 
 the wants of astronomy have called forth, contribute 
 to facility and precision in making and recording ob- 
 servations, and although this accomplishment may not 
 demand a very high order of intelligence, it is still 
 a mental exercise of much educational value, inasmuch 
 as it is altogether unlike any process of reasoning 
 by abstract symbols, and may serve as a corrective 
 of the effect of too exclusive an attention to reason- 
 ing of that kind. The amount of mathematical 
 knowledge which a complete understanding of Practical 
 Astronomy requires is not more than what every 
 candidate for Mathematical Honors is expected to 
 
PREFACE. IX 
 
 possess, and accordingly the subject may be regarded 
 as being within the reach of candidates of all degrees 
 of proficiency. For these reasons I think it is much 
 to be desired that Lectures on Practical Astronomy 
 should always form a part of University teaching, 
 and that the subject should at the same time receive 
 due recognition in the Senate House Examination. 
 
 I am of opinion that a Professor of Astronomy, 
 or Lecturer, who may undertake to lecture in this 
 University on Practical Astronomy, if he should 
 recommend selected portions of a Treatise like the 
 present for the students' reading, and give only a 
 few Lectures, would be able to convey sufficient 
 instruction to candidates for Mathematical Honors, 
 without requiring a disproportionate expenditure of 
 time on this one branch of only one of the many 
 subjects that engage their attention. The Lectures 
 might be given partly in the Astronomical Lecture- 
 Room in the Museum Buildings, to take advantage of 
 the collections of instruments and models contained 
 either in the Cabinet of the Plumian Professor's private 
 room or in the Astronomical Apparatus Room, and 
 partly at the Observatory for the purpose of pointing 
 out the construction and mounting of the several in- 
 struments, and the methods of taking observations. 
 
 The contents of this Volume to page 337 are 
 limited to the consideration of what is done with 
 fixed instruments in a fixed observatory, and as 
 having, consequently, exclusive relation to the 
 foundations of exact Astronomical Science, might be 
 appropriately separated from all other parts of Prac- 
 tical Astronomy, and form a distinct Volume. But 
 because in the before mentioned Syllabus additional 
 
PREFACE. 
 
 subjects were inserted which might be ranged under 
 the heads of Observations with Transportable Instru- 
 ments, and Miscellaneous Astronomical Information 
 not given in the previous part of the Treatise, which 
 also in delivering the Lectures formed the concluding 
 portion of the Course, I have included them in the 
 last two Sections of the present Volume, although 
 on account of failing health and strength, I have 
 not been able to treat of them so fully and in as 
 much detail as the subjects comprised in the funda- 
 mental portion. It will, however, I think, be found 
 that The Theodolite and The Sextant, two portable 
 instruments of essential use in the Sciences respec- 
 tively of Geography and Navigation which are of so 
 much national importance at the present time, have 
 been adequately handled. Also due consideration has 
 been given to the methods recently employed for 
 determining the Solar Parallax, and, in particular, 
 inferences deduced from the observations of the 
 Transit of Venus across the Sun's disk, on Decem- 
 ber 8, 1874, have been brought into notice. 
 
 CAMBRIDGE, 
 
 May 8, 1879. 
 
CONTENTS. 
 
 N. B. The numbers refer to Articles, except where Pages are expressly signified. 
 
 INTEODUCTION PAGES 119. 
 
 Art. 1, division of Astronomy into Practical, Spherical, and Gravita- 
 tional : the Lectures embrace only the first division. 2, data obtained by 
 observations with fixed instruments in a fixed Observatory constitute the 
 Foundation of astronomical science. 3, two kinds of astronomical observa- 
 tion ; how related to the contents of the Nautical Almanac, and the Sciences 
 of Geography and Navigation. 4, antecedent knowledge required for 
 understanding the Lectures. 5, general statement of the immediate! 
 purpose of observations in a permanent Observatory, namely, to determine i 
 celestial positions by means of spherical coordinates in Eight Ascension I 
 and Declination. 6, names of the principal fixed instruments employed 
 for this purpose, and general account of what they respectively effect. 
 7, reasons for treating of the Transit-instrument, the Mural Circle, and the 
 Transit- Circle separately. 89, remarks on the Cambridge Observatory 
 and its site ; processes for drawing a meridian line, fixing the positions of 
 the piers of a transit-instrument, and placing a meridian mark. 
 
 Art. 11, optical conditions to be satisfied by an Astronomical Telescope. 
 12 13. definition of the optical centre of a single, or compound, lens. 14, 
 analogous optical centre of a circular spherical mirror. 15 16, optical axes 
 of pencils of light refracted by a simple or compound lens, or reflected by a 
 mirror. 17 19, method of adjusting and centering the two lenses of an 
 achromatic object-glass ; exemplified in the instance of the object-glass of 
 the Northumberland Telescope. 20 23, The principal eye-pieces of an 
 Astronomical Telescope; Huyghens's, Eamsden's, the four-glass erecting 
 eye-piece, and the diagonal eye-piece. 
 
 Art. 24 (1) (15), short notices of modern works relating to Practical 
 Astronomy; authors: Woodhouse, Pearson, Coddington, Airy, A. E. Grant 
 (Cambridge), E. Grant, Loomis (New York), Brunnow (translation in part 
 by E. Main), E. Main (Spherical Astronomy), P. T. Main (Plane Astronomy), 
 Brunnow (the whole translated by the Author), Godfray, Chambers (two 
 works), Chauvenet (Philadelphia and London), F. W. Simms (published by 
 Troughton and Simms). 24 (16), additional sources of information ; 
 Herschel's "Introduction to Astronomy" and "Outlines of Astronomy"; 
 the Articles " Mural Circle and " Equatorial" by Sheepshanks in the 
 Penny Encyclopaedia ; Introductions to the volumes of Cambridge Observa- 
 
xii CONTENTS. 
 
 tions of 1836, 1837, and 1838; the volumes of Greenwich Observations 
 from 1811, by Pond and Airy ; references to those volumes (1842 1871) 
 which contain particular descriptions of Greenwich instruments. 25, 
 general relation between Instrumental Observation and Practical Astro- 
 nomy ; treatment of each instrument under four Divisions. 
 
 THE TKANSIT INSTKUMENT PAGES 20 118. 
 
 I. Description of the Parts and Mounting of the Instrument . PAGES 2023. 
 
 Art. 27, the Piers ; the pivots of the transverse axis, and the supporting 
 Y's. 28 29, test of the cylindrical form of the pivots ; effect of wear of the 
 pivots by friction, requiring correction ; counterpoises to diminish the 
 friction. 30, description of Eamsden's eye-piece, and adaptation for use in 
 taking transit-observations. 31 32, the single-wire and double-wire 
 Micrometers. 33, the field of view of the Telescope, as seen with a 
 Eamsden's eye-piece ; the vertical transit-wires and the horizontal wires ; 
 reading of the Micrometer. 34 36, the external parts of the eye-end of the 
 Telescope; the setting-circles, their uses, and correction of their index- 
 errors ; the action of the tangent-screw and clamp of a setting-circle, dis- 
 tinguished from other applications ; adjustments of the wire-frame by angular 
 motion about the telescope-axis, shifting it horizontally, and fixing it at the 
 geometrical focus of the object-glass. 37 38, methods of illuminating the 
 field of view for seeing the wires on a bright ground, and of illuminating 
 the wires for seeing them on a dark ground. 
 
 IE. The mechanical adjustments of a Transit-Instrument and corrections of 
 Errors by calculation , ~ . . . . . PAGES 33 102. 
 
 Art. 40, definitions of axis of motion, line of collimation, plane of colli- 
 mation, and optical axis, the pivots being cylindrical. 41 42, the plane of 
 collimation to be adjusted to the meridian either mechanically, or virtually 
 by calculation ; the latter method preferable. 43, preliminary adjustments 
 of the middle wire and the micrometer-wire. 44, definitions of collimation, 
 level, and azimuth errors. 45 47, arrangements in the case of the 
 Cambridge Transit-instrument for correcting the three errors; special 
 means of correcting the level and azimuth errors of the Greenwich and 
 Cambridge Transit-Circles. 48 50, mechanical correction of collimation 
 error by a meridian mark and reversing the instrument. 51 52, calculation 
 of uncorrected collimation error ; effect of employing two marks, one north- 
 ward and the other southward. 53 56, description and applications of 
 collimators; substitution of a collimator for a meridian mark; Gauss's 
 method of obtaining collimation error by two horizontal collimators without 
 reversion. 57 58, determination of the sign of the collimation-correction. 
 59, calculation of the value in arc of the micrometer-revolution. 6061, 
 observations and calculations for ascertaining the equatorial time-intervals 
 of the several wires from the mid-wire and from the mean of all. 62, 
 example of calculating collimation-error from bisections of the two marks in 
 reverse positions of the instrument. 63, error of pointing due to diurnal 
 aberration ; formula for calculating the correction ; application to the lati- 
 
CONTENTS. Xlll 
 
 tude of the Cambridge Observatory. 6566, method of obtaining collima- 
 tion- error by transits of Polaris and reversing the instrument ; investigation 
 of a formula for calculating the correction. 
 
 Art. 67, construction of the spirit-level ; radius of curvature of its axis, 
 and measure of its sensibility; case of the Cambridge Level. 68 73, 
 adjustment of a spirit-level ; levelling by adjusted marks ; by fixed adjusted 
 scales; by movable scales unadjusted; calculation of level-error in scale- 
 intervals, the pivots being cylindrical and equal. 74 76, correction for 
 inequality of cylindrical pivots ; determination of the value in arc of the 
 scale -interval by a mural circle, or by a bubble-trier ; sign of the correction 
 for unequal cylindrical pivots. 78 79, example of the calculation of the 
 level-correction, and determination of its sign; list of corrections for 
 apparent inequality of the pivots obtained in 18281854, giving evidence of 
 wear of the pivots ; discontinuance of the use of the spirit-level. 
 
 Arts. 80 83, the collimating eye-piece, its optical construction, and 
 mode of its application to the Telescope; supersedes the plumb-line, or 
 spirit-level, for determining the vertical direction; its use for placing the 
 wire-frame at the geometrical focus of the object-glass. 84 86, simul- 
 taneous determinations of collimation and level errors by the collimating 
 eye-piece and either reversing the instrument, or employing north and south 
 collimators ; the latter method practised with the Greenwich and Cambridge 
 Transit-Circles. 87, level error independently obtainable by direct and 
 reflection observations of transits of Polaris. 
 
 Arts. 88 90, azimuth error ; necessarily obtained by astronomical obser- 
 vation ; the process of observing, and general formula of calculation ; 
 modifications of the formula in four cases. 91, mechanical correction of 
 azimuth error, and determination of the distance of a fixed mark from the 
 meridian. 92, formulae for calculating the time-corrections of the three 
 errors of collimation, level, and azimuth. 93 97, definition of the total 
 error of collimation (see errata), the pivots being equal and cylindrical ; 
 special means of obtaining its value when the pivots are unequal and of any 
 irregular form nearly cylindrical .; investigation of a formula for calculating 
 the change of collimation-error due to deviation from the cylindrical form ; 
 the time-correction for such deviation ; determinations of the constants 
 involved in the formula ; example of the calculation of formulas for both 
 positions of the illumination-end of the axis. 102, formation of. a mean 
 table of pivot-corrections from special measures taken in three years, the 
 argument being P.D. ; inference of the correction for any P. D. by interpola- 
 tion; an account of the method of obtaining the pivot-corrections of the 
 Cambridge Transit-Circle (in a Note to Art. 102). 
 
 Arts. 103106, the Astronomical Clock, with mercurial pendulum ; mode 
 of altering the clock's rate; correction of imperfect compensation by adding 
 or subtracting mercury ; formula for calculating the required amount. 
 
 III. Methods of taking Transit Observations, and calculations of Correc- 
 tions [of certain transits as observed]. . * . . PAGES 102 126. 
 
XIV CONTEXTS. 
 
 Art. 107, adjustment of the setting circles by the collimating eye-piece, 
 or by a known star ; setting for an object of given P.D. 108 112, methods 
 of observing transits of the Sun, the Moon full or nearly full, Jupiter, Mars, 
 the Eing or Body of Saturn, Mercury, Venus, and very small planets ; 
 transits of stars very near the Pole, taken either at the several wires, or at 
 the micrometer-wire in arbitrary positions, or at an additional set of wires 
 separated by small intervals (the case of the Greenwich and Cambridge 
 Transit-circles). 113 115, deduction of time of transit of Centre from the 
 observed time of transit of Limb ; correction of transit-time of a Limb for 
 defect of illumination, in observations of the Moon, Venus (horned and 
 gibbous), Mars, and Jupiter ; investigations of formulae for calculating the 
 corrections. 116 126, reduction of the mean of transits across some of 
 the wires to transit across the mean of all the wires ; previous determination 
 of the interval of each wire from the mean of all for any star or moving 
 body ; auxiliary table of intervals for an equatorial star, and for Polaris and 
 5 Ursae Minoris ; reduction to the mean of the wires of transit- times taken 
 at dark bars for observing faint objects ; investigation of factors for 
 obtaining the correction of broken transits of the Sun, Moon, and Planets 
 from that for a star of the same P.D. ; additional factor for the Moon to 
 include the effect of parallax; (the measure of the Moon's diameter by 
 transits not affected by parallax) ; rule for deriving from the intervals calcu- 
 lated for omitted wires the reduction of the mean of the observed times to 
 the mean of all the wires. 127 128, investigation of a formula for 
 calculating level-correction from transits of Polaris taken at part of the 
 wires directly, and at part by reflection ; the result independent of collima- 
 tion and azimuth errors; deduction thence of collimation-error from 
 measures by the collimating eye-piece; the method by horizontal collimators 
 preferable. 
 
 Arts. 129 131, registering of transit-observations by means of a galvanic 
 circuit ; description of the apparatus, and of the chronographic operation ; 
 advantages of this method as respects accuracy, and diminution of difference 
 of personal equation. 132 135, discussion relative to personal equation both 
 in galvanic and in eye-and-ear observations ; notice of personal discordance 
 in taking transits of first and second limbs. 
 
 IV. Calculation of Apparent Right Ascensions from the transit-observations, and 
 of the Mean Eight Ascensions of Stars . . . . PAGES 126148. 
 
 Arts. 136147, historical dissertation on the determination of Eight 
 Ascensions reckoned from the First Point of Aries; Flamsteed's method, and 
 formation of his Historia Calestls ; Bradley's more perfect method through 
 discovery of the effects of Aberration and Nutation ; formulae for reducing 
 Apparent to Mean places of Stars, and for calculating their Annual 
 Variations, deduced by Bessel from researches in his Fundamenta Astronomic 
 and Tabula Regiomontance ; corrections of the constants of Bessel's formulae 
 by 0. Struve, Dr Peters, W. Struve, and Leverrier ; Peters's formulae for 
 calculating the Annual Precessions in E.A. and P.D. ; the formulae for 
 
CONTENTS. XV 
 
 reduction to Mean E.A. and P.D. adopted in the Naut. Aim. ; Airy's 
 modification of the same for avoiding negative signs. 
 
 Arts. 148 150, determination of the error of the clock on true sidereal 
 time of the place of observation; effected by means of a catalogue of 
 fundamental stars ; origination of the catalogues of such stars contained in 
 the Naut. Aim. ; deductions therefrom of the E.A. of the fundamental stars 
 used for the reduction of the Cambridge transit-observations. 151 154, 
 calculation of the clock's daily rate and the clock-error ; rules for finding the 
 correction for clock- error to be applied to the clock-time of any meridian 
 transit ; allowance for difference of personal equations when the transits are 
 not all taken by the same observer ; inference of the mean K.A. of a star 
 from all its apparent E.A. obtained in the course of a year ; calculation of 
 the star's mean E.A. at a given epoch from the results of several years' 
 observations ; formation of a catalogue of the mean E.A. of the stars, with 
 the annual variations and proper motions ; an enumeration of the principal 
 Catalogues of Fixed Stars, in a Note to Art. 154 (page 146). 
 
 Art. 155, final determination of the mean error of the assumed E.A. of 
 the fundamental stars ; only obtainable by combining Circle and Transit 
 observations of the Sun on the meridian. 156, example of the complete 
 reduction of the transit-observation of a star. 
 
 Description of the Cambridge Transit -Instrument, as represented in 
 Plate I. . . PAGE 149. 
 
 THE MUEAL CIECLE . PAGES 148233. 
 
 I. Description of the Parts and Mounting of the Instrument . PAGES 151 159. 
 
 Art. 158, special description of the Mural Circle of the Cambridge 
 Observatory. 159, protection from the Sun's heat by a screen. 160 161, 
 details and advantages of the tangent-screw and clamp attached to the 
 Cambridge Circle. 162, construction and mounting of a micrometer- 
 microscope for reading off the Circle graduation ; its field of view, cross-wires, 
 toothed comb, and zero-indicator. 163, the field of view of the Telescope, the 
 five vertical wires, fixed horizontal wire and parallel micrometer- wire, toothed 
 comb, and zero -indicator coinciding with the fixed wire; external parts of the 
 eye-end, namely, means of adjusting the fixed wire, or micrometer-wire 
 horizontally, the micrometer-head and opposite spring, apparatus for at- 
 taching the Telescope to the limb of the Circle and shifting the position, 
 screws for correction of collimation-error, and screw-adjustment of the wire- 
 frame to the geometrical focus of the object-glass. 164, illumination of the 
 field of view, setting-circle, pointer for setting by a calculated Circle-reading. 
 165, the pivots of the Cambridge Circle, counterpoise action by friction- 
 wheels, means of correcting mechanically level and azimuth errors. 
 
 II. The adjustments of a Mural Circle, and corrections of instrumental 
 
 errors PAGES 160187. 
 
 Arts. 166 167, mechanical adjustments to the plane of the meridian 
 (sufficiently accurate ; processes of fixing the "position of the wall, and of 
 
XVI CONTENTS. 
 
 mounting the instrument, for the application of such adjustments. 168 
 170, description of Eamsden's Ghost Apparatus, and method of using it for 
 levelling the axis of motion of a mural circle. 171, mechanical correction 
 of a level-error obtained by direct and reflection transits of Polaris. 172, 
 correction of collimation-error by the collimating eye-piece and screw-adjust- 
 ment, after correcting the level-error ; corrections of collimation and level 
 errors by means of two horizontal Collimators and the collimating eye-piece ; 
 mechanical correction of azimuth error found by transits of stars. 173 
 174, temporary use of the Circle Telescope for transit-observations, after 
 numerical determinations of the three errors ; usual mode of testing the 
 adjustment of its plane of collimation to the meridian. 175 179, adjust- 
 ments of the pointings of the axes, and positions of the combs, of the 
 micrometer-microscopes ; setting of the microscopes ; definition of the error 
 of Euns ; the microscope reading of a Circle-graduation ; calculation of the 
 correction, positive or negative, of an error of Bun. 180 182, the Circle- 
 reading and application of total correction for Kuns ; the Pointer-reading ; 
 concluded Circle-reading; proof that with the use of six microscopes the 
 concluded reading is not affected by eccentricity of the graduation, and 
 irregularity of the forms of the pivots. 
 
 Arts. 183 184, equatorial adjustment of the fixed horizontal wire for 
 reference-position of the observations of P. D. ; use of the micrometer- wire 
 alone for measures of P. D. ; readings for coincidence of the micrometer- 
 wire with the fixed wire ; determination of the arc-value of the micrometer 
 revolution, for reducing bisections with the micrometer-wire to the 
 reference position ; elimination, by alternate measures, of the disturbing 
 effect of atmospheric refraction in making the determination by a meridian 
 mark ; general formula applicable to alternate measures (in a Note to Art. 
 185, page 178). 187, discussion of residual errors by which Circle-readings 
 may be affected. 
 
 Arts. 188 191, definition of Zenith Point, or Index Error; infer- 
 ence therefrom of Zenith Distances ; method of finding the Zenith Point 
 by a vertical, or a horizontal, floating collimator ; applications of the col- 
 limating eye-piece for the same purpose, with and without a fixed wire ; 
 formula for coincidence of the micrometer-wire with its image when a fixed 
 wire is used. 
 
 DESCRIPTIONS, ACCOMPANYING PLATE II., OF THE GEEENWICH AND CAMBRIDGE MUKAL 
 CIRCLES . . . . . * -.. . . . PAGES 186, 187. 
 
 HI. Methods of observing with the Mural Circle, and Calculation of Apparent 
 Zenith Distances PAGES 187225 
 
 Arts. 192 196, use of a setting-circle with spirit-level; adjustment 
 of the index for setting in a direct observation ; additional index for setting 
 in a reflection observation ; uses of two setting-circles of a Transit-circle 
 which is not reversed. 197 201, the direct observation of a star ; note of 
 the place of bisection in the field, the star not being near the Pole ; note of 
 
CONTENTS. XVll 
 
 the clock-time of bisection for a star near the Pole ; formulae for reduction 
 to the meridian for curvature of path in the two cases ; determination of 
 the sign of the correction. 202 203, the observation of a star by reflection ; 
 process of taking the double observation ; placing a mercury-trough, and 
 clearing the surface of the mercury, for the reflection-observation ; cor- 
 rection for curvature of path, the same as that for the direct observation 
 with opposite sign. 204, additional correction for reduction to the meridian 
 in observations of the P.D. of moving bodies ; determinations of its amount 
 and sign in the cases of the Sun and Planets. 205 206, method of 
 observing both Limbs of the Sun at the same transit with the Mural Circle ; 
 the practice with the Cambridge and Greenwich Transit-circles. 
 
 Arts. 207208, mode of observing the P.D. of a Limb of the Moon ; 
 reduction to the meridian, requiring two additional factors on account of the 
 Moon's orbital motion and parallax; observations for P.D. of a N. or S. 
 Limb, and B.A. of a first or second Limb, by one observer at the same 
 meridian passage ; bisections, or transits, of opposite limbs nearly full by a 
 Transit-circle with assistance. 209213, Circle-observations of Planets 
 having sensible disks ; Jupiter, the globe of Saturn, Mars, Venus ; bisections 
 of the estimated centres of other planets ; investigation of corrections for 
 defect of illumination of gibbous and horned disks ; inference of apparent 
 semi-diameters ; amount and sign of the correction for the Moon when 
 nearly full ; general rule for determining which limb is defective. 214, 
 bisection, at the edge of a broad micrometer-bar, of an object too faint for 
 illumination of the field, and reduction of the micrometer-reading to the 
 reference-position. 
 
 Arts. 215 218, determination of apparent Zenith Distances ; correction 
 for Eefraction by BessePs Tables ; local Zenith and Polar Distances ; 
 discordance of zenith points deduced from reflection and direct observations 
 of stars ; method of correcting apparent Z. D. for discordance of zenith 
 points. 219 220, calculation of the colatitude of an Observatory from cor- 
 rected Z. D. of the same star above and below Pole ; estimation of the weights 
 of results given by observations of different stars ; concluded colatitude of the 
 Cambridge Observatory from observations in 1833, 1836, 1837, and 1838. 
 221228, particulars of an experiment for determining the effect of 
 mechanical flexure on the circle-readings, made by two collimators mounted 
 for collimating with each other and with the circle Telescope at all Zenith 
 Distances. 229, result of the experiment, shewing that mechanical flexure 
 does not account for the discordance of zenith points exhibited by the 
 Cambridge Mural Circle ; conjectural reasons for ascribing it to the large 
 size of the Circle. 230, errors of graduation ; evidence that they do not 
 sensibly affect the indications of the Cambridge Circle. 
 
 IV. Calculations of the Mean Polar Distances of Stars, and of the Geocentric 
 Polar Distances of the Sun, Moon, and Planets. . . PAGES 225 233. 
 
 Art. 231, the apparent P.D. of a Star, and calculation of its mean P.D. 
 at the beginning of the year ; application of corrections for error of adopted 
 
 C. I 
 
XV111 CONTENTS. 
 
 colatitude and discordance of zenith points to the mean of all such mean 
 P.D. of the star obtained in the year; the mean P.D. concluded from 
 observations in several years ; complete formation of a Catalogue of mean 
 places of stars for a given epoch. 
 
 Art. 232, calculated local P. D. of a bisected N. or S. Limb of the Sun, 
 Moon, or a Planet; deduction of the Geocentric P.D. by correcting for 
 Parallax; special formula for calculating the Moon's parallax; formula for 
 any other body. 233 235, inference of geocentric P. D. of centre, and of 
 geocentric measures of diameter, from bisections of opposite limbs ; calcula- 
 tion of the Moon's geocentric diameter from transits of both limbs taken 
 on the meridian, or out of the meridian with an Equatorial; correction 
 thereby of the value assumed in the Naut. Aim. ; confirmation of the tabular 
 value of the Sun's diameter by Transit and Circle observations ; derivation 
 of the semidiameter of Venus from the micrometer measures of 1838 and 
 1839, by an investigation which eliminates the effect of irradiation; 
 consequent correction of the value adopted in the Naut. Aim. 236, correction 
 of the observed P. D. of the Moon for error of position of the plane of 
 collimation. 
 
 Art. 237 (pages 230 233), examples of the complete reduction of Circle 
 observations of a Star, and of the Sun. 
 
 THE TRANSIT CIRCLE . . . . . . PAGES 233256. 
 
 Arts. 238 239, notices respecting the Greenwich and Cambridge Transit 
 Circles; statements relative to the latter (see Note in page 234), and extracts 
 from the account of it given in the M. N. of the R. A. S. 240245, particular 
 descriptions, accompanying Plate in., of the various parts and appendages 
 of the Instrument, the sets of wires, the two collimators, the piers for the 
 mounting, the graduation and microscopes, the clamping and slow-motion 
 apparatus, the illuminations, and generally of the manipulation and 
 arrangements for taking simultaneous observations of R. A. and P. D. 246, 
 special means employed for obtaining corrections of R. A. observations for 
 deviations of the pivots from the cylindrical form. 
 
 Arts. 247 252, calculations for deducing from the observations of the 
 Sun made in the course of a year with a Transit-instrument and Mural 
 Circle, or with a Transit Circle, together with data from Solar Tables, the 
 exact position of the First Point of Aries, the Obliquity of the Ecliptic, and 
 the mean error of the assumed R. A. of the fundamental stars ; conversion 
 of errors in R. A. and P. D. into errors in Longitude and Ecliptic Polar 
 Distance. 253258, derivation of data for exactly measuring time-intervals 
 and determining epochs from meridional observations; definitions of a 
 sidereal day and a mean solar day ; formula for computing the Sun's Mean 
 Longitude at the Greenwich mean noon of January 1, 1800 + t, derived from 
 results obtained by Bessel by comparing meridian observations of the Sun 
 separated by a long interval ; consequent formula for computing the sidereal 
 time of mean noon ; analogous formula for the epoch of Greenwich mean noon 
 of January 1, 1850+ 1, from Leverrier's Solar Tables ; example of calculating 
 by each set of formulae the sidereal time of a given mean noon ; rule for 
 
CONTENTS. XIX 
 
 computing the mean time of transit of the first point of Aries. 259 261, 
 rules for converting the sidereal time of a place into its mean time, and the 
 reverse operation; conversion of local sidereal time into Greenwich mean 
 time, the Longitude of the place being known. 262264, principle of 
 determining terrestrial Longitude by astronomical means; comparison for 
 that purpose of simultaneous sidereal times at two localities by transfer of 
 chronometers ; by galvanic signals ; determination by both methods of the 
 Longitude of the Cambridge Observatory. 265, concluding remarks on the 
 relation of the results of meridian observations to Physical Astronomy. 
 
 Statement of the subjects treated of in the subsequent portion of the Lectures 
 
 PAGES 256257. 
 
 THE EQUATOKIAL . . . _. . . '.-'" PAGES 257 316. 
 
 Art. 266, general statement of the purposes to which an Equatorial 
 is adapted; its uses in taking observations out of the meridian, especially 
 differential measures of B. A. and P.D. 
 
 I. Description of the parts and mode of mounting . . PAGES 258 262. 
 
 Arts. 267 271, the Five-feet Equatorial of the Cambridge Observatory; 
 indication of its parts and mounting by reference to Figure 40; the 
 piers supporting the instrument, place and support of the clock, the upper 
 and lower pivots and Y's, the frame bearing the Telescope, the polar axis, 
 the declination-circle and two microscopes, the declination- axis, the hour- 
 circle and two microscopes, the graduations and slow-motion apparatus of 
 the two circles; the wires and illumination of the field, and the Finder. 
 272 273, construction of a diagonal eye-piece for conveniently observing 
 objects near the Pole. 
 
 II. The Adjustments of an Equatorial .... PAGES 263 276. 
 
 Arts. 274 275, mechanical corrections of instrumental errors ; determi- 
 nations of the errors by observations of stars; the errors arranged in 
 the logical order of the corrections, namely, index error of the declination- 
 circle, error of the angular elevation of the polar axis, the deviation of that 
 axis from the plane of the meridian, the collimation error, the error of 
 position of the declination-axis, and the index error of the hour-circle. 276, 
 adjustments that may be made mechanically without observations of stars, 
 placing the declination-axis at right angles to the polar axis by a swinging 
 level (the construction and mode of application of which are shewn by Fig. 
 44) ; correcting collimation error by a mark; correcting the index error of the 
 hour-circle by the swinging level. 277, test of the stability of the polar axis 
 by a swinging level with its axis of motion parallel to the telescope-axis. 
 278, method of adjusting the position of the polar axis of an Equatorial 
 which has no declination-circle ; formulte for calculating the deviation of the 
 instrumental Pole southward and westward. 279, the other adjustments 
 of an Equatorial for taking observations of K.A. and P.D. the same 
 as those of the Transit and Mural Circle, or Transit-circle. 
 
XX CONTENTS. 
 
 III. Methods of observing with an Equatorial and elimination of instrumental 
 
 errors PAGES 276280. 
 
 Arts. 280 282, setting for observing an object; the process of observing, 
 the same as that with a Transit-circle; correction for Runs of the declina- 
 tion-microscopes; investigation of formulae for correcting, if required, 
 for error of position of the polar axis; advantage of observing in reverse 
 positions of the declination-circle. 283 285, independent determination 
 of celestial places by an Equatorial; not as accurate as the differential 
 method by comparisons with known stars; details of the differential 
 observation when the declination -circle is read, and when the objects are 
 bisected by a micrometer- wire. 
 
 IV. Final calculation of R.A. and P.D. from Equatorial Observations 
 
 PAGES 280288. 
 
 Arts. 286 290, formulae for calculating corrections for refractions 
 in absolute determinations of E. A. and P.D. with an Equatorial; deductions 
 therefrom of formulae applicable to differential observations; description of 
 Eamsden's Refraction-piece, and its use in shortening the calculations 
 of refraction-corrections. 291, the reductions of Equatorial observations of 
 Limbs to observations of Centres, the same as for meridian observations. 
 292 294, investigation of formulae for calculating the corrections of 
 Equatorial observations of R.A. and P.D. for Parallax; special investigation 
 of formulae for calculating exactly the Moon's parallax in R. A. and P. D. ; 
 formulae expressing the ratio of the Moon's geocentric semidiameter to an 
 apparent or measured semidiameter. 
 
 THE NORTHUMBERLAND EQUATORIAL OF THE CAMBRIDGE OBSERVATORY (REPRE- 
 SENTED IN PLATES IV. AND V.) PAGES 288296. 
 
 Arts. 295 300, historical notices respecting the completion of the Instru- 
 ment and erection of the Dome ; descriptions of the Telescope, and the construc- 
 tion of the Polar Frame for supporting it ; the attachment of a Declination- sec- 
 tor for measuring small differences of P. D. ; the Hour-Circle and connected 
 Clock; method of measuring differences of R.A. with the instrument carried 
 by clock-movement. 301 302, more particular account of the parts and 
 appendages of the Telescope and Dome, by reference to Plates IV. and V., the 
 former of which represents the eye-end of the Telescope and Double-wire- 
 micrometer, and generally the observing-apparatus within reach of the 
 observer, and the other a north-and- south section of the Dome and Instru- 
 ment, with the movable chair, the observer in position, and the apparatus 
 for slow-motion in P. D. and hour-angle. 303, determinations of the values 
 of the revolution of the Telescope-micrometer, the Sector microscope-micro- 
 meter, and the interval between the graduations of the Sector. 
 
 The counterpoise mounting of an Equatorial . . . PAGES 296 301. 
 
 Arts. 304307, the instrumental, arrangements, advantages, and disad- 
 vantages of this form of mounting; a mode of counterpoise mounting 
 
CONTENTS. XXI 
 
 (shewn by Figure 50) suitable for out-door sweeping for objects; processes of 
 sweeping when the object is recognisable by its physical aspect, and when it 
 appears as a star. 
 
 Cther observations made with an Equatorial . . . PAGES 301 316. 
 
 Arts. 308 315, observations for parallax of Mars in opposition, at two 
 places or at one; elongations of a satellite from its primary; measures of the 
 Moon's diameter ; eclipses of a star, or a Planet, or the Sun, by the Moon ; 
 calculations of results from the several eclipses. 316 322, measures of 
 relative positions of the components of double, triple and multiple stars ; use 
 of the double- wire micrometer; two examples given of measures by the double- 
 wire micrometer; method of measuring by the double-image micrometer (re- 
 presented in Plate IV., figs. 2 and 3). 323 325, measures of diameters 
 of Planets by the double-wire and double-image micrometers. 326, equa- 
 torial observations for discovering comets and small planets, and for 
 furnishing data for calculating ephemerides after discovery. 
 
 OTHER FIXED INSTRUMENTS OF AN OBSERVATORY . PAGES 316337. 
 THE ALTITUDE AND AZIMUTH INSTKUMENT , . , . PAGES 317 329. 
 
 Arts. 327 328, the principle of the Mounting (shewn by Fig. 53) ; the 
 use of the Altazimuth of the Greenwich Observatory for observations of the 
 Moon. 329 338, methods of observing separately azimuths and zenith 
 distances, and applying instrumental corrections ; dependence of the lunar 
 observations on catalogued places of stars. 339 342, reduction of the 
 observations for comparison with theoretical places; corrections for semi- 
 diameter and parallax ; conversion of tabular errors in azimuth and altitude 
 into errors in R. A. and P. D. 
 
 THE FIXED ZENITH SECTOR . , * . . . PAGES 329 333. 
 
 Arts. 343 346, adaptation of the instrument for measuring small zenith 
 distances; applied at Greenwich in measuring the zenith distances of 
 7 Draconis for determining the constants of aberration and nutation ; 
 Bradley's Zenith Sector ; Pond's ; Airy's Reflex Zenith Tube ; particular 
 description of the process of observing with the last-named instrument. 
 
 THE TBANSIT IN THE PRIME VERTICAL PAGES 333 337. 
 
 Arts. 347 350, Repsold's instrument ; its construction, and the process 
 of using it employed by Professor W. Struve for obtaining the constants of 
 aberration and nutation. 351, Bessel's method of determining the latitude 
 of an Observatory by a Transit in the prime vertical. 
 
 OBSERVATIONS WITH TRANSPORTABLE INSTRUMENTS 
 
 PAGES 337366. 
 
 Arts. 352359, Ramsden's Zenith Sector; Airy's Zenith Sector; the 
 portable Altazimuth; the portable Transit-instrument. 360 369, the 
 Theodolite ; description accompanying Figure 55 ; its adjustments ; use of a 
 
XX11 CONTENTS. 
 
 compass-needle ; application of the Theodolite in large trigonometrical 
 surveys; exemplification in measuring by base-line and triangulation the 
 differences in Longitude and Latitude between the transit-instrument of 
 the Cambridge Observatory and a clock in the Museum building. 370 376, 
 Eamsden's and Troughton's Sextants ; description by reference to Figure 56; 
 the graduation and the adjustments ; the use of the Sextant at sea for 
 obtaining the time of day by equal altitudes of the Sun, the latitude of 
 the place, and measures of Lunar Distances for calculating the longitude ; 
 observations with a sextant on land by means of an artificial horizon. 
 377 379, constructions of, and methods of observing with, Troughton's 
 Reflecting Circle and Borda's Eepeating Circle. 
 
 MISCELLANEOUS ADDITIONAL SUBJECTS . . . PAGES 366 394 
 
 Art. 381, different methods of determining terrestrial longitude ; by 
 Transfer of Chronometers ; by Galvanic Signals, applied in finding the 
 longitudes of Cambridge and of Paris from Greenwich ; by Lunar Distances 
 (formula for calculating to second differences tabular Lunar Distances, with 
 an example of the calculation) ; by observations of occupations of stars and 
 planets by the Moon ; by transit-observations of the Moon and Moon- 
 culminating Stars ; by observations of Eclipses of Jupiter's Satellites ; and 
 by Trigonometrical Surveys ; comparison of the chronometric and geodetic 
 determinations of the longitude of Valentia. 382 385, Solar Parallax ; 
 method of finding it by Transits of Venus across the Sun's disk ; allowance 
 for the black drop ; investigation of formulas for calculation in the cases 
 of Delisle's method and Halley's method ; the results deduced from the 
 processes hitherto employed for determining Solar Parallax. 
 
 Arts. 386 390, Eeticles; differential observations of E.A. and P.D. 
 made with Lacaille's Eeticle; with Valtz's Eeticle; with a Eing Micrometer. 
 Boguslawski's method by transits of two known stars and an unknown object 
 across a single wire. 391 399, the Vernier ; the Transit-reducer, for the 
 time-reduction of a transit-observation to the meridian; the use of Huyghens's 
 eye-piece for observing physical phenomena ; magnifying powers ; the Dy- 
 nameter, described by reference to Figures 61 and 62 ; interpolations to 
 second differences and to fourth differences. 
 
 APPENDIX containing investigations of formulas for ascertaining the magni- 
 fication and brightness of an image produced by any combination of lenses 
 and mirrors . PAGES 395400. 
 
ERKATA. 
 
 Page 84, beginning of Art. 93, for This angle read The bracketed angle. 
 
 ,, 115, line 6 from the bottom, omit, In this calculation the arcs p and P 
 should not be substituted for their sines. See the foot-note in page 
 201. 
 
 118, line 6 from the bottom, for azimuth read level. 
 
 131, line 1, the Note proposed to be introduced was omitted, as being of 
 too theoretical a character. The theory referred to is given in the 
 Philosophical Magazine for April 1872, pp. 289295. 
 
 201, line 3, for z - < read Z - e. 
 
f'B R 
 
 UN i 
 
 GAL 
 
 LECTURES 
 
 OX 
 
 PRACTICAL ASTBfWWY 
 
 ERRATUM. 
 
 Page 356, lias 21, for 52'. 13'. 32",3 N. read 52. 12'. 10",9 N. 
 
 LVVU parts oy oemg immediately concerned witlTtEe means 
 and methods of taking observations of celestial objects, is at the 
 same time related to them as the foundation 1 on which they 
 rest. Spherical Astronomy and Physical Astronomy admit 
 of actual application only so far as they depend on results 
 obtained by observations made with Astronomical Instruments. 
 3. Astronomical observations are of two kinds : (1) obser- 
 vations made at a permanent Observatory with fixed instru- 
 
 1 The work entitled Fundamenta Astronomic, produced in the year 1818 by 
 Bessel, the eminent astronomer of Konigsberg, consists mainly of deductions 
 from observations of Stars and the Sun, made by Bradley at Greenwich, in the 
 years 17501762. 
 
 C. 1 
 
LIBRAE 
 
 UNIT 
 
 s GALII A. 
 
 LECTURES 
 
 PEACTICAL ASTRONOMY. 
 
 INTRODUCTION. 
 
 1. THE science of Astronomy may be regarded as composed 
 of three distinct parts: Practical Astronomy, relating to the con- 
 struction and use of Astronomical Instruments; Spherical As- 
 tronomy, consisting for the most part of deductions, according 
 to the rules of Spherical Trigonometry, from data furnished by 
 Practical Astronomy, apart from any dynamical considerations ; 
 Physical Astronomy, or calculations of the motions of the Sun, 
 Moon, and Planets, depending essentially on the hypothesis of 
 Universal Gravitation. The Lectures will embrace only subjects 
 which may be considered to pertain to the first part. 
 
 2. Practical Astronomy, while it is distinguished from the 
 other two parts by being immediately concerned with the means 
 and methods of taking observations of celestial objects, is at the 
 same time related to them as the foundation 1 on which they 
 rest. Spherical Astronomy and Physical Astronomy admit 
 of actual application only so far as they depend on results 
 obtained by observations made with Astronomical Instruments. 
 
 3. Astronomical observations are of two kinds : (1) obser- 
 vations made at a permanent Observatory with fixed instru- 
 
 1 The work entitled Fundamenta Astronomies, produced in the year 1818 by 
 Bessel, the eminent astronomer of Konigsberg, consists mainly of deductions 
 from observations of Stars and the Sun, made by Bradley at Greenwich, in the 
 years 17501762. 
 
 C. 1 
 
2 PRACTICAL ASTRONOMY. 
 
 ments, for the purposes, generally, of determining and arranging 
 in Catalogues exact positions of fixed stars, for use as points of 
 reference in other observations, or of ascertaining at recorded 
 times the positions of moving bodies in order thereby to test 
 and correct their physical theories ; (2) observations made either 
 with instruments placed in temporary observatories, and with 
 theodolites, for geodetical surveys and mapping the earth's sur- 
 face, or with portable instruments for nautical purposes. The 
 sciences of Geography and Navigation are respectively depen- 
 dent on the two kinds of observation in class (2), conjoined with 
 data furnished by means of observations that come under class 
 (1), such data, for instance, as those contained in the Nautical 
 Almanac. We may also consider to be included in class (2) 
 observations of eclipses and occultations of celestial objects, 
 measures of the diameters of Sun, Moon, and Planets, and mea- 
 sures of the relative positions of the components of double and 
 multiple stars. Although the observations of the first class, as 
 being fundamental, will be chiefly treated of in these Lectures, 
 all those of the other class will receive some share of attention. 
 
 4. For understanding the Lectures some previous know- 
 ledge, of no great amount, is required. I shall suppose the 
 student to be acquainted with the terms by which certain 
 points and great circles of the celestial sphere are designated, 
 as given, for instance, in the introduction to Hymers's Astronomy, 
 or in Chapters I. and II. of Godfray's Treatise on Astronomy. 
 A few propositions of Spherical Trigonometry and Analytical 
 Geometry will have to be employed, and in explaining the 
 modes of using telescopes and microscopes for astronomical 
 observation certain propositions of Geometrical Optics will be 
 taken for granted. The proofs of two optical propositions of 
 practical importance, one relating to the magnifying power of a 
 telescope, and the other to the brightness of the image formed 
 in its field of view, will be given in an Appendix. 
 
 5. It may be asserted generally that what is effected in 
 Practical Astronomy is the measurement of celestial arcs by 
 means of clocks, or graduated instruments, or by micrometers. 
 The measurement of an arc by the intervention of an astro- 
 
INTRODUCTION. 3 
 
 nomical clock depends on the assumed uniformity of the 
 rotation of the earth about its axis. A position on the celestial 
 sphere is assigned by the measurement of two arcs, or spherical 
 co-ordinates, one of which is reckoned from an assumed origin 
 on a great circle of the sphere to the point of intersection of 
 this great circle with another passing through its pole and 
 through the position itself, and the other co-ordinate is the 
 arc of the second great circle intercepted between the position 
 and the aforesaid point of intersection. The co-ordinates are 
 Right Ascension and Declination if the pole be that of the 
 equator, and Longitude and Latitude if it be the pole of 
 the ecliptic. Instead of Declination and Latitude, their com- 
 plements, called respectively North Polar Distance and Ecliptic 
 Polar Distance, are frequently employed as co-ordinates of 
 position. 
 
 6. The principal instruments used in a permanent Obser- 
 vatory for determining exact celestial positions are the Transit 
 Instrument, the Mural Circle, the Transit-Circle (a combination 
 of the Transit Instrument and Mural Circle), the Equatorial, 
 and the Altitude and Azimuth Instrument. The parts and 
 appendages of these instruments are appropriate to the pur- 
 poses they are designed to answer. The Transit Instrument 
 measures Right Ascension by intervals of time, and serves to 
 determine epochs of time ; the Mural Circle measures by its 
 graduation Declination, or North Polar Distance ; the Transit 
 Circle is capable of giving at the same time both Right 
 Ascension and Declination, and of effecting all that is done 
 by the separate use of the Transit Instrument and the Mural 
 Circle. The observations with these three instruments are 
 made in the meridian of the place of observation. The Equa- 
 torial is adapted to ascertain simultaneously both the Right 
 Ascension and the Declination of an object out of the meri- 
 dian. The same assertion is true of the Alt-azimuth Instru- 
 ment, the Altitude and Azimuth immediately observed 
 being converted by calculation into Right Ascension and 
 Declination. The Longitude and Latitude of an object are not 
 directly observed, but are deduced by calculating according 
 
 12 
 
4 PRACTICAL ASTRONOMY. 
 
 to rules of Spherical Trigonometry from its Right Ascension 
 and Declination obtained by means of instrumental observa- 
 tion. 
 
 7. The use of the transit circle is preferable to observing 
 separately with a transit instrument and a mural circle, because 
 the latter method requires two observers, and circumstances 
 may occur under which only one of the co-ordinates of position 
 is obtained, whereas with the transit circle it is generally possible 
 for a single observer to obtain both co-ordinates. But although 
 the transit circle has this advantage in practice, I propose to treat 
 of the transit instrument and mural circle, at first, as separate 
 instruments, and afterwards to consider them as combined to 
 form the transit circle. This course is adopted for the sake of 
 simplicity and distinctness in exhibiting the principles of the 
 Adjustments and Corrections which are required for ensuring 
 accuracy in taking meridional observations. 
 
 8. It is not necessary to give in detail the description of an 
 observatory to those who have the opportunity any day of 
 inspecting the Cambridge Observatory. This institution, 
 although not to be classed with national observatories like that 
 at Greenwich, fulfils the main requisites of an observatory as 
 respects site, rooms appropriated to instruments and the work 
 of calculation, and residences for observers, and in being furnished 
 with powerful instruments of the best quality. The situation 
 of an observatory ought to be such as to present no obstacle to 
 pointing a telescope to a celestial object whatever be its 
 position in the heavens 1 . It is, however, very rarely possible to 
 observe satisfactorily an object at an altitude less than two 
 
 1 The Cambridge Observatory was erected in 1822-4, one-third of the cost 
 being obtained by subscriptions, and the rest granted from the University 
 Chest. The chief promoters of the undertaking were Professor Woodhouse, 
 Mr Catton of St John's College, and Mr Peacock, afterwards Lowndean Pro- 
 fessor. The first Director, Professor Woodhouse, was succeeded in 1828 by 
 Mr Airy, now Astronomer Royal, whom I succeeded in 1836. The present 
 Director, Professor Adams, was appointed in 1861. The site of the Observatory 
 sufficiently fulfils the condition of commanding an unobstructed view of the 
 heavens ; but occasionally I found it necessary to cut off the tops of trees in the 
 surrounding plantations to secure observations of certain comets. 
 
INTRODUCTION. 5 
 
 degrees, on account of the disturbing effects of refraction, and 
 the prevalence of mists, near the horizon. 
 
 9. When the Cambridge Observatory was erected it was 
 thought necessary to arrange that the telescope of the transit 
 instrument should be capable of pointing to a fixed meridian 
 mark, and accordingly the position of the instrument was 
 determined by satisfying this condition with respect to a mark 
 on the tower of Grantch ester church, situated to the south of 
 the observatory at the distance of about 2J miles. This was 
 done for a first approximation by bisecting the spire of the tower 
 by the middle wire of the telescope of a small transit instru- 
 ment, and comparing the Sun's transit across the wire with trite 
 time of apparent noon brought up by the chronometer used for 
 taking the transit from Mr Catton's observatory at St John's Col- 
 lege. Nearer approximations were made by transit observations 
 of high and low stars with the same small instrument 1 . Probably 
 the simplest method of determining the position of the piers of 
 a transit instrument, the telescope of which is required ta point 
 to a mark in a given position, would be to employ a theodolite, 
 placed a little to the north or south of the mark, as an alt- 
 azimuth instrument for finding by the method of equal altitudes 
 of the Sun or a star the direction of the meridian passing 
 through the mark. Then by pointing the telescope in this 
 direction, and depressing it to the horizon, a position might be 
 fixed upon, such that if the piers be placed equally distant 
 from it and at a distance from each other suitable for 
 mounting the instrument, and if their azimuths be adjusted 
 till the image of the mark is seen at the middle of the field 
 of view of the mounted telescope, the required conditions will 
 be satisfied. 
 
 In the actual state of Practical Astronomy meridian marks 
 for adjusting instruments in the meridian can be dispensed 
 with. There are, however, certain purposes (which will be 
 subsequently indicated), for which they may occasionally be 
 
 1 This statement is derived from the " account of the transit instrument 
 made by Mr Dollond, and lately put up at the Cambridge Observatory," given 
 by Professor Woodhouse in the Philosophical Transactions for 1825 (p. 419). 
 
6 PRACTICAL ASTRONOMY. 
 
 made use of. The use of the Greenwich marks placed by Pond 
 at Blackwall and at Chingford was discontinued by the present 
 Astronomer Royal in 1836 1 . 
 
 10. The account given in Art. 9 of methods of directing 
 the telescope of a transit instrument to a meridian mark was 
 scarcely logical, inasmuch as these methods depend on employ- 
 ing astronomical instruments the descriptions of which, and of 
 the modes of observing with them, properly belong to a 
 subsequent stage of the Lectures. If it were required to draw, 
 ab initio, a meridian line through a given spot, for the purpose 
 of marking out positions for the piers of a transit instrument, 
 this might be done by setting up a pole vertically at the spot, 
 drawing around it several horizontal circles, and noting the 
 points of coincidence of the extremity of the shadow of the pole 
 with these circles before and after noon. Then if the two 
 points on each circle be joined by a chord, the mean of the 
 directions of the middle points of the chords from the pole will 
 be approximately the direction of the meridian line. The 
 experiment would be best made about midsummer, when the 
 Sun's diurnal path is high, and the change of Declination small. 
 This process would probably be sufficiently exact for a first 
 determination of the azimuths of the piers ; and also for fixing 
 upon the position of a near meridian mark which, by being put 
 in coincidence with the focus of an achromatic compound lens, 
 might by the intervention of this lens be distinctly seen in the 
 transit-telescope. A meridian mark might thus be set up 
 without having recourse to an astronomical instrument 2 . 
 
 1 These marks are both on the north side of the Observatory, that at 
 Blackwall at a distance of about two miles, and that at Chiugford "on the 
 Essex hills " about eleven miles. The latter on account of its distance could 
 rarely be seen, and the sight of the other was often prevented by ships on the 
 Thames. For a south mark Pond had recourse to a small transit -telescope 
 used as a Collimator on the principle, now very generally adopted, of collimating 
 with two telescopes pointed towards each other. What is said in Woodhouse's 
 Astronomy (Vol. i. p. 8-1) respecting the Greenwich meridian marks is altogether 
 inaccurate. (See Pond's Introduction to the Greenwich Observations of 1834.) 
 
 2 Admiral Smyth joined to this method of viewing a meridian mark apparatus 
 for adjusting the position of the mark by screws. After adjusting it to the 
 
INTRODUCTION. 7 
 
 11. We may now proceed to the consideration of the 
 optical conditions which must be satisfied by the telescope and 
 eye-pieces of an astronomical instrument which is to be 
 employed for the accurate measurement of celestial arcs. The 
 remarks will apply, at first, only to refracting telescopes. The 
 object-glass of a Refractor is composed of two lenses, the outer 
 one a double convex lens, and the inner one a double concave 
 lens. The adjacent surfaces of the two lenses have exactly 
 equal and opposite curvatures and generally are in actual 
 contact, and the curvature of the second surface of the concave 
 lens is small compared to that of the first surface. The outer 
 lens is of crown-glass, the other of flint-glass. The forms and 
 refractive powers of the glasses are such as satisfy, as far as 
 may be, the conditions of correcting both spherical aberration 
 and colour. It is not possible with two lenses completely to 
 correct colour; but it is of importance to remark that the 
 residual colour is least injurious when it has a purplish tint, 
 being composed of rays lying towards the violet end of the 
 spectrum, because the light of these is much less intense than 
 that of those towards the red end. 
 
 12. We have next to define exactly the point which is 
 -Called the optical centre of the object-glass, on the existence and 
 
 position of which the purposes to which the astronomical 
 telescope is applied essentially depend. First, it must be 
 understood that the axis of a lens is the straight line which, 
 passing through its middle point, cuts the two surfaces at right 
 angles, and that the axis of a system of lenses is the straight 
 line which similarly cuts at right angles the two surfaces of 
 each of the components of the system. Now by Geometrical 
 Optics it is shewn that a beam of light diverging from a point 
 and incident on the whole of the surface of any lens (i. e. a 
 centrical beam), on emerging from the lens very approximately 
 converges towards, or diverges from, a point so situated that the 
 following rule holds good : The straight line joining this point 
 
 meridian in the first instance, and fastening it in its position, to prevent 
 disturbance the screws were withdrawn. (Smyth's Celestial Cycle, Vol. i., 
 p. 331.) 
 
8 PRACTICAL ASTRONOMY. 
 
 and the point of original divergence (which may be called the 
 axis of the beam) cuts the axis of the lens at a certain point 
 which is very approximately the same for all small inclinations 
 of the former axis to the other. This point is the optical centre 
 of the lens. 
 
 13. If instead of being divergent the incident centrical 
 beam had been convergent, its axis would still have passed 
 through an optical centre. Hence in the case of the combination 
 of two lenses in juxtaposition, such as those which form the 
 object-glass of an astronomical telescope, there will be an optical 
 centre of each of the lenses, the centrical beam emerging from 
 the first lens being an incident centrical beam with respect to 
 the other. But these centres will be points on the common axis 
 of the lenses so near each other, that for small inclinations we may 
 suppose without sensible error that the axis of the beam incident 
 on the first lens, and that of the beam emergent from the second, 
 are coincident in direction, and that consequently the compound 
 glass has an invariable optical centre. This supposition has 
 not been found by experience to be in any sensible degree 
 erroneous. 
 
 14 Analogous considerations are applicable when a centri- 
 cal beam is incident on a spherical mirror, whether convex or 
 concave, the contour of which is a circle. For a mirror of this 
 form, the centre of its curvature takes the place of the optical 
 centre defined in Art. 12, and the straight line joining the focus 
 of an incident beam and the focus of the same beam after 
 reflection passes quam proxime through that centre of 
 curvature* 
 
 15. From what has been said (Arts. 12 and 13) respecting 
 the incidence of a diverging or converging beam on a lens, it is 
 evident that if the beam be conceived to be composed of 
 indefinitely small pencils of light and we select one of them, the 
 straight line joining the focus, actual or virtual, of this pencil 
 before incidence, and the focus, actual or virtual, of the same 
 pencil after emergence, will equally pass through the optical 
 centre of the lens. This is true also for a compound lens like 
 the object-glass of an astronomical telescope. 
 
INTRODUCTION. 
 
 16. Similarly in the case of a spherical mirror, the straight 
 line joining the focus of an incident pencil and the focus of the 
 same pencil after reflection passes through the centre of the 
 spherical surface. 
 
 17. Generally the two lenses of an object-glass are joined 
 together, and the compound glass is put in position in the tube 
 of the telescope, by the optician once for all, and the observer 
 has not the means of altering the adjustments. Professor Airy, 
 who, as my predecessor at the Cambridge Observatory, designed 
 and superintended the mounting of the telescope of the 
 Northumberland Equatorial, provided apparatus whereby the 
 observer could himself execute the adjustments of the object- 
 glass. This I succeeded in doing in the first instance by myself, 
 as mentioned in p. 34 of the Astronomer Royal's "Account of 
 the Northumberland Equatoreal and Dome," which forms 
 Appendix II. of the Volume of Cambridge Observations for the 
 year 1843 (Vol. XV). Subsequently (in 1848), having occasion 
 to take out the two glasses for the purpose of cleaning them 
 (which the above-mentioned apparatus allowed of doing), I 
 called in the assistance of Mr Simms for replacing them, and 
 had the opportunity of witnessing, and taking part in, the 
 whole operation. Considering that an account of the process 
 might be generally useful to practical astronomers, as indicating 
 means of testing, or judging of, the performance of their 
 telescopes, I thought it might be appropriately given in detail 
 in this Introduction. 
 
 18. The mechanism of the apparatus was arranged so as 
 to enable the observer to execute three movements by the 
 intervention of iron rods extending from the object-glass to the 
 eye-end of the telescope, giving the means, at the same time 
 that the movements were produced, of seeing the effects they 
 had on the image of a bright star. These movements were (1) 
 a rotation of the crown-glass lens about the axis of the telescope 
 for the purpose of observing the effects of different relative 
 positions of the surfaces of the two lenses, and choosing that 
 which appeared to give the best effect as to definition ; (2) a 
 tilting of the whole object-glass in any required direction, by 
 
10 PRACTICAL ASTRONOMY. 
 
 turning three screws tapped in projections from the object-glass 
 cell and holding it in position; (3) motions of translation, in 
 two directions at right angles to each other, of the crown-glass 
 lens relatively to the flint-glass lens for the purpose of centering 
 the lenses. The mechanical means of producing these move- 
 ments being provided, the process of adjusting the object-glass 
 is such as follows. 
 
 19. First the image of a bright star (as Arcturus) is looked 
 at with the eye-piece well adjusted to focus in order to see 
 whether there is a determination of the light of the image 
 towards one direction. If so, the appearance (technically 
 called a tail) has to be corrected tentatively by tilting the 
 object-glass. The rule is, that if the tail is towards the left 
 hand, the object-glass requires to be thrown off on the right 
 hand. The tail being thus corrected, the image is next looked 
 at much out of focus by pushing in, or drawing out, the eye- 
 piece a considerable distance from its normal position. There 
 will then be seen in the midst of a confused coloured image 
 a line of light of the nature of a caustic formed by the inter- 
 section of rays of different refrangibilities. This caustic is 
 required to be an exact circle, and if it be of any other form, it 
 must be brought, as nearly as may be possible by the judgment 
 of the eye, to the circular form by the rectangular transverse 
 movements of the crown-lens upon the flint-lens. After this 
 has been done and the image is looked at in focus, there should 
 be a well-defined point from which the light of the image is 
 radiated in all directions, or about which concentric bright rings 
 are formed. The rings are more conspicuous the smaller the 
 telescope, and in one so large as the Northumberland telescope 
 (of llf inches aperture) they are scarcely noticeable, while the 
 radiations in the case of bright stars are very intense. These 
 spurious optical effects, the theory of which does not appear to 
 be well understood, do not sensibly interfere with fixing upon 
 the central point of the image for bisection. It is advisable, 
 after proceeding so far, to turn again the crown-glass about the 
 axis of the telescope in order to choose the relative position of 
 the glasses most favourable for good definition, and then with the 
 
INTRODUCTION. 11 
 
 glasses in the selected position to repeat the processes of tilting 
 and centering for a final adjustment. This, if I remember 
 rightly, was the course we adopted in adjusting the object-glass 
 of the Northumberland telescope. Very little optical effect was 
 discernible by causing the crown-glass to revolve relatively to 
 the other. 
 
 20. The principal eye-pieces of the astronomical telescope 
 are Huyghens's, or the Negative eye-piece, and Ramsden's, or 
 the Positive eye-piece. Occasionally use is made of an Erecting 
 eye-piece, and a Diagonal eye piece. I shall not consider it 
 necessary to give in these Lectures mathematical details 
 respecting the forms and arrangement of the lenses of eye- 
 pieces, this part of the subject being fully treated of in standard 
 works on Geometrical Optics 1 . It will suffice for my purpose 
 to make only the following remarks. 
 
 21. Huyghens's eye-piece is sometimes called " the Achro- 
 matic eye-piece/' because the condition of achromatism is very 
 approximately fulfilled when, as is practically the case, the 
 directions of the pencils are nearly parallel to the common axis 
 of the two lenses. This eye-piece is, consequently, suitable for 
 examining and recording the physical features of celestial 
 objects, and, in fact, is generally employed for that purpose; 
 but because the image of the object is formed between the 
 lenses, it is not one that can be adapted to taking observations 
 which require measures of time or space. For such observations 
 the observer must be able to see distinctly, in coincidence with 
 the optical image of the celestial object, fine wires, or threads, 
 stretched across a stop fixed in the telescope-tube and limiting 
 the field of view, and also the edges of the stop which form the 
 boundary of the field. This condition can be satisfied by a 
 Ramsden's eye-piece, the image formed by the telescope's object- 
 glass being situated close to the inner glass of the eye-piece on 
 the side towards the object-glass, and being distinctly visible in 
 
 1 The requisite mathematical investigations relating to lenses and mirrors 
 are well given in Coddington's Treatise on Optics, Part I., and the parts and 
 uses of various kinds of telescopes and eye-pieces are fully described, with 
 accompanying figures, in Chapter ix., Part II., of the same work. 
 
12 PRACTICAL ASTRONOMY. 
 
 that position by adjusting the eye-piece to focus. Accordingly 
 this arrangement allows of seeing the image of the celestial 
 object in coincidence with wires or diaphragms, at the geo- 
 metrical focus of the object-glass, adapted for taking the desired 
 astronomical measures. Ramsden's eye-piece is not achromatic 
 like Huyghens's ; if, however, care be taken to look at the 
 image or wire as nearly as may be in the direction of the 
 optical axis, all injurious appearance of colour due to the eye- 
 piece is got rid of. To enable the observer to do this the eye- 
 piece is carried by a slide movable in grooves laterally by 
 hand, and can thus be placed exactly opposite the wire or 
 image looked at. By this contrivance the extent of the field of 
 view is in effect enlarged, and the image can be seen distinctly 
 during the whole of its transit across the field defined by the 
 stop. 
 
 22. The erecting eye-piece, consisting of four glasses, is 
 generally used with telescopes intended chiefly for looking at 
 terrestrial objects. The way in which it has been modified for 
 converting a telescope into a double-image micrometer will be 
 taken notice of in the course of these Lectures. 
 
 23. A diagonal eye-piece is furnished with a plane metallic 
 reflector inclined at an angle of 45 to the optical axis of the 
 telescope for the purpose of changing the direction of vision 
 through 90. The two lenses of this eye-piece are placed at the 
 ends of a broken tube consisting of two straight parts at right 
 angles to each other, and the reflector is at the angle of their 
 junction. By these means the observer is able to get sight of an 
 object under circumstances in which the parts of the instrument 
 may not allow of his looking directly along the axis of the tele- 
 scope. An eye-piece of this kind is mostly required for small 
 portable instruments. In some cases of the mounting of an 
 equatorial it is needed for looking at the Pole star. 
 
 24. I propose to conclude this Introduction with giving 
 some account of works on Practical Astronomy to which 
 the student might have recourse for information on the 
 subject as a whole, or on particular parts of it. It is hoped 
 that the list, although it has no pretension to be considered 
 
INTRODUCTION. 13 
 
 complete, may, with the accompanying remarks, be found to be 
 useful. 
 
 (1) Woodhouse's Treatise on Astronomy Theoretical and 
 Practical. A new Edition. Cambridge : Vol. I. Part I. (1821) 
 and Part II. (1824). Part I. treats of the Fixed Stars, and Part 
 II. gives the Theories of the Sun, Moon, and Planets, as 
 founded 0:1 Kepler's Laws. [Vol. II., published in 1818, contains 
 exclusively Physical Astronomy.] The principles involved in 
 making Astronomical Observations, and in reducing them for 
 subsequent use, are clearly stated in this work; but the accounts 
 of Astronomical Instruments, and of the methods of observing, 
 fall much behind the improved state of Practical Astronomy at 
 the present time. In Chapter xxxvni. (p. 754) there is given 
 an investigation of formula? for calculating the Sun's Parallax 
 from observations, according to Halley's method, of a transit 
 of Venus over the Sun's disk, with application to that of 
 1769. 
 
 (2) Pearson's Introduction to Practical Astronomy. Lon- 
 don : printed for the Author. Vol. I. (1824) and Vol. II. (1829), 
 of quarto size, with a volume of Plates. The first volume con- 
 sists of Tables computed for facilitating the Reduction of Celestial 
 Observations, with explanations of their construction and use. 
 The second volume contains "descriptions of the various Instru- 
 ments that have ' been usefully employed in determining the 
 places of the heavenly bodies, with an account of the methods 
 of adjusting them and using them." The descriptions of Instru- 
 ments, fixed, portable, and auxiliary, are illustrated by nume- 
 rous Plates, and, regard being had to the date of the publica- 
 tion, are very complete. 
 
 (3) Coddington's System of Optics. Cambridge : Part I. 
 (1829) and Part II. (1830). The first Part is entitled "A 
 Treatise on the Reflexion and Refraction of Light," and, as the 
 author acknowledges, "contains the substance of Professor Airy's 
 Papers on the Achromatism and the Spherical Aberration of 
 Eye-pieces, published in the Cambridge Philosophical Transac- 
 tions." Part II., which consists for the most part of applications 
 of mathematical results obtained in the first part, and is illus- 
 
14 PRACTICAL ASTRONOMY. 
 
 trated with excellent engravings, gives much information useful 
 to the practical astronomer. (See foot-note to Art. 20.) 
 
 (4) " Six Lectures on Astronomy, delivered at the Temper- 
 ance Hall, Ipswich, in the month of March, 1848," by George 
 Biddell Airy, Astronomer Royal. London: 1849,2nd Edition. 
 These Lectures, remarkable as respects the method and clear- 
 ness of the explanations, seem to prove the possibility of 
 conveying a large amount of astronomical information, practical, 
 theoretical, and physical, to an audience not possessing the 
 advantage of a mathematical education. 
 
 (5) "Plane Astronomy, including explanations of Celestial 
 Phenomena and descriptions of the principal Astronomical 
 Instruments," Part I. By the Rev. A. R. Grant, Fellow and 
 Assistant Tutor of Trinity College (Cambridge : 1850). This 
 work, of which only Part I. was published, is mentioned here 
 as containing a good account of the striding spirit-level, and of 
 the principles on which its indications of level depend. 
 
 (6) "History of Physical Astronomy from the earliest ages 
 to the middle of the nineteenth century." By Robert Grant, 
 F.R.A.S. London: 1852. Although the History is devoted 
 mainly to Physical Astronomy, some portions of the work, 
 especially Chapters xvm XXL, consist of historical notices 
 respecting successive advances made in the construction and 
 use of Astronomical Instruments, and the additions to Astrono- 
 mical Science thereby effected. These notices, which I shall take 
 occasion to refer to in the course of the Lectures, are particu- 
 larly interesting as indicating how the foundation of Physical 
 Astronomy has been laid in the results obtained by Observational 
 Astronomy, and by what means, in proportion as theoretical 
 calculations made on the hypothesis of universal gravitation 
 called for observations in greater number and of greater ac- 
 curacy, the demand was met by improved instruments and 
 modes of observing, and by a diligent and intelligent use of 
 the apparatus provided by the skill of the instrument-maker 
 for taking exact observations. 
 
 (7) "An Introduction to Practical Astronomy, with a col- 
 lection of Astronomical Tables." By Elias Loomis, LL.D., Pro- 
 
' : / *'C 
 
 INTRODUCTION. 15 
 
 fessor of Mathematics and Natural Philosophy in the University 
 of the city of New York. New York : 1855. In Chapters X. 
 and XI. of this treatise are given at considerable length inves- 
 tigations of rules for calculating Eclipses of the Moon, Eclipses 
 of the Sun, and Occultations of Stars by the Moon, accompanied 
 by examples of the application of the rules. In an example of 
 the calculation of the beginning and end of a Solar Eclipse, and 
 in two examples of the calculation of the time of an occultation, 
 Bessel's method of computation is adopted, and the arithmetical 
 details are fully exhibited. 
 
 (8) Briinnow's Spherical Astronomy, translated by the Rev. 
 Robert Main, M.A., F.R.S., Radcliffe Observer at Oxford. Part 
 I., including the Chapters on Parallax, Refraction, Aberration, 
 Precession and Nutation. Cambridge : 1860. Mr Main did 
 not produce a translation of the remainder of the work. The 
 whole, as translated into English by the author himself, will 
 presently be remarked upon. 
 
 (9) "Practical and Spherical Astronomy, for the use chiefly 
 of Students in the Universities." By the Rev. Robert Main. 
 Cambridge : 1863. Chapter II. of this publication treats of 
 Instruments and corrections of instrumental errors, and having 
 been composed by an author who has had large experience in 
 Practical Astronomy, may be depended upon as being free from 
 inaccuracies and defects which would seem to be inevitable 
 when the writer has had no practice in the use of astronomical 
 instruments. The major part of the work is devoted to Spheri- 
 cal Astronomy and deductions from Kepler's Laws of Planetary 
 Motion. The department of Practical Astronomy might, as it 
 seemed to me, as respects both the actual state of the science, and 
 the reading of Cambridge mathematical students, receive with 
 advantage fuller and more extensive treatment. 
 
 (10) "An Introduction to Plane Astronomy, for the use of 
 Colleges and Schools." By Philip Thomas Main, M.A., Fellow 
 of St John's College, Cambridge. Cambridge: 1865. This 
 elementary work, which, as to arrangement, follows the above- 
 mentioned treatise on Practical and Spherical Astronomy by Mr 
 Main's father, seems well adapted to answer the purpose, intended 
 
1C PRACTICAL ASTKONOMY. 
 
 by the author, "of a text-book for that part of the subject which 
 is required in the first three days of the Examination for 
 Mathematical Honours." 
 
 (11) Spherical Astronomy. By F. Briinnow, Ph. Dr., trans- 
 lated by the author from the second German edition. London, 
 Asher and Co., 1865. The first edition was the one translated 
 in part by Mr Main, as mentioned above (No. 8). Previous to 
 Dr Briinnow's translation of the second edition, the work had 
 been "considerably enlarged." As its title implies, it treats 
 of Spherical Astronomy apart from physical considerations, and 
 makes no reference to Kepler's Laws. In the seventh and last 
 section, Practical Astronomy is included under the head of 
 " Theory of the Astronomical Instruments." Although, as being 
 a German work, it differs in composition and details from 
 English treatises on the same subjects, it contains much that 
 the English astronomer might derive advantage from. 
 
 (12) "A Treatise on Astronomy for the use of Colleges and 
 Schools." By Hugh Godfray, M.A., of St John's College. Cam- 
 bridge and London : 1866. The purpose of this publication is 
 of the same kind as that already mentioned relatively to No. 
 10; but as it takes in a wider range of subjects, "embracing 
 all those branches of astronomy which have, from time to time, 
 been recommended by the Board of Mathematical Studies," it 
 is suitable for the reading of Candidates for Mathematical 
 Honours who are preparing for the Examination in the last 
 five days. The descriptions in Chapter II. of the construction 
 of clocks for astronomical use, and the account given at the 
 end of Chapter VI. of the principle of Rochon's Double Image 
 Crystal Micrometer, may be mentioned as conveying informa- 
 tion not usually met with in elementary treatises on Astronomy. 
 
 (13) "A Hand-book of Descriptive and Practical Astro- 
 nomy," London: 1861, and "Descriptive Astronomy," Oxford, 
 at the Clarendon Press: 1867. By George F. Chambers, F.RA.S. 
 These two works, the contents of which are partly identical, the 
 second being mainly an expansion of the first, convey a very 
 large amount of miscellaneous astronomical information. In 
 the first, the subject of Book VIII. is "Astronomical Instru- 
 
INTRODUCTION. 17 
 
 ments," and in the other the subject of Book VII., consisting 
 of nine Chapters, is "Practical Astronomy." These two volumes 
 are remarkable for containing a very large number of carefully 
 executed diagrams, and illustrative representations of celestial 
 objects, amounting in the first to 160, and in the later volume 
 to 224. 
 
 (14) "A Manual of Spherical and Practical Astronomy : 
 embracing the general problems of Spherical Astronomy, the 
 special applications to Nautical Astronomy, and the theory and 
 use of fixed and portable Astronomical Instruments; with an 
 appendix on the method of least squares." By William Chau- 
 venet, Professor of Mathematics and Astronomy in Washington 
 University, Saint Louis. Volumes I. and II. Philadelphia and 
 London: 1863. The first volume contains "Spherical Astro- 
 nomy;" the other "the theory and use of Astronomical Instru- 
 ments" and "the method of least squares." The contents of 
 the two volumes form a complete Treatise on Observational 
 Astronomy, properly making no mention of Kepler's Laws. 
 The American method of recording transits by the intervention 
 of a galvanic circuit is described in Arts. 71 77 of Vol. II. 
 The appendix to Vol. II., consisting wholly of a very elaborate 
 discussion of the method of least squares, occupies not less than 
 ninety pages. 
 
 (15) "A Treatise on the Principal Mathematical Instru- 
 ments employed in Surveying, Levelling, and Astronomy, 
 explaining their construction, adjustments, and use; with an 
 Appendix and Tables." By Frederick W. Simms, F.R.A.S., 
 F.G.S., Civil Engineer. 8th edition. Troughton and Simms, 
 138 Fleet Street, London: 1850. This publication is entirely 
 of a practical character, giving very useful information respecting 
 the construction and manipulation of the various instruments 
 employed in surveying and levelling. It does not, however, 
 describe the form of theodolite which, as constructed by Mr 
 James Simms of Fleet Street, is now in general use for Trigono- 
 metrical Surveys. I shall have occasion to speak of this 
 serviceable instrument in the course of the Lectures. The 
 instruments referred to in the title of the work as being 
 
 c. 2 
 
18 PRACTICAL ASTRONOMY. 
 
 employed in Astronomy are those of a portable kind, as the 
 sextant, the reflecting circle, the portable transit, and the 
 portable altitude and azimuth instrument. The subject of the 
 Appendix is, Protracting and Plotting, &c., in surveys, and the 
 instruments employed for such purposes. 
 
 (16) In addition to the sources of information indicated 
 in the foregoing list, there are some others which may properly 
 be adverted to here. Herschel's " Outlines of Astronomy" is 
 principally devoted to Physical Astronomy ; but Chapters I., II., 
 III., of this work, as well as " The Introduction to Astronomy" 
 by the same author, forming a Volume of Lardner's Cabinet 
 Cyclopaedia (1833), treat of Practical Astronomy, and may be 
 read with advantage. The Articles " Mural Circle " and 
 "Equatoreal" in the Penny Encyclopaedia, written by Richaid 
 Sheepshanks, an experienced astronomer, although they are brief, 
 are intelligible and much to the purpose, especially that on the 
 Equatorial. The Introductions to the Cambridge Observations 
 of 1836 and 1837 describe in detail the methods of observing 
 with the meridian instruments of the Cambridge Observatory, 
 and the Introduction to the Volume for 1838 contains an 
 account of the adjustments of the Northumberland Equatorial 
 and the uses of its mechanical appliances. Information deriva- 
 ble from all the different sources that have now been mentioned 
 would be amply sufficient for the Cambridge student, as far as 
 regards the educational effect of a knowledge of this department 
 of science ; but if he desires to pursue the subject beyond the 
 requirements of a university examination, and to acquire a 
 thorough acquaintance with the successive modern improve- 
 ments in the construction of instruments, the modes of observing, 
 and the reductions of the observations, he cannot do better than 
 have recourse to the Volumes of Greenwich Observations 
 published in the interval, beginning at 1811, during which the 
 Royal Observatory has been under the direction of Pond and 
 his immediate successor, the present Astronomer Royal. Of 
 the principal instruments in actual use at Greenwich, descrip- 
 tions, copiously illustrated by plates, are given in several of the 
 Annual Volumes, as here indicated : 
 
INTRODUCTION. 19 
 
 The Altitude and Azimuth Instrument is described in pp. iv 
 xvi of the Introduction to the Volume of Observations of 
 1847, the plates being at the end of the Introduction, and a 
 perspective view of the instrument (taken from the " Illustrated 
 News" for the week ending Oct. 2, 1847, p. 221) forming a 
 frontispiece to the Volume. 
 
 The descriptions of the Transit Circle, the Reflex Zenith Tube, 
 and the Great Equatorial, with the plates, constitute respectively 
 Appendix I. to the Volume for 1852, Appendix I. to that for 
 1854, and Appendix III. to the one for 1868. 
 
 A history and description, with figures, of the Water 
 Telescope (which there will subsequently be occasion to speak 
 of) are given in the Introduction to the Volume for 1871, pp. 
 cxix cxxxii. This instrument, mounted for a special purpose, 
 is no longer required for use \ 
 
 From the consideration I have given to the contents of the 
 elementary works on Practical Astronomy used as text-books, I 
 have come to the conclusion that, while other parts of the 
 subject have been satisfactorily handled, there is generally a 
 want of precision and completeness as respects the principle and 
 uses of the colli mating eye-piece, and the adjustments and 
 calculations relating to equatorial observations. These faults I 
 shall endeavour to rectify by explanations given in the course 
 of the Lectures. 
 
 25. PRACTICAL ASTRONOMY may be regarded as including 
 whatever is directly effected by means of INSTRUMENTS adapted 
 for observations of celestial objects. Lectures on this subject 
 ought therefore to give an account (1) of the instruments, as to 
 the modes of mounting them, their component parts, and the 
 purposes for which the different parts were designed ; (2) of the 
 adjustments that are required to be either actually or virtually 
 
 1 A plan of the buildings and grounds of the Greenwich Observatory (1863, 
 August), with explanation and history, forms Appendix II. to the Observations 
 of 1862. 
 
 22 
 
20 PRACTICAL ASTRONOMY. 
 
 effected before the instruments are used for taking observations ; 
 (3) of the methods of exact observation ; (4) of the calculations 
 that are proper for making the immediate results of observation 
 available for the intended purposes. The Lectures on each 
 principal instrument will be given, both as to order and matter, 
 in conformity with these four divisions. 
 
 THE TRANSIT INSTRUMENT. 
 
 26. A Transit Instrument is specially employed for record- 
 ing the times, as shewn by a regulated Astronomical Clock, of 
 the passage of a celestial object across the meridian of a given 
 spot, and its construction and mounting are arranged so as to 
 be suitable for fulfilling this purpose. 
 
 I. DESCRIPTION OF THE PARTS AND MOUNTING OF THE 
 INSTRUMENT. 
 
 27. The telescope is furnished with an achromatic 1 object- 
 glass, and a set of Ramsden's eye-pieces, and attached to the 
 middle part of its tube there is a transverse axis, the ends of 
 which are formed into cylindrical pivots. When the telescope 
 is mounted these pivots rest in two grooves, usually called Y's, 
 formed by two intersecting planes, and inserted, with their 
 angles vertically downwards, into the two piers which support 
 the instrument. In the model represented by Figure 1, a, a 
 are the piers, 5, b the two arms of the transverse axis, c, c the 
 positions of the pivots resting in their Y's, d the object-end and 
 e the eye-end of the telescope tube, and/, /are the positions of 
 two microscopes pointing towards the ends of the pivots, to be 
 
 1 In many of Dollond's telescopes, the achromatic object-glass was composed 
 of three lenses. The improvement as to achromatism effected by the third lens 
 being found not to compensate for the additional loss of light, object-glasses of 
 two lenses are now universally adopted. The object-glass of the Dollond tele- 
 scope of the Cambridge Transit Instrument consists of two lenses ; that of the 
 equatorially mounted 46-in. Dollond had originally three lenses, but in conse- 
 quence of two of the lenses having been accidentally broken, it was replaced by 
 a two-lens object-glass. 
 
THE TRANSIT INSTRUMENT. 
 
 21 
 
 used for a purpose which will be indicated under the head of 
 Division III. The piers are usually of massive stone, resting on 
 a firm foundation in the ground, and sometimes on a common 
 
 Fig. 1. 
 
 slab. They are found to be steadier when the sub-stratum is 
 sand than when it consists of clay or rock. 
 
 28. Supposing the pivots, as first turned, to be exactly 
 cylindrical, whatever material they are made of the friction 
 between them and the Y's must in some degree produce con- 
 tinual deviation from that form, the use of the instrument being 
 such that the surfaces of the pivots are not equally worn at all 
 points. The observer ought, therefore, to have the means of 
 testing from time to time the cylindrical form of the pivots, and 
 of correcting any error by which the observations may be affected 
 in consequence of change of form. Unless this can be done the 
 logic of the use of the transit instrument is not perfect. The form 
 of the pivots may to a certain extent be tested by making use 
 of a striding spirit-level, and applying the Y's of its feet to 
 the pivots in a manner that will be indicated under Division II. 
 If on applying the level, its readings undergo no change while 
 the instrument is turned through a complete revolution about 
 its axis, we may conclude that both the pivots are cylindrical, 
 whether or not they are equal, excepting in the very improbable 
 
22 PRACTICAL ASTEONOMY. 
 
 case of the alterations of form being so exactly alike that their 
 effect is merely a change of position of the level without change 
 of its inclination to the horizon. This experiment, however, 
 cannot be completely made, because there is a considerable 
 angular range of the pointing of the telescope through which 
 its tube prevents application of the striding level to the pivots. 
 The level-readings may still be adequate to determine whether 
 the pivots change form, and to some extent may furnish data 
 for calculating the amount of change. But in Division III. it 
 will be shewn how this source of error may be completely got 
 rid of by optical means. At present it is sufficient to remark 
 that under all circumstances it is desirable to diminish the 
 friction between the pivots and the Y's by taking off some of 
 the weight of the instrument. 
 
 29. This is done by means of Counterpoises, acting either 
 by leverage or by steel springs. The counterpoises of the Cam- 
 bridge Transit Instrument were originally of the former kind, 
 one end of the lever, in the form of a Y, pressing upwards 
 against a collar adjacent to the pivot, and the other end carry- 
 ing a box containing shot to allow of regulating the weight. 
 Under this arrangement it was possible the instrument might 
 happen to be sustained by resting partly on a face of the Y of 
 the counterpoise, and partly on a face of a Y of the pier ; which 
 I once discovered to be actually the case 1 . On account of the 
 unsatisfactory action of these counterpoises I afterwards substi- 
 tuted for them spring-counterpoises fastened to the inner faces 
 of the piers, and pressing upwards on collars near the pivots by 
 means of friction-wheels. The Transit Circle of the Cambridge 
 Observatory (which occupies the place of the original Transit 
 Instrument) has lever-counterpoises acting similarly by means of 
 friction-wheels, but in place of springs the wheels are sustained 
 by rods resting on arms of the levers, the opposite arms carrying 
 
 1 The lever-counterpoises might have the sustaining Y at the lower extremity 
 of a bar hanging from the end of one arm of the lever, the weight being 
 adjustible on the other arm. (See the Figure in p. 45 of Mr Godfray's Treatise 
 on Astronomy,) 
 
THE TRANSIT INSTRUMENT. 
 
 the weights, and the levers passing through holes in the piers 
 nearly on a level with the floor of the room. 
 
 30. Figure 2 represents any section of a Ramsden's eye- 
 piece by a plane through its axis, together with the course in 
 that plane of a pencil of rays converging from the object-glass 
 to the focus r. If this focus be the position of any point of a 
 
 Fig. 2. 
 
 wire, or diaphragm, situated "in the transverse plane coincident 
 with the stops ss which limit the field of view, the figure shews 
 how by means of the two lenses the eye at E sees distinctly the 
 point of the wire at r, and in coincidence with it the image of 
 the celestial object from which the pencil of rays originally 
 proceeded. Figure 3 exhibits the sliding-piece in grooves, men- 
 tioned in Art. 21, which carries the eye-piece, and by being 
 shifted horizontally, enables the observer to see distinctly 
 through an aperture at c the image of a star or other celestial 
 object in successive positions during its transit across the field 
 of view of the telescope, and also a series of fixed wires at the 
 instants the image crosses them. 
 
 31. Before proceeding farther in the description of auxiliary 
 apparatus, it will be proper to introduce an account of the prin- 
 ciple and construction of the Filar Micrometer 1 , which has 
 
 1 The first invention and use of this Micrometer may certainly be ascribed 
 to William Gascoigne, who, "while only eighteen years of age appears to have 
 been actively engaged in observing the celestial bodies, and in advancing the 
 state of optics and practical astronomy. 1 ' He was killed at, the battle of Marston 
 Moor on July 2, 1644, in the twenty-fourth year of his age. (See Grant's 
 History of Physical Astronomy, pp. 449 453.) 
 
PRACTICAL ASTRONOMY. 
 
 been applied in an important manner not only in observations 
 taken with the Transit Instrument, but also, under different 
 forms, in the observational use of other astronomical instru- 
 ments. The principle of the Micrometer is to measure very 
 small spaces by the intervention of much larger spaces, which 
 by mechanical construction are made to have to the first a fixed 
 ratio, and are capable of being easily read off from a graduated 
 scale. Figure 4 represents a double-wire Micrometer as con- 
 structed by Troughton. A is one of the micrometer-heads, the 
 
 Fig. 4. 
 
 form of which is symmetrical with respect to an axis ; aa is an 
 attached circle graduated from to 100 on its rim, and movable 
 about the axis by turning the milled head bb ; c is the micro- 
 meter-screw working in the micrometer-head, and passing into 
 the hollow terminal d\ the screw is fastened to the fork eee, which 
 slides within a box of which gg,gg are projected sides, and carries 
 with it the micrometer-wire / crossing from arm to arm trans- 
 versely ; h is the place of the index by which the readings of the 
 graduation are taken : the screw being of the form called right- 
 handed, which is that universally adopted by mechanists, if the 
 circle be turned in the direction for which the index-readings 
 of the graduation increase the opposite to that in which the 
 graduation-numbers in the figure increase the fork and wire will 
 be drawn towards the micrometer-head, so that the distance of 
 the wire from the micrometer-head is less the greater the micro- 
 meter-reading. At k there is an antagonistic compressed spiral 
 spring, which, when the graduation is turned in the direction 
 
THE TRANSIT INSTRUMENT. 25 
 
 of decreasing index-readings, pushes the fork and wire^rom the 
 micrometer-head. This construction is consequently adapted 
 to measuring any small space through which the wire is moved 
 by turning the milled-head, by recording the larger space simul- 
 taneously described by a given line of the graduation ; and it is 
 evident that the ratio of the former space to the other is equal 
 to that of the interval between the threads of the screw to the 
 circumference of the graduated circle. The comb II serves to 
 indicate the integral number of revolutions, the fraction of a 
 revolution is read off in graduation intervals, and the deepest 
 indenture in the comb shews approximately the position of the 
 wire at the zero reading. The exact performance of a screw 
 depends essentially on the equality of the intervals between 
 consecutive threads. A "drunken" screw is one defective in this 
 respect. Although this condition is usually fulfilled by the 
 mechanist with wonderful accuracy, as a matter of precaution 
 the micrometer-measurement of spaces requiring a large number 
 of turns of the screw should, as far as possible, be avoided. 
 
 32. The foregoing description may be sufficient as respects 
 the single-wire micrometer. When there is a second wire (as 
 m in Fig. 4), this is drawn towards the micrometer-head B 
 in exactly the same way as the other is drawn towards A, the 
 second graduated circle being turned in the direction of in- 
 creasing index-readings of the scale marked on its rim. The 
 fork of this micrometer slides within the fork of the first, and 
 the two wires can pass each other in close proximity, so as to 
 be both sufficiently in focus at the same time. 
 
 33. The field of view of the Transit Telescope, as seen with 
 a Ramsden's eye-piece, is shewn in Figure 5, in which are ex- 
 hibited the vertical wires (usually of spider's web) at which the 
 Transit observations are taken, and crossing them horizontally, 
 two wires mid-way between which the image of the object is 
 made to traverse the field by adjusting the pointing of the 
 telescope, it being of some consequence that the transits should 
 all be taken at the same parts of the wires and field. The 
 number of vertical wires (seven in the figure) is always uneven 
 in order that the mean of the times of transit across all the 
 
26 
 
 PRACTICAL ASTRONOMY. 
 
 wires may be nearly the same as that across the middle wire. 
 The additional wire mn is a micrometer-wire, movable, as ex- 
 plained in Art. 31, by turning the milled-head a of the micro- 
 
 Fig. 5. 
 
 meter-screw, and by the action of an antagonist spring at b. 
 The fractional part of a reading of micrometer-revolutions is 
 taken from the graduated circle c by means of the index d, and 
 the integral number of revolutions is shewn by the indents of 
 the comb e e, the place of the wire for zero reading being 
 approximately marked by the centre of a small circular hole in 
 the comb. 
 
 34. Figure 6 represents those parts of the apparatus per- 
 taining to the e} r e-end of the Transit-telescope which are seen 
 externally. I proceed now to describe these in detail and to 
 mention their respective uses. The two small circles a&, ab', 
 attached to the telescope tube near the eye-end, are setting- 
 circles. Their use is principally to enable the observer to point 
 the telescope in readiness for observing the transit of an object 
 of which he already knows approximately the Eight Ascension 
 and clock-time of Transit, and the Zenith Distance, or North 
 Polar Distance. Each circle is graduated expressly for this purpose, 
 and is provided with a small spirit-level movable about an axis 
 through the centre of the graduation, and carrying at one end 
 an index and vernier by means of which the angular position 
 
THE TRANSIT INSTRUMENT. 27 
 
 of the level may be set to any required graduation-reading. In 
 the figure, cd is the spirit-level of one of the setting-circles, the 
 vernier being attached to the end d. The position of the zero 
 
 Fig. 6. 
 
 reading of the setting-circle is so adjusted that when the tele- 
 scope points to the zenith, and the bubble of the spirit-level is 
 in its middle position, the index reading is equal to the co-lati- 
 tude of the Observatory. Hence for any other pointing of the 
 telescope the index-reading for bubble in mid-position is North 
 Polar Distance. This arrangement for setting for an object in 
 North Polar Distance is made suitable to the graduation of 
 either circle when on the West side, the graduations being such 
 that when by reversing the instrument that on the East side is 
 transferred to the West side, the process of setting is precisely 
 the same as before. For this reason the setting is always done 
 
28 PRACTICAL ASTRONOMY. 
 
 on the West side 1 . Occasionally, when the interval between the 
 transits of two objects put down for observation allows of little 
 time for setting for the second after the transit of the first, the 
 reading of the East circle should, to save time, be prepared 
 beforehand for setting for the second observation. This may 
 be conveniently done by first setting for the second observation 
 with the West circle, and then, without moving the telescope, 
 clamping the East level with bubble in mid-position; after 
 which the setting for the first observation may be performed in 
 the usual manner, and immediately after the transit across the 
 seven wires the telescope may be pointed for the second obser- 
 vation by turning it till the bubble of the East circle is in mid- 
 position. Before making the above-mentioned uses of the 
 setting-circles, their spirit-levels require to be adjusted by a 
 process which will come under consideration in Division III. 
 Figure 7 is an enlarged representation of the setting-circle 
 
 Fig. 7. 
 
 1 Formerly, each setting-circle of the Cambridge Transit Instrument had 
 two verniers, one for setting in zenith distance, and the other for setting in 
 north polar distance, and the graduations were suitable for setting on the west 
 side. As one of these verniers might be dispensed with, the second one was 
 removed, and in order that the other might point to north polar distance, each 
 circle was re- fixed, after being shifted about its centre through an angle equal 
 to the co-latitude, and that on the west side was used as before, for setting. 
 The setting-circles of the Cambridge Transit-circle also have each one vernier for 
 setting in N. P. D., but as this instrument is not used in reverse positions, 
 their graduations are exactly alike, and either circle may be chosen for setting. 
 
THE TKANSIT INSTRUMENT. 29 
 
 of a meridian Instrument: as and y are screws with capstan- 
 heads for adjusting the spirit-level. The action of the tangent- 
 screw cgc and clamp / is explained in the next article. 
 
 35. To make the index-reading of the setting-circle equal 
 to a given angle, which it is necessary to do before setting the 
 telescope for observing a selected object, the observer makes 
 use of a tangent-screw and clamp (the position of which in 
 Figure 6 is at o), which serves to give slow motion to the 
 vernier for exactly setting the index to the given angle. As 
 this- apparatus for slow motion is frequently applied in the use 
 of astronomical instruments, I propose to state here the prin- 
 ciple of its construction and the mode of its action. Its essen- 
 tial parts may be generally described as follows by reference to 
 Figure 8. A fine screw aa, with milled-heads bb at the two 
 
 ends, is supported by a nut c, in which its stem, detained by 
 flanges, is caused to revolve by turning either milled-head, as 
 may happen to be most convenient. Another nut o is tapped 
 for the screw to work in, and is fixed to a plate which overlaps 
 a little the limb ff of the instrument on the farther side. At 
 the same time another plate gg is in contact, partly with the 
 first plate, and partly with the limb on the nearer side, over- 
 lapping to the same extent as the first plate, and to allow of 
 both contacts a portion of the thickness of the second plate 
 equal to the thickness of the limb is cut away. The farther 
 plate is tapped for the action of a screw which passes through 
 a hole in the plate gg, and has its head and flange in the posi- 
 tion h. By turning this screw the surface of the second plate is 
 brought up to that of the first in such manner that the limb is 
 nipped tightly between the two overlapping parts. When the 
 pressure is taken off by turning back the clamping-screw h, the 
 
30 PRACTICAL ASTRONOMY. 
 
 two plates are separated by springs and at the same time are 
 kept in position by suitable detents, in order to allow of large 
 movements either of the tangent-screw relative to the limb, 
 or of the limb relative to the tangent-screw. Supposing now 
 the clamping to have been effected by means of the screw- 
 head h, by turning one of the heads bb the- nut o will approach 
 or recede from the nut c, according to the direction of the 
 turning. These arrangements being understood, it will be seen 
 that there are two cases of the action of a tangent-screw and 
 clamp. In one case the supporting nut c is fixed to a plate in 
 rigid connection with a vernier which is to be used either for 
 reading off from the graduated limb, or for setting the reading 
 to a given angle. This is the usual case of the tangent-screw 
 of the setting circle of a meridian instrument. Since the limb 
 of the circle is fixed relatively to the telescope-tube, the nut o 
 and the plates, when clamped to the limb, are also fixed, and 
 consequently turning either head b causes the nut c to move 
 relatively to the limb. In the instance represented by Fig. 6 
 this nut is rigidly connected with the tongue p, which conse- 
 quently moves with it, carrying the spirit-level cd and the 
 vernier d. In the other case, the supporting nut c is fastened 
 to a wall, or immovable plate, and turning bb gives the slow 
 motion to the nut o, the clamping plates, and the clamped limb 
 ff. This is the kind of action of the tangent-screw of a Mural 
 Circle, or Transit Circle. What is said above may suffice for 
 explaining generally the principle and use of the tangent-screw 
 and clamp. The details of the construction differ according to 
 the purpose for which the instrument is designed. The tangent- 
 screw of the Mural Circle, for instance, is constructed very 
 differently from that of a Ramsden's Sextant, as will be shewn 
 when those instruments are under consideration. 
 
 36. Again in Figure 6, the screw-head seen projecting at/" 
 belongs to the tangent-screw of the setting-circle on the farther 
 side of the eye-end. At g, two screws acting oppositely on a 
 projecting piece of the frame which carries the micrometer-wire, 
 and thereby causing angular motion of the frame about the 
 axis of the telescope, serve for fixing this wire in the position of 
 
^ 
 
 THE TRANSIT INSTRUMENT. 
 
 
 31 
 
 exact parallelism to the middle wire. At h there is an adjusting 
 apparatus, like that at g, by the action of which on a piece pro- 
 jecting from an interior tube which carries the two wire-frames 
 and the eye-piece, this tube can be shifted longitudinally, and the 
 
 Fig- 6. 
 
 wires be placed at the distance from the object-glass of its 
 geometrical focus. The particular reason that the observer 
 should have the means of making this adjustment will be ex- 
 plained in Division II. At the extremity of the eye-end the 
 figure represents the eye-piece E inserted in a plate movable 
 in grooves (as mentioned in Arts. 21 and 30), together with the 
 places of the micrometer-head .1 and the opposite spring m. In 
 addition to the parts above mentioned there are two opposite 
 screws with capstan -heads (not exhibited in the figure), which 
 work in the frame to which the transit-wires are attached, and 
 answer the purpose of shifting it horizontally and fixing it in 
 any required position. 
 
32 PRACTICAL ASTRONOMY. 
 
 37. In order to see the wires distinctly at night-time for 
 observing transits, it is necessary to illumine the field of view, 
 that the wires may appear dark on a bright ground. The illu- 
 mination may be effected and its amount regulated in the 
 following manner. The light of a lamp, put opposite the end 
 of one of the pivots through which a perforation has been made, 
 falls upon a reflector which is situated where the axis of motion 
 intersects the axis of the telescope, and, being inclined to the 
 latter at an angle of 45, sends the reflected light to the eye- 
 end, and thus illumines the field. The reflector is ring-shaped 
 in order to allow of free passage of the pencils of light from the 
 object-glass, and its surface is gilded for the purpose of increas- 
 ing the intensity of the scattered light reflected from it. The 
 degree of illumination is regulated by varying the size of the 
 aperture at the end of the pivot, which is done by means of 
 rack-work and wheels, moved by turning the milled-head at n, 
 which is within reach of the observer when he is looking 
 through the eye-piece. In the Greenwich Transit Circle, as 
 also in that of the Cambridge Observatory, the illumination can 
 be gradually diminished to the point of extinction by means of 
 apparatus provided by Mr James Simms for enabling the ob- 
 server to vary the inclination of the reflector. 
 
 38. When the light of the object to be observed is very 
 feeble, it may happen that the least amount of illumination 
 required for seeing the wires with sufficient distinctness for the 
 observation is too strong for distinguishing the object. In that 
 case recourse is had to illumining the wires on a dark field. To 
 do this the illuminating light has to pass in the direction from 
 the eye-glass to the wires. The Cambridge Transit Instrument 
 was furnished with the means of suspending a small lamp in 
 such manner that its light fell in a slanting direction on the 
 wires so as to make them visible on a dark ground. This 
 method is liable to the objection that as one side of a wire is 
 illumined more than the other, the axis of the line of illumina- 
 tion does not exactly coincide with the axis of the wire. This 
 fault might be got rid of by having two lamps, one on each 
 side of the system of wires : but at best this method is practi- 
 
THE TRANSIT INSTRUMENT. 33 
 
 cally inconvenient. It has been proposed to substitute for the 
 spider-lines fine lines engraven on plate glass. These lines would 
 ordinarily appear dark on a bright ground ; but on applying at 
 night-time the light of a lamp to the edge of the piece of glass, 
 they would be seen, by the light scattered from innumerable 
 facets, as bright lines on a dark ground. I do not know that 
 this ingenious method has been put in practice. I have used 
 with success for the same purpose the collimating eye-piece (of 
 which a description will subsequently be given), by holding in 
 the hand a lamp so that its light reflected from the plate-glass, 
 situated between the eye and the wires and inclined to the axis 
 at an angle of 45, is incident on the wires. This and the pre- 
 ceding method are objectionable on account of the loss of light 
 from the faint object caused by its passage through the plate- 
 glass. The most complete method appears to be one invented 
 by Mr Simms, according to which by continuing to vary the 
 inclination of the reflector beyond the point of extinction men- 
 tioned in Art. 37, the light of the lamp is thrown on the wires, 
 the field remaining dark. The apparatus for this purpose has 
 been supplemented by furnishing the observer with means of 
 regulating by turning a milled-head the amount of the illumi- 
 nation of the wires. 
 
 39. From the foregoing descriptions of the mounting and 
 the parts of a Transit Instrument we may proceed to treat of it 
 under the second Division ; that is, to consider what preliminary 
 arrangements have to be made in order that it may effect the 
 immediate purpose of determining the clock-time of transit of 
 a celestial body across the meridian passing through its centre 
 of figure, which meridian will be taken to be the reference 
 meridian of the Observatory. 
 
 II. THE ADJUSTMENTS OF A TRANSIT INSTRUMENT AND 
 CORRECTION OF ITS ERRORS BY CALCULATION. 
 
 40. It will at first be supposed that the pivots are exactly 
 equal and cylindrical, and that their axes are in the same 
 
 c. 3 
 
34 PRACTICAL ASTRONOMY. 
 
 straight line (see Art. 27). It is evident that on this suppo- 
 sition the axis of motion coincides with the common axis of the 
 pivots, and that it will have the same position in space in 
 reverse positions of the instrument, if only the Y's in which the 
 pivots rest retain the same position. It has been stated (Arts. 
 12, 13, 15) that the axes of all pencils of rays that go to form 
 the image looked at with the eye-piece pass through the optical 
 centre of the object-glass, so that a plane through this point 
 cutting the axis of motion at right angles has not only a fixed 
 position relative to the instrument, but is also related in a 
 definite manner to the directions of the . pencils of rays that 
 come from a celestial object. A line of collimation may be 
 defined to be the axis of any pencil of convergent rays which 
 forms the image of a point of the object at the geometrical 
 focus of the object-glass, and its direction, as far as concerns 
 transit observations, is determined by its inclination to the 
 above-mentioned plane. This, in fact, is the plane to which 
 collimation is referred, and, for brevity, will be called the plane 
 of collimation. There is no line which can properly be named 
 the optical axis of collimation ; but a straight line drawn from 
 the optical centre of the object-glass to the point of the plane of 
 collimation which is mid-way between the two horizontal lines 
 represented in Figure 5, is in some sense an optical axis, 
 because, for the sake of uniformity and exactness, the images 
 whose transits are taken should all be made to cross the field 
 nearly through this point. If, however, the wire-frames be 
 accurately adjusted, crossing a little higher or lower will be of 
 no consequence. 
 
 41. That a Transit Instrument may answer the purpose to 
 which it is applied, the optical axis (defined as in the preceding 
 article) must either be made by mechanical adjustments to move 
 in the plane of the meridian, or its small deviation from that 
 plane must be ascertained for the pointing of the telescope 
 in each observation, in order that the error of the observed time 
 of transit thence arising may be corrected by calculation in the 
 reduction of the observations, which is virtually to correct the 
 errors of an unadjusted instrument. 
 
THE TRANSIT INSTRUMENT. 35 
 
 42. The method of correcting for instrumental errors by 
 calculation is preferable for two reasons to that by mechanical 
 adjustment: (1) because it dispenses with frequent turning of 
 screws, which, as tending to unsteadiness, should as far as 
 possible be avoided ; (2) because the data for the calculation are 
 generally obtainable with great precision by the use of optical 
 instruments for magnifying. 
 
 43. In certain circumstances mechanical adjustment is 
 always employed ; namely, when it is sufficiently exact, and not 
 liable to be disturbed, and at the same time the method by 
 calculation is not readily available. This is the case with 
 respect to adjusting the middle transit- wire in order to make it 
 coincident with a plane perpendicular to the axis of motion, and 
 the micrometer-wire, so that it shall be exactly parallel to this 
 fixed wire. These two adjustments, which it will be proper to 
 consider first of all, may be performed by the following processes. 
 For the first adjustment there should be added to the apparatus 
 at h provided for shifting longitudinally the tube which carries 
 the eye-piece and wire-frames (see Fig. 6 and Art. 35), two 
 opposite screws which, by acting transversely on the same 
 projecting piece, would give the means of turning that tube 
 about its axis, and fixing it in any required angular position. 
 Then, assuming that the axis of motion has been roughly 
 adjusted to horizontally, and the wire-frames have been 
 approximately put into position, that carrying the system of 
 seven wires being fixed by opposite screws, the middle wire is to 
 be brought to bisect (by turning the screw for azimuthal adjust- 
 ment of the axis of motion, as hereafter spoken of) a well- 
 defined fixed point, which may be some point of a terrestrial 
 object, or a meridian mark, or, what is still better, the intersec- 
 tion of crossing wires at the focus of a collimator. Now by 
 means of the horizontal adjustment by the opposite screws 
 referred to at the end of Art. 36, and the angular adjustment 
 just mentioned, it is always possible to make the bisected point 
 run exactly along the middle-wire from end to end, while the 
 telescope is shifted about the axis of motion. The position of 
 the wire when this condition is fulfilled is coincident with a 
 
 32 
 
36 PRACTICAL ASTRONOMY. 
 
 plane perpendicular to that axis. (I have found that the Pole- 
 star, by reason of its slow diurnal motion, may be used when on 
 the meridian for testing the accuracy of this adjustment.) 
 This being done, the micrometer- wire is to be placed exactly 
 parallel to the middle wire, partly by its micrometer-movement, 
 and partly by employing the apparatus at g (Fig. 6) for shifting 
 the wire about the axis of the telescope and fixing it in the 
 required position 1 . The parallelism is effected when by the 
 two movements a fine line of light of uniform breadth is formed 
 between the two wires. 
 
 44. When these adjustments of the wires have been 
 effected, three other adjustments, necessary and sufficient for 
 the exact performance of a Transit Instrument, having, as 
 supposed in Art. 40, equal cylindrical pivots, may be proceeded 
 with. (1) The middle wire is required to be coincident with 
 the plane of collimation defined in Art. 40 ; (2) the plane of 
 collimation must be vertical, and therefore pass through the 
 zenith of the point of its intersection with the axis of the 
 instrument ; (3) this vertical plane must also pass through the 
 pole of the heavens in order that it may be coincident with the 
 meridian of that central point. The first of these conditions 
 has reference to the instrument, the second to the earth, and 
 the third to the heavens. Deviations from fulfilment of these 
 conditions are instrumental errors, which have either to be cor- 
 rected by mechanical means, or to be allowed for by calculation 
 (see Art. 41). As it is proposed to indicate both methods of 
 correction, it will be proper to introduce here a description 
 of the apparatus employed for correcting the three errors 
 mechanically. 
 
 45. For correcting the error of collimation, that is, the 
 
 1 In the Cambridge Transit Instrument no apparatus was provided for the 
 rotational movement of the tube which carries the wire-frame, in consequence 
 of which, having to take out the tube for the purpose of inserting the apparatus 
 at h, I was obliged, in replacing it, to make the angular adjustment by hand in 
 the best way I could. The Transit Circle now in use is furnished with means of 
 making both the longitudinal and the rotational adjustment, two sets of an' 
 tagonist screws acting for that purpose upon the projecting piece at h. 
 
THE TRANSIT INSTRUMENT. 
 
 37 
 
 deviation of the middle wire from the plane of collimation, the 
 movement of the wire-frame by capstan-headed screws, men- 
 tioned before in Art. 36, is provided. For correction of the error 
 of level, or deviation of the axis of motion from a horizontal 
 plane, the Y in which one of the pivots rests is movable 
 vertically, and can be fixed in position, by adjusting screws ; 
 and for correction of the error of azimuth, or deviation of the 
 plane of collimation, corrected for level error, from the plane of 
 the meridian, the other Y is movable horizontally, and can be 
 fixed in position, by antagonist adjusting screws. The mechani- 
 cal arrangements made for these two movements in the mount- 
 ing of the Cambridge Transit Instrument are exhibited in 
 Figures 9 and 10, as seen when the brass plates that concealed 
 them were removed 1 . 
 
 Fig. 9. 
 
 46. The vertical adjustments were made by turning with 
 spikes the capstan-headed screws at a and b (Fig. 9), and the 
 azimuth adjustments by turning the antagonist screws at c and 
 
 1 As this apparatus has been dismounted, I had the opportunity of examining 
 it before drawing the Figures 9 and 10. These are, for the most part, copied 
 from figures contained in the second plate to Woodhouse's account of the 
 instrument (before referred to in a note to Art. 9), which, as no explanation 
 accompanied them, were scarcely intelligible without inspection of the ap- 
 paratus. 
 
38 
 
 PRACTICAL ASTRONOMY. 
 
 d (Fig 10). The former adjusting screws were maintained in 
 bearing by the spiral springs at g and f. The capstan-heads 
 were graduated in such manner that after completing the 
 
 Fig. 10. 
 
 adjustment of the axis, the index-readings of the two heads for 
 movement in altitude should be the same, as also those of the 
 two heads for movement in azimuth. This appears to have 
 been arranged in order to avoid the strain which such action of 
 two screws as that which the above figures indicate is evidently 
 liable to produce if they should be turned unequally. Accord- 
 ingly after turning the screws for either the altitude or the 
 azimuth adjustment, the index readings of both screws were 
 always left the same. (It would have been better if the 
 altitude adjustment had been effected by a single capstan- 
 headed screw placed between the spiral springs at g and f 
 (Fig. 9), supposing these to be of considerable strength, and if for 
 the azimuth adjustment one of the screws at c and d, instead of 
 working in the movable plate A'A\ had simply abutted on this 
 plate for fixing it, the movement being effected by the screw at the 
 other end.) The vertical movement of AA took place along the 
 fixed edges pq and rs, and the horizontal movement of A' A' 
 along the fixed edge tu, on which it was made to bear by the 
 steel springs and screws g and h (Fig. 10). I found that the plate 
 BB was made to be movable in contact with A A about a horizon- 
 tal axis ee supported by the latter, and that the like was the case 
 
THE TRANSIT INSTRUMENT. 39 
 
 relatively to the plates B' B f and A' A'. This was done in order 
 that by the weight of the instrument the pivots might be 
 made to rest in perfect contact with the Y's ; and for the same 
 reason the pieces C and C', in which the Y's are cut, admitted 
 of being shifted horizontally till the pivots obtained their 
 proper bearing, after which they were fixed in position by the 
 capstan-headed screws m and n. Since, however, with all the 
 above-mentioned precautions it was not certain that strain, or 
 unsteadiness, was wholly prevented, I consider the mode of 
 adjustment described in the next paragraph, which dispenses 
 with screw-movement, to be much preferable. 
 
 47. The foregoing explanations may suffice to indicate 
 generally what is required to be done for correcting the three 
 errors, whether the Transit Instrument be small or large, portable 
 or fixed. But the details of the mechanical means may differ 
 according to the size or purpose of the instrument. In the 
 Transit Circles of the Greenwich and Cambridge Observatories, 
 the wire-frame is not fixed in position by capstan-headed 
 screws, but is movable by a micrometer- screw with graduated 
 head and antagonist spring, and the use of the single 
 micrometer-wire is dispensed with. Also the pivots are made 
 to rest in portions of cylindrical surfaces exactly fitting them, 
 and instead of employing adjusting screws for correcting the 
 errors of level and azimuth when they have become large, this 
 is done by simply scraping these surfaces. The fitting portions 
 extend on each side through 15, and their centres are about 
 120 apart. 
 
 Correction of Collimation- Error by mechanical adjustment. 
 
 48. We may now proceed to consider particular processes 
 by which the three errors may be corrected, beginning with the 
 mechanical correction of the error of collimation. The method 
 for a long time generally adopted for making this correction 
 was to set up a fixed terrestrial mark very near the meridian, 
 and at a distance from the telescope so great that the rays from 
 any point of it entering the object-glass would be very nearly 
 parallel (see Art. 9). The Cambridge mark, which was on the 
 
40 
 
 PRACTICAL ASTRONOMY. 
 
 tower of Grantchester church (distant 2J miles), was a cross in 
 the form of X 1 , and its angular distance West from the meridian 
 was about 14". Under these circumstances the middle wire 
 could be brought to bisect the image of the cross by means of 
 the screws for azimuth adjustment (Art. 46). Then on reversing 
 the instrument 2 by exchanging the positions of the pivots in the 
 
 Y's, the wire would usually be found to be separated from the 
 cross. 
 
 49. Figures 11 and 12 represent what is seen before and 
 
 1 For this form I substituted a small circular disk,- held by fine iron wires 
 at the centre of a wooden circular frame which was supported by two legs 
 equally inclined to the vertical through the disk. This mark was also used for 
 horizontal bisection. 
 
 2 The Cambridge transit instrument was reversed by being lifted from the 
 Y's by a system of pulleys hanging from the end of an iron bar, which was 
 drawn out for the occasion so far as to bring the point of suspension vertically 
 over the centre of the instrument. Leathern straps which passed under the 
 transverse arms being attached to the pulley-frame, the lifting was effected by 
 pulling the rope by hand. This method I found to be attended with some risk, 
 having discovered (in good time) that the check which limited the drawing out 
 of the bar required to be made fast, having gradually given way. A better 
 method is adopted for lifting the present Transit-circle, a much heavier instru- 
 ment, this being done by a jack-screw machine, which stands usually at a corner 
 of the room, and by means of rails and a turn-table can be brought into a 
 position proper for acting from below on the transverse arms. There is rarely 
 occasion for this operation, the observations being all taken, not necessarily, 
 but for greater convenience, in the same position of the instrument, and the 
 error of collimation being found, without reversing, by processes which will 
 shortly be described. 
 
THE TRANSIT INSTRUMENT. 41 
 
 after the reversion. As the line of collimation (Art. 40) of 
 any point of the wire makes equal angles with the plane of 
 collimation before and after reversion on opposite sides, it 
 follows that this plane passes through a point c midway 
 between the bisected cross and the wire in its new position D'd'. 
 Hence to correct the error of collimation of the middle wire, it 
 is only required to move the wire after the reversion half-way 
 towards the image of the cross by means of the screws for 
 collimation-adjustment (Art. 45). If this movement is effected 
 by capstan-headed screws, as is usually the case in small and 
 portable instruments, it can only be done tentatively, and after 
 a first trial the same process should be repeated till the 
 wire appears to bisect the cross both before and after the 
 reversion. When this is the case the correction has been made 
 with as much accuracy as is attainable by mechanical means. 
 
 50. If the telescope has a micrometer-wire (as mn in 
 Figure 5), the mechanical correction of collimation-error can 
 be more readily effected and with greater precision. In that 
 case the mark is bisected by the micrometer-wire before 
 reversing, and the micrometer-reading is recorded; and the 
 same steps are taken after the reversion. If in Figures 11 and 
 12 Dd and D'd' represent the micrometer-wire in the two 
 positions, the difference of the two readings will evidently 
 measure the distance of D'd 1 from the cross, or twice the colli- 
 mation error. The mean of the readings will be the micrometer- 
 reading for the mid-position c, and if the micrometer-head be 
 set to this reading, the micrometer-wire will actually be placed 
 in the plane of collimation. Consequently the mid-wire will be 
 corrected for collimation-error by being brought by the screws 
 for horizontal adjustment of the wire-frame into coincidence 
 with the micrometer- wire according to the judgment of the eye, 
 which may be done with very considerable accuracy. If, how- 
 ever, the wire-frame be movable by a micrometer (as mentioned 
 in Art. 47), the mid-wire is at once placed in the plane of colli- 
 mation by making the micrometer-reading equal to the mean 
 of the two recorded readings. 
 
42 PKACTICAL ASTRONOMY. 
 
 Calculation of uncorrected Collimation- Error. 
 
 51. In the use of large meridian instruments the processes 
 of mechanical correction are only employed for securing that 
 the errors (which from one cause or another are always chang- 
 ing) shall not exceed very small amounts, no attempt being 
 made to maintain them at zero by adjustments. Hence it is 
 necessary from time to time to calculate the actual amounts, 
 for the purpose of getting rid of their effects on the results of 
 the observations. Proceeding, accordingly, to indicate how 
 collimation-error may be calculated, I shall, at first, suppose 
 that the measured quantities are expressed in micrometer- 
 revolutions, and afterwards shew how to determine the value 
 in celestial arc of the space the wire is carried over by one 
 revolution of the micrometer-head. It has been ascertained 
 (Art. 50) that the micrometer-reading for coincidence with the 
 plane of collimation is the mean of the micrometer-readings for 
 bisection of the mark before and after reversion. (For the sake 
 of precision each adopted micrometer-reading is usually the 
 mean of the readings for, at least, six bisections.) The coin- 
 cidence of the micrometer-wire with the mid-wire is not judged 
 of in the way mentioned in Art. 50, but is determined by just 
 excluding the line of light between the two wires three or more 
 times on one side and an equal number of times on the other 
 side, and taking the mean of the micrometer-readings for all 
 the positions for the adopted coincidence-reading. The colli- 
 mation-error of the mid-wire expressed in micrometer-revolu- 
 tions is the difference between this coincidence-reading and 
 that for coincidence with the plane of collimation. In the case 
 of a Transit Circle which has no micrometer-wire, and is not 
 reversed, the collimation-error is calculated by means of colli- 
 mators, as will be presently shewn. 
 
 52. On the supposition that the pivots are cylindrical and 
 equal, the determination of collimation-error by this method is 
 complete unless the operation of reversing the instrument pro- 
 duces horizontal displacement of the Y's. As it is possible this 
 might happen, especially with such apparatus for supporting 
 
THE TRANSIT INSTRUMENT. 43 
 
 the Y's as that described in Art. 46, the observer ought always 
 to employ two marks, one southward and the other northward, 
 because the effect of horizontal angular displacement of the 
 axis is thereby eliminated, as is proved by the following reason- 
 ing. The telescope being horizontal and pointing southward, 
 suppose the eye-end to be shifted, by the displacement, towards 
 the east. Then the image of the south mark will be shifted, 
 relatively to the mid-wire, towards the West through a space 
 which measures the angular displacement. So when the tele- 
 scope is turned to point northwards, and the eye-end is conse- 
 quently shifted, by the displacement, westward, the image of 
 the north mark is shifted, relatively to the mid-wire, through 
 an equal space towards the East. Hence if r t and r z be the 
 micrometer-readings for bisection of the marks before the re- 
 version, and r[ and r a f those for bisection after reversion, we 
 may infer from what has just been said that the latter would 
 have been r,' a and r 2 ' + a if there had been no displacement. 
 
 Hence the reading for plane of collimation is ^ (r x + r/ a) by 
 one mark, and - (r a -f r a ' + a) by the other, and the mean of the 
 
 two determinations is ^ (r t + r/ -f r 2 + r a '), which is the mean 
 
 of the four readings actually recorded. This mean, therefore, 
 is the true reading for coincidence with the plane of collimation, 
 notwithstanding any small angular displacement of the axis. 
 The same result would evidently have been obtained if the 
 shifting of the eye-end, the telescope pointing southward, had 
 been towards the West. 
 
 53. Instead of a second distant mark on the opposite side 
 r>f the instrument (an arrangement which it would rarely be 
 possible to make), at the Cambridge Observatory, as well as at 
 Greenwich (see Note to Art. 9), a small transit telescope was 
 formerly set up near the telescope of the transit instrument, to 
 serve for a Collimator. The two telescopes being so placed 
 that they pointed directly towards each other, an image of the 
 wires of the small telescope, which are supposed to be accurately 
 
PRACTICAL ASTRONOMY. 
 
 placed at its geometrical focus, was formed at the geometrical 
 focus of the large telescope, and a definite point of this image 
 was selected for bisection. The selected point, as being a focus 
 of convergence of parallel rays incident on the object-glass, was 
 equivalent to the image of a mark at an infinite distance. It 
 is obvious that if two collimators, one towards the north and 
 the other towards the south, were employed in this manner, a 
 fixed terrestrial mark would not be required. Gauss introduced 
 the improvement, now generally adopted, of setting up two 
 collimators so that each could be a collimator to the other as 
 well as to the intermediate telescope, by which means the 
 error of collimation can be calculated without reversing the 
 instrument, as may be shewn by the following argument. 
 
 54. In Figure 13 A and B represent the two collimators, 
 and C the transit telescope between them. The optical centres 
 of the three object-glasses are respectively at o, o, and o", and the 
 positions e, e, e" , marked by the intersections of crossing lines, 
 are their geometrical foci. It will be supposed that the system 
 of wires of the telescope C is placed exactly at the focal dis- 
 tance d'e", so that rays proceeding from a point of the wires 
 situated on or very near the optical axis will emerge from the 
 object-glass as parallel rays. This adjustment, usually performed 
 by the instrument-maker, might be effected by the observer 
 himself by using the screw^adjustment at h mentioned in Art. 36, 
 
THE TRANSIT INSTRUMENT. 45 
 
 the wire-frame being thereby placed in such a position that 
 the mid-wire, where it is intercepted between the horizontal 
 wires, and the image of a star or planet crossing it, are both 
 seen distinctly. (A better method of making this adjustment 
 will be indicated when the collimating eye-piece is under con- 
 sideration.) The additional figures aa and bb' represent, within 
 the interior circles, the fields of view of A and B as seen by 
 eye-pieces at c and d. The wires thus exhibited are used for 
 collimating, and it is necessary that they should be exactly at 
 the geometrical foci of the collimators. This condition may be 
 satisfied as follows. After adjusting, as above stated, the wire- 
 frame of 0, A and are made to point towards each other, with 
 their optical axes approximately in the same straight line, by 
 screw-movements provided for roughly adjusting the optical 
 axis of A both vertically and horizontally. An image of the 
 wires of C is thus formed at the geometrical focus of A, being 
 due to parallel rays issuing from the object-glass of and 
 incident on that of A. The wires of A are then shifted (by an 
 external ring-screw, or other apparatus) till they are seen 
 distinctly through the eye-piece at c coincidently with the 
 above-named image, in which case the required adjustment is 
 effected. The wires of B have to be adjusted to focus by a like 
 process. The mid-wire of C being assumed to be accurately 
 parallel to the plane of collimation (see Art. 43), the vertical 
 wires of A and B have to be adjusted so as to be parallel to 
 that wire. This may be done by means of the horizontal and 
 vertical adjustments of their optical axes above spoken of, and 
 by unclamping, moving rotationally by hand, and then clamping, 
 the tubes in which the wires are fixed. No other means of 
 making the rotational adjustment is required, this being suffi- 
 ciently exact if the collimating operation be performed accord- 
 ing to the following process. 
 
 55. The wires and comb in the two fields of view aa, bb' 
 (Fig. 13) are exactly alike ; but the micrometer-movement in 
 the former is horizontal and is used for collimating in right 
 ascension, whilst that in the other is vertical and might be used 
 for collimating in declination, and for determining, in the case 
 
46 PRACTICAL ASTRONOMY. 
 
 of a Mural Circle or a Transit Circle, the effect of its flexure. 
 It would suffice for all purposes to furnish only one of the 
 collimators with a micrometer, if means were provided of ro- 
 tating its wires through 90. I adopted an arrangement by 
 which this could be done by hand with sufficient exactness, 
 and was thus able, by means of two collimators, only one of 
 which had a micrometer, to correct the collimation-error of the 
 Mural Circle, and also to calculate corrections for the effect of 
 its horizontal flexure. It would, however, be preferable to 
 furnish each collimator with a micrometer in the manner ex- 
 hibited in Fig. 13. (This is the case in the collimators of 
 the Cambridge Transit-circle.) In the field of each collima- 
 tor are represented two parallel wires, separated by a small 
 interval, and movable together by turning the micrometer-head. 
 Supposing the parallel wires of B to be put nearly in the middle 
 of the field, as indicated by the zero of the comb, the point of 
 the vertical wire which bisects the interval between them may 
 be taken for e. ' If B has no micrometer the point e' is marked 
 by the intersection of the vertical wire by a horizontal wire 
 through the middle of the field. By preliminary operations 
 the optical axes of A, C, and B should be placed nearly in the 
 same straight line, and the optical axes of A and B nearly in 
 the plane of collimation of C. These arrangements, which may 
 be made by means of the vertical and horizontal adjustments of 
 the optical axes of A and B, require no great precision. Also in 
 the tube of the telescope C an aperture is required to be made 
 through which, when the axis of the telescope is vertical, the 
 collimators can point directly towards each other. By this 
 means an image of e is formed at the geometrical focus of A, 
 and by the adjustments of its axis may be brought nearly to 
 the centre of its field. These precautions prevent any appre- 
 ciable error arising from defect of parallelism of the vertical 
 wires of A and B to the mid-wire of C. The parallel wires of 
 A may now be moved till the image of e exactly bisects the 
 interval between them. This may be done with great precision 
 by the judgment of the eye. For greater certainty it is usual 
 to make six or more such trials, to record the micrometer-read- 
 
THE TRANSIT INSTRUMENT. 47 
 
 ing after each, and finally to set the micrometer to the mean 
 of all the readings. The point e in A may then be taken 
 to be that point of the horizontal wire which is midway 
 between the parallel wires. The telescope C being next pointed 
 to A, let the image of e be formed at e" on or near the optical 
 axis. This image is to be bisected six or more times by the 
 micrometer- wire of C (if the system of wires be not movable by 
 a micrometer), and the mean of the readings taken. Exactly 
 the same process is to be gone through with respect to the point 
 e of the collimator B. Now although the collimation between 
 A and B does not place eo and e'o in the same straight line, it 
 secures that their directions shall be parallel. Hence if the line 
 e"o" determined by the collimation between A and C be inclined 
 to the plane of collimation of C, that determined by collimating 
 between B and C will be equally inclined on the opposite side. 
 Hence the mean between the two readings for bisection of the 
 two points e" will be the micrometer-reading for the plane of 
 collimation, and the difference between this reading and that 
 for coincidence with the mid-wire (found by the method indi- 
 cated in Art. 51), will be the measure of the collimation-error of 
 the mid-wire in micrometer-revolutions. 
 
 56. If the wire-frame be movable by a micrometer, the 
 process for finding the micrometer-reading for plane of colli- 
 mation is the same as before, the mid-wire being used in place 
 of the separate micrometer-wire. It is the practice at Green- 
 wich to obtain the micrometer-reading for collimation-plane 
 every day, after each operation to set the micrometer to a 
 certain convenient reading that may be presumed to be little 
 different from the true reading, and to adopt for true reading 
 the mean of several determinations, usually those of a week. 
 Then the adopted error of collimation is the difference between 
 this adopted reading and that for the arbitrary position occupied 
 by the wires when the transits are taken. The error of colli- 
 mation for the transit observations made with the Cambridge 
 Transit Circle is determined on the same principle. 
 
 57. The amount of collimation-error having been found, it 
 is next required to determine the sign with which it is to be 
 
48 PRACTICAL ASTRONOMY. 
 
 applied for correcting its effect on the observed times of transit. 
 If the instrument be reversed from time to time, as was formerly 
 the practice at Cambridge, the sign of the collimation-correction 
 depends on the position of the transverse axis as to East or 
 West, which, therefore, has to be noted. This was done by 
 recording after each reversion ' Illumination East,' or ' Illumina- 
 tion West/ according as the pivot which is perforated for 
 admitting the light for illuminating the field of view was left 
 eastward or westward. (It would have been simpler to indicate 
 the position by the eastward or westward direction of the 
 micrometer- head, which happened to be opposite to that of the 
 perforated pivot.) Let us suppose the micrometer-head to be 
 westward. Then taking into account that a micrometer-reading 
 is always greater the nearer the micrometer-wire is to the head, 
 it will be seen that the mid-wire is to the east or west of the 
 plane of collimation according as the reading for coincidence 
 with it is less or greater than that for plane of collimation. 
 Hence since the image of a star above pole crosses the field 
 from west to east, in the case of the less coincidence reading it 
 crosses the mid- wire after crossing the plane of collimation, that 
 is, too late, and the correction of the time of transit is negative. 
 In the contrary case of the coincidence-reading being greater 
 than that for plane of collimation, the correction is positive. 
 The signs of the correction for a star below pole are opposite to 
 these, because its image crosses the field in the opposite direc- 
 tion. The signs of the corrections for micrometer-head eastward 
 are evidently contrary, under the same circumstances, to those 
 for micrometer-head westward. 
 
 58. The micrometer-head of the Greenwich Transit Circle is 
 eastward, and the instrument is not reversed. Hence for a star 
 above pole the sign of the collimation-correction is always the 
 same as that of the algebraic excess of the reading for collima- 
 tion-plane above that for the position assigned to the wire-frame 
 (see Art. 56). In the case of the Cambridge Transit Circle the 
 micrometer-head is always westward ', and consequently the 
 signs of the collimation-correction s, under the same circum- 
 stances, are opposite to those for the Greenwich instrument. 
 
THE TRANSIT INSTRUMENT. 
 
 49 
 
 59. It is now required to express the amount of the error, 
 as obtained by the foregoing processes, in terms of the arc of a 
 great circle ; that is, to determine the value in arc of one revolu- 
 tion of the micrometer. For this purpose it will be necessary to 
 anticipate what has to be said under Division III. so far as to 
 assume that the observer is able to record with great precision 
 the clock -time of the instant of the passage of a star across the 
 micrometer- wire. In Figure 14, P is the pole of the heavens, 
 
 Fig. 14. 
 
 and PN is a circle of north polar distance in or very near the 
 meridian, and coincident with the micrometer-wire. A transit 
 of Polaris (the pole star) is taken at the micrometer-wire in 
 this position, and after moving the wire to the position bab' by 
 turning the micrometer-head through an integral number (n) of 
 revolutions, the instant of transit at o- is in like manner taken. 
 (Any other star the north polar distance of which is approxi- 
 mately known and sufficiently small to allow by its slow movement 
 of time for this operation, might be selected.) Per is the star's 
 polar distance, and <rN is an arc of a great circle at right angles 
 to PN. Let x be the value of one revolution of the micrometer 
 in seconds of space. Then <rN= nx. Hence if Per = 8, and the 
 angle <rPN= 15, t being the interval between the transits in 
 seconds of time, we have by the right-angled spherical triangle 
 <rPN, sin nx = sin 8 sin 15& Let R be put for 206265", the 
 number of seconds in an arc equal to radius. Then, nx being 
 small, R sin nx = nx nearly, and hence 
 
 R . 
 = sin 
 n 
 
 sn 
 
 c. 
 
50 PRACTICAL ASTRONOMY. 
 
 If the arc lot be not larger than it is required to be for 
 making an exact determination, we should have E sin 15= lot 
 -nearly, and the formula becomes 
 
 x sin 
 ft 
 
 which would generally be sufficiently approximate. The value 
 of 8, if not known by a contemporaneous observation of the 
 star's polar distance, might for certain stars be taken from the 
 Table of Apparent Places of cireumpolar stars in the Nautical 
 Almanac, The adopted value of x, if used for arcs exceeding 
 that for one micrometer-revolution, should be obtained from a 
 formula in which n is a considerable integer 1 . 
 
 60. The foregoing processes only determine the error of 
 collimation of the middle wire, whereas the transits are 
 generally taken at all the wires. We must, therefore, find the 
 collimation-correction for the position corresponding to the 
 mean of all the clock-times of transit, which is, in fact, to take 
 into account the collimation error of each of the wires. This 
 mean correction may be inferred from that for the middle wire 
 as follows. Recurring to Figure 14, suppose PN to be coinci- 
 dent with the mid-wire, bcrb' now to be the position of any 
 other wire, and transits of Polaris, or some other known star 
 near the pole, to be taken at both wires. Then, a- being the 
 point of transit of bab' by the star, the arc crJV^ of a great circle 
 perpendicular to PN measures the horizontal, or equatorial, 
 interval between the wires, because it is the projection on the 
 heavens of the rectilinear perpendicular let fall from the point <r 
 on the mid-wire. Hence putting loa for the arc crN, S for the 
 star's polar distance Per, and t for the interval between the 
 transits in seconds of time, we have by Spherical Trigonometry, 
 
 sin 15 =sin sin lot, 
 from which formula a, the interval between the wires in seconds 
 
 1 The value 17",06, used in all instances of the reduction of micrometer- 
 measures taken with the Cambridge transit instrument, was adopted by Mr Airy 
 in 1829, and subsequently proved by myself to be sufficiently exact. In no ca? 
 is it used for arcs that are not very small. 
 
THE TRANSIT INSTRUMENT. 51 
 
 of time for an equatorial star, may be calculated. The same 
 calculation is required for each of the wires, and as accuracy in 
 these determinations is important, they should be mean results 
 from repeated observations. 
 
 61. Naming the wires A, B, C, D, E, F, Gf, and supposing 
 a star above pole to cross them in this order when the micro- 
 meter-head is westward, let a, 6, c, be the calculated 
 equatorial intervals of the first three, and e, f, g those of the 
 last three, from the middle wire Z>, and let x be the unknown 
 interval of D from the mean of all. Then 
 
 (a* -a) + (x- b) +(x -c) +x+ (x + e) + (x+f) + (x + g) = 0; 
 /- x = j(a + b + e-e-f-g). 
 
 According as this result is positive or negative, the star passes 
 the mean of the wires after or before passing the mid- wire, and 
 the sign of the correction to be added to the collimation-correc- 
 tion for mid-wire is plus or minus. For the transit of a star 
 below pole the signs have to be changed ; and when the 
 micrometer-head is eastward, the sign is the opposite to that 
 under the same circumstances for micrometer-head westward. 
 
 After determining the intervals a, Z>, c, e } f, g by a 
 large number of observations, and calculating the value of x, 
 the equatorial intervals (x a), (x 6), &c. of the several wires 
 from the mean of all may be readily calculated. 
 
 62. I proceed next to give an instance of the numerical 
 calculation of the correction of error of collimation for the mean 
 of the wires, taking, first, the case of a determination by 
 employing a North mark and a South mark, and reversing the 
 instrument (see Arts. 51, 52, 53). The selected instance was 
 actually made use of in the reduction of some observations 
 taken with the Cambridge transit instrument. It is to be 
 understood that each of the subjoined micrometer-readings for 
 bisection of a mark consists of an integral number of complete 
 revolutions of the micrometer (signified by the letter r), together 
 with decimal parts of one revolution (see Art. 31), and 
 
52 PRACTICAL ASTRONOMY. 
 
 that it is the mean of the recorded readings for, at least, six 
 bisections. 
 
 Bisection of Bisection of 
 
 South Mark. North Mark. 
 
 r, r. 
 
 Micrometer-head eastward 25,784 17,948 
 
 Micrometer-head westward 22,626 ...... 30,495 
 
 Mean of each set of two readings 24,205 24,221 
 
 Mean of the four readings, or concluded j 
 reading for plane of colliniation / 
 Micrometer-reading for coincidence with D 24,156 
 
 Difference 0,057 
 
 Hence a.s l r = 17",06, the error of collimation of D = 0,057 
 x 17",06 = 0",97. As the micrometer-readings increase towards 
 the micrometer-head, the true plane of collimation after the 
 reversion was more westward than D, and the images of stars 
 passed D after passing the plane of collimation. Hence, 
 micrometer-head westward, the collimation-correction for D 
 was 0",97. Also it was Imown from previous calculation 
 that at the time the me&n of the wires was more westward 
 than D by 0",51, or that the value of x (Art. 61) was + 0",51. 
 Hence by the rules in Art. 61, the collimation-correction for the 
 mean of the wires = - 0",97 + '0", 51 = - 0",46, for transits after 
 the reversion, and + 0",46 for transits before the reversion. 
 
 63. It is usual to include in the correction of the error of 
 collimation that of the small error due to diurnal aberration. 
 The effect of this aberration is to throw the apparent places of 
 celestial objects in the direction of the earth's rotation, and 
 therefore eastward. The transits of stars above pole being 
 thereby delayed, the sign of the correction for such positions is 
 minus. The amount of the correction may be inferred from 
 that for the annual aberration by multiplying the latter by the 
 ratio of the velocity of the observer due to the earth's rotation 
 to the mean velocity of the earth in its orbit. Let H be the 
 mean radius of the ear.th's orbit, a the earth's equatorial radius, 
 r the geocentric radius of the place of observation, X the 
 
- /t 
 
 THE TRANSIT INSTRUMENT. 53 
 
 [/ . 
 
 geocentric latitude, and n the number of sidereal days in the 
 
 sidereal year. Then that ratio is -^ cos X, or -^ . - cos X. Taking 
 
 JtC -tl Q> 
 
 8",95 to be the Sun's Equatorial Horizontal Parallax, this being 
 
 Wlfj 
 
 the value most recently obtained, it will be found that -= is the 
 
 number whose logarithm is 8,20118, or, as often expressed for 
 brevity, [8,20118]. Hence adopting 20",445 for the constant of 
 annual aberration, as determined b} T Struve, the general ex- 
 pression for the correction for diurnal aberration is [9,51176] 
 
 - cos X. For the Cambridge Observatory - = [9,99909] and 
 a & 
 
 X = 5 2. 2'. Hence the correction for that Observatory, or any 
 place having the same latitude, is 0",20. 
 
 64. The method of determining collimation-error by two 
 collimators, one to the North and the other to the South, with- 
 out reversing the instrument, is virtually to make use of two 
 marks at an infinite distance, and to secure that the axes of the 
 beams by which they are seen in the transit telescope shall be 
 strictly parallel, and approximately in the same straight line 
 (see Arts. 54 and 55, arid Fig. 13). In this case the micrometer- 
 reading for plane of collimation is simply the mean between the 
 two readings for bisection of the marks, whether by a micrometer- 
 wire or the mid-wire, and the amount and sign of the collimation- 
 correction for the mid-wire are found by the processes explained 
 in Arts. 55 58. Then the additional correction for the mean of 
 the wires might be determined precisely in the manner stated in 
 Arts. 60 and 61. According, however, to present practice with the 
 Greenwich and Cambridge Transit-circles, this last operation is 
 not performed, the transit-times at the several wires being in all 
 cases reduced to times of transit across the middle wire. 
 
 65. The foregoing methods of determining collimation- 
 error are such as could hardly be put in practice except at well- 
 appointed fixed observatories. There are, besides, at such 
 observatories means of obtaining exact determinations of the 
 error of collimation by making use of the collimating eye-piece, 
 but as these are methods requiring independent determinations 
 
PKACTICAL ASTRONOMY. 
 
 of the Level error, they cannot be fully treated of till the 
 correction of that error is under consideration. I propose, 
 therefore, at present only to indicate a process which might be 
 available for a portable, or temporarily mounted, transit instru- 
 ment, or alt-azimuth instrument. In this method the instru- 
 ment is, first, to be placed so that its plane of collirnation is 
 approximately coincident with the plane of the meridian. This 
 might be done in the case of the transit instrument, possibly 
 with sufficient accuracy, by using a spirit-level for horizontally 
 of the axis of motion, and determining the direction of the 
 meridian by a compass, or by the process described in Art. 10. 
 Instead of this roughly approximate, but independent, method, 
 if the Nautical Almanac and a timed chronometer were at 
 hand, the time at which Polaris would be on the meridian 
 might be ascertained, and by pointing the telescope to the star 
 at that time, the axis being levelled, the instrument would be 
 placed in the required position. For an alt-azimuth the 
 process would be to adjust nearly the vertical and horizontal 
 axes of motion by spirit-levels, and then determine the 
 meridian direction by equal altitudes of the same star before 
 and after its passage across the meridian. 
 
 66. The instrument having been thus put in position with 
 sufficient exactness, transits of Polaris, or some other slow- 
 moving star, are to be taken in the following manner. In 
 Figure 15, ADD' A is the star's path above pole, as seen in the 
 
 Fig. 15. 
 
 field of view, Aa is one of the transit wires, and Del is the 
 middle wire. Transits are taken at Aa and Del, after which 
 the instrument is reversed. After the reversion, whereby Aa 
 
THE TRANSIT INSTRUMENT. 55 
 
 comes to A' a and Dd to D'd', a transit is taken at A' a. Let 6 
 be the time-interval from A to D, and ff that from D to .A'. 
 Then, according to the figure, & ' - 6 is the time of describing 
 DD', and if 8 be the star's polar distance, 
 
 half the small arc DD' = + 15 (^"^) s i n 8, 
 
 Z 
 
 which is the correction of collimation of Dd. If the interval 6 be 
 greater than &, the positions of Z>d and D'd! would be reversed, 
 and the sign of the collimation- correction, as also that of the 
 above formula, would be changed. When the star is below pole, the 
 apparent path is in the opposite direction, and the sign of the 
 formula is changed on this account. Hence for the position the 
 telescope had before the reversion, the general expression for 
 the collimation-correction of Dd is 
 
 sm 
 
 the + or sign applying according as the star was above or below 
 pole. For the position after the reversion the same formula is 
 to be used, with the signs changed. 
 
 Mechanical correction of Level Error by the Spirit-level. 
 
 67. Level Error, which may be defined to be deviation of 
 an axis of motion from horizontality, may be corrected either 
 by means of a plumb-line, or a Spirit-level. The former method 
 will eventually come under consideration, but since, as respects 
 the transit instrument, the other is most commonly used, it will 
 suffice to give here only an account of the construction of the 
 Spirit-level) and the mode of using it. It consists mainly of a 
 cylindrical piece of glass in which is formed a uniform bore, 
 nearly filled with spirit or ether, and closed at both ends. The 
 axis of the bore is slightly curved, so as to be a small part of a 
 circle of large diameter. There exist mechanical means of 
 satisfying with great exactness the conditions required for 
 trustworthy indications of the Level, namely, that the trans- 
 verse section of the bore be uniform, and its axis be of uniform 
 
56 
 
 PRACTICAL ASTRONOMY. 
 
 curvature in one plane. The sensibility of the Level is inversely 
 proportional to the curvature of the bore, and it is, therefore, 
 desirable to obtain a measure of the amount of this curvature. 
 The following process gives the means of doing this. In 
 Figure 16, is the centre of curvature of the upper boundary 
 
 Fig. 16. 
 
 Q' 
 
 of a section of the bore by a vertical plane containing the axis, 
 and Oc drawn vertically upwards passes through the central 
 point of the bubble ab, which, as being the space not filled with 
 the fluid, occupies the uppermost part of the bore. The tube 
 being supposed to receive a small angular displacement in the 
 direction from A towards B, whereby the bubble is made to 
 pass from the position ab' relative to the bore to the position 
 ab, let Oc' be drawn bisecting the original interval between the 
 ends a' and b' of the bubble. The motion may be assumed to 
 take place wholly about the centre of curvature 0, because if 
 that point were carried by the disturbance to some position 0', 
 
THE TRANSIT INSTRUMENT. 57 
 
 it may be conceived to be brought back to by a motion of 
 translation, without change of the place of the bubble in the 
 tube. From these explanations it will be seen that 
 
 da = b'b = c'c, 
 
 and that if Oc = r and the z c Oc or, we have ra = cc'. Hence r 
 may be calculated if a and cc' be given. Taking, for example, 
 the Spirit-level of the Cambridge transit instrument, let cc be 
 the interval between the scale divisions, which is one-tenth of 
 an inch. Then, the corresponding value of a in arc having been 
 ascertained to be 1",3 (by means that will be subsequently 
 stated), we have for calculating the value of r, 
 
 O in ,l x 206265 
 
 r = 
 
 1,3 
 
 which will be found to be 1322 feet. This length, the inverse 
 of which measures the curvature of the axis of the bore, indicates 
 a high degree of sensibility of the Level 1 . 
 
 68. In order that a Spirit-level may be applied for 
 measuring small inclinations to a horizontal plane, it is ne- 
 cessary that the position of the bubble which corresponds to 
 horizontality should either be actually marked, or should be 
 implicitly determined. This may be called the adjustment of 
 the Level. Let us, at first, suppose the Level to have an 
 adjusting screw and marks, but no scale; which is the case of 
 most of the Spirit-levels pertaining to small or auxiliary in- 
 struments. In Figure 17 (1) and (2), such a Level is represented . 
 in reverse positions, with marks on the glass tube and an 
 adjusting screw at 8. In (1) the Level is applied to the 
 plane db, m and n are the bubble-ends, after the bubble is 
 brought by turning the screw (which alters only the inclination 
 of the glass) to be, according to the judgment of the eye, in 
 mid-position relatively to the marks, and ac is a straight line 
 
 i In the observations made by W. Struve for determining the constant of 
 aberration with a transit instrument in the prime vertical, a Spirit-level was 
 used, the sensibility of which was measured by a radius of 1527 feet, the scale- 
 interval being one French ligne (= 0,0888 in.) and its value in arc not sensibly 
 differing from 1". (See Astranamisclie NacJirichtcn, 1844, No. 468, col. 202.) 
 
58 
 
 PRACTICAL ASTRONOMY. 
 
 drawn parallel to the tangent to the bubble at its lowest point, 
 and therefore horizontal. Conceiving the line ac to be fixed, 
 relative to the brass case containing the glass, we may take ad 
 
 Fig. 17. 
 
 (1) 
 
 c 
 
 
 
 
 
 a,' 
 
 
 (2) 
 
 
 d 
 
 / 
 
 i 
 
 (3) 
 
 and cb to be legs supporting the Level ; and drawing be hori- 
 zontally, the angle dbe of inclination of the plane to the horizon, 
 which the Level is required to indicate, will be subtended by de 
 the excess of cb above ad. Now let the Level be reversed. 
 Then, as shewn in (2), on applying it to the same plane db, 
 ac comes to ca', ad to a'b, cb to cd, and the bubble-ends move 
 up to the positions m' and n. This movement of the bubble, 
 being due to the angular change of position of ac, measures the 
 inclination of ac to the horizon. Draw be horizontal, and 
 bf parallel, to ac. Then fd = cd a'b = cb ad = de = de. 
 Consequently the angle fbe, which is equal to the inclination 
 of ca' to the horizon, is double the L dbe of inclination of the 
 
THE TRANSIT INSTRUMENT. 59 
 
 plane. Hence if the bubble be brought half-way back by 
 turning the screw, its deviation from mid-position will measure 
 an inclination to the horizon equal to that of the plane, and 
 will, therefore, indicate truly. If after this adjustment the 
 Level be applied to a plane in reverse positions, the places of 
 mark and bubble-end, both on the right-hand and on the left- 
 hand, should be the same after as before the reversion. If this 
 condition be not satisfactorily fulfilled, the operation for ad- 
 justing the Level must be repeated. It is evident that after 
 completing the adjustment, a plane is shewn to be horizontal 
 if, when the Level is applied to it, the bubble stands in a 
 central position relatively to the marks. 
 
 69. We have now to take the case of a Level which, 
 instead of having marks, has scales of equal parts so placed 
 that the positions of the bubble-ends may be read off in scale- 
 intervals. Figure 17 (3) and (4) will answer the purpose of 
 describing the construction and mode of using such a Level, 
 although the drawings were not made to represent any par- 
 ticular instance. There are two ways of arranging the scales 
 for recording positions of the bubble-ends : (1) the graduation 
 is along the edge of a single ivory plate fastened to the brass 
 case of the Level in such manner that the edge is parallel and 
 close to the axis of symmetry of the part of the surface of the 
 glass tube which is left uncovered for seeing the bubble. In 
 this arrangement the graduation-readings increase alike in both 
 directions from a fixed zero which is situated at, or near, the 
 centre of the aperture. This is the case of most Levels not of 
 large size. (2) Two scales a and 6 are graduated, as represented 
 in Figure 17 (3), along their edges, and can be shifted by hand 
 on the bar cd, which is fastened to the brass covering of the 
 Level. According to this arrangement, which allows of putting 
 the scales on each occasion of levelling in positions suitable for 
 reading off, a small number of graduation-intervals suffices. 
 On each scale the graduation-numbers increase in the direction 
 from the middle of the bubble towards the end. In the cases 
 of both arrangements the graduation on one side of the ivory is 
 the counterpart of that on the other, for the sake of reading off 
 
60 PRACTICAL ASTRONOMY. 
 
 alike in reverse positions of the Level without change of position 
 of the observer. 
 
 70. Taking first the case of a fixed graduation-scale, let x 
 and y be the scale- readings for the bubble-ends before inverting 
 the Level, and let x and y be the readings in the same direc- 
 tions as x and y respectively after the inversion. Then if h be 
 the scale-measure of the angle made by the plane to which the 
 Level is applied with a horizontal plane, and the index errors 
 for the arbitrarily selected positions of the scales be a and b, we 
 shall have from the readings before and after the inversion 
 respectively, 
 
 (6 + y) and 2A = (6 + x) - (a H- y'), 
 
 the reading for one bubble-end being as much increased or 
 diminished by a change of level as that for the other is dimin- 
 ished or increased, and the index errors being the same for the 
 same halves of the scale in both positions of the Level. By 
 adding the two equations a and b are eliminated, and the result 
 is 
 
 h - 1 ( x ^^ _ #jj~ y\ 
 
 9 \ 2 2 / * 
 
 Hence, as the bubble runs towards more elevated parts, ac- 
 cording as h is positive or negative, the plane is inclined 
 upwards or downwards in the direction in which the x'& are 
 measured. If by turning the adjusting screws (in the manner 
 indicated below in Art. 71) the reading for the bubble-end in 
 
 x -f- x 
 the ^-direction be made = , the reading for that in the 
 
 At i fk t 
 
 ^-direction becomes ; and accordingly the Level is ad- 
 z 
 
 justed so that on applying it to any plane, half the excess of 
 one reading above the other measures the angle of elevation of 
 the plane on the side of the greater reading, and the plane 
 would be shewn to be horizontal by equal readings. 
 
 71. Figure 17 (4), which represents a vertical section of 
 that support of the Level in which the screw e works, shews the 
 
THE TRANSIT INSTRUMENT. 61 
 
 apparatus by which a change of inclination of the glass tube is 
 effected. A piece of brass/ in close contact with the vertical 
 sides of the concealed aperture gh, is attached to one end of the 
 glass tube, and is held in position by the screw e, and an 
 antagonist screw, the capstan-head of which is situated within 
 a rectangular aperture cut through the brass support. It is 
 evident that by this contrivance, when one screw is turned to 
 take off its bearing, by the action of the other the piece /, 
 together with the connected end of the tube, may be either 
 elevated or depressed. After the bubble has been brought by 
 these operations into the position proper for the adjustment of 
 the Level (see Art. 70), both screws should be made to bear on 
 f. The same figure exhibits also the form of the Y at the foot 
 of each support of a Level of this kind, which i-s called a striding 
 Level. 
 
 72. When there are two movable scales, after taking, as 
 before, bubble-end readings in reverse positions of the Level, 
 we can, by shifting the scales^ make the reading in the x - 
 
 x -\- x' 
 direction equal to ^ , and that in the ^/-direction equal to 
 
 ^~. Then so long as the scales retain the positions thus 
 
 m 
 
 given to them, the level error might be inferred from scale-read- 
 ings for the bubble-ends just as in the case of the adjusted fixed 
 scale considered in Art. 70. Suppose now the Y's at the feet of 
 the Level (Art. 71) to be applied to the cylindrical pivots of a 
 transit instrument, and from bubble-end readings thus obtained, 
 let the positions of the scales be altered in the manner just 
 indicated. Then by turning the screws provided for correcting 
 the Level error of the instrument (see Art. 46), the readings for 
 the bubble-ends may be made equal; which being done, the 
 Level error would be, at least approximately, corrected. It 
 might, however, happen that after these operations the bubble 
 would not be in mid-position relatively to the aperture in the 
 brass ; and since it is desirable, as regards the process of 
 levelling, that this condition should be nearly fulfilled, the 
 adjusting screws of the Level have to be turned for this purpose. 
 
G2 PRACTICAL ASTRONOMY. 
 
 After these initial approximate adjustments both of the Level 
 and of the axis of motion of the transit instrument, there is no 
 need, so long as the Level error continues of small amount, to 
 take into consideration index errors of the scales, provided always 
 the Level error be calculated, according to the formula for h 
 given in Art. 70, from readings taken in reverse positions of 
 the Level. After any pair of such readings, the places of the 
 scales might be altered ad libitum for the sake of convenience 
 in reading off. 
 
 73. On the supposition that the pivots of a transit 
 instrument are exactly cylindrical and equal, the process for 
 determining the Level error by a Spirit-level is unaffected by any 
 small inequality either of the angles of the Y's of the piers, or 
 of the angles of the Y's of the Level. For if the former angles 
 are unequal, the axis of motion will still have a fixed position, 
 the same after as before a reversion, and its inclination to a 
 horizontal plane is determined by the usual mode of applying 
 the Level ; and if the angles of the Y's of the Level are unequal, 
 the displacement of the bubble due to that circumstance in 
 either the a? or the y direction before the inversion of the Level 
 will be just equal, and, as respects the graduation, opposite to 
 the displacement after inversion. Hence the sums x + x and 
 y + y, and consequently the calculation of Level error, will be the 
 same as if the inequality did not exist. This reasoning shews 
 the necessity of always determining Level error by readings 
 taken in reverse positions of the Level, as the angles of its Y's 
 can only be supposed to be approximately equal. 
 
 74. But it is possible that the determination may be 
 affected by inequality in the size of the cylindrical pivots, and to 
 make the logic of the method complete, it is required to indicate 
 means of correcting any error that may be due to this cause. 
 The purpose of the following reasoning is to obtain a formula for 
 calculating such correction. In Figure 18, ABE' A is a trans- 
 verse section of either cylindrical pivot, cutting its axis in ; 
 the z BHA (= 2a) is the angle of the Y of the Level, and the 
 z BH'A (= 2a') is the angle of the Y of the pier (both supposed 
 to be known by actual measurement), and a common vertical 
 
THE TRANSIT INSTRUMENT. 
 
 63 
 
 line HOH' bisects both angles. Hence, HA being a tangent to 
 the circle at A, OH= OA cosec a. So also OH' = OA cosec a'. 
 
 Let S . OA be the difference of the radii of the pivots, S . OH' 
 the consequent difference of elevation of the axes of the pivots, 
 and S . OH the consequent difference of elevation of the 
 corresponding ends of the Level above the common axis of the 
 pivots. Then, 
 
 S . OH _ S . AO cosec a _ cosec a _ , 
 8.0H' = S.AO coseca' ~ coseca' = ' 
 
 k being a known quantity because a and a' are known. Next 
 let x + y be the true Level error of the axis of the transit 
 instrument in scale-measure before reversing it, y being the part 
 due to inequality of the pivots. Then as only this part changes 
 sign by the reversion (Art. 73), the true Level error after the 
 reversion is x y. Now 8 . OH', as expressed in scale-measure, is 
 the portion of Level error due to inequality of the pivots, and is 
 therefore equal to y. Hence 8. OH, expressed in the same 
 measure, is ky. By this quantity the effect of the inequality of 
 the pivots on the inclination of the axis is apparently increased, 
 because of the simultaneous effect in the same direction on the 
 inclination of the Level. Let, therefore, m and n be apparent 
 Level errors obtained, by calculating according to the formula for 
 
64 PRACTICAL ASTRONOMY. 
 
 h in Art. 70, from indications of the Spirit-level in inverse 
 positions recorded before and after the reversion of the instru- 
 ment. Then, since x -f y and x y are the true Level errors 
 before and after the reversion, we shall have 
 
 n=xy ky. 
 
 These equations give for the value of ky, ^r7^--j\, which is the 
 
 2 (L + k) 
 
 formula for calculating the correction for inequality of the 
 pivots, inasmuch as ky subtracted from m gives the true level 
 error before the reversion, and added to n gives the same after 
 the reversion. As the pivots are made by mechanical construc- 
 tion as nearly equal as possible, this correction will at first be 
 of very small amount, and will afterwards only change slowly 
 by unequal wear of the pivots. It should, therefore, be deter- 
 mined by repeated trials, levellings being taken for that purpose 
 before and after each reversion, and a value of the correction 
 (apart from its sign) should be calculated from each trial. The 
 adopted value may then be the mean of several determinations. 
 The proper sign of the correction, as respects the transit obser- 
 vations, will presently come under consideration. 
 
 75. As the formula for h in Art. 70 gives the Level error in 
 scale-intervals, it is necessary, in order to express this quantity 
 in celestial arc, to find the value in arc of the unit of the scale. 
 In an observatory furnished with a Mural Circle, the readiest 
 method of doing this is to attach the spirit-level firmly to the 
 top of the circle, and after noting the scale-readings of the 
 bubble-ends in two positions of the circle separated by an arc 
 ascertained by microscopic reading of the graduation (as here- 
 after described) in each position, to compare this arc with the 
 difference of the Level readings. The value of the scale-interval 
 may then be calculated by a simple proportion. (In this man- 
 ner Mr Airy obtained on October 10, 1834, the value 1",3 
 mentioned in Art. 67, having previously adopted 1",0 on the 
 authority of the maker of the Level.) It is evident that with 
 the same means the regularity of the indications of a Level 
 might be tested by comparisons of its readings with a series of 
 corresponding circle readings differing by a constant arc. 
 
THE TRANSIT INSTRUMENT. 65 
 
 76. If these means should not be available, the required 
 information may be obtained, although less perfectly, by an 
 instrument called a Bubble-trier, constructed in the following 
 manner. A brass bar, resting at one end on two legs at the 
 extremities of a short cross-piece forming with it a T-shaped 
 frame, is tapped at the other end for a screw to work in, which 
 serves for a third leg. The intermediate part is furnished with 
 two Y-supports on which the glass tube with its attached scale 
 is placed. When the instrument rests with its three feet on 
 a horizontal plane, the screw by being turned alters the eleva- 
 tion of the end in which it works, and consequently the inclina- 
 tion of the tube to the plane and the place of the bubble. Also 
 to the stem of the screw is attached a graduated circular disk, 
 the angular movement of which is read off from the graduation 
 by the vertical edge of a brass index contiguous to the circum- 
 ference of the disk and fixed to the bar. The purpose of the 
 instrument is to compare arbitrary equal angular movements of 
 the disk as indicated by the graduation, and by consequence 
 equal differences of inclination of the Level, with the corre- 
 sponding differences of scale-readings for the bubble-ends. If 
 the latter differences be equal as well as the other, the perform- 
 ance of the Level is satisfactory. The maker of the instrument 
 also arranges the graduation of the disk, with reference to the 
 ratio of the interval between the turns of the screw to the length 
 of the bar, in such manner that differences of readings from the 
 disk give the actual differences in arc of the inclination of the 
 Level. It is evident that by this apparatus the value of the 
 scale-interval in arc may also be determined. 
 
 77. The sign of the correction for inequality of the pivots 
 is determined by the following considerations. In the expres- 
 
 n \ WY) u 77 I 
 
 sion , n ; for Tcy obtained in Art. 74, m and n are the values 
 A (^1 H~ K>) 
 
 of the level-correction h before and after the reversion of the 
 instrument (as obtained by the process exhibited by the example 
 given in the next article), and the difference between m and n 
 depends only on the inequality of the pivots. According as m 
 c, 5 
 
66 PKACTICAL ASTRONOMY. 
 
 is greater or less than n, the plane of collimation is more east- 
 ward before than after the reversion, and the pivot that was 
 westward before the reversion is greater or less than the other, 
 inasmuch as the eastward deviation of that plane is greater the 
 greater the westward pivot. This rule determines which is 
 the greater pivot. For instance, as respects the Cambridge 
 instrument, it was found by many trials made before and after 
 the date of the subjoined example, that the smaller pivot is 
 that at the end perforated for the illumination apparatus. When 
 therefore this end is eastward, the inequality increases the 
 apparent amount of the level-correction (h) partly by increasing 
 the inclination of the instrument and partly by increasing the 
 inclination of the Level. The latter part only requires to be 
 corrected, which is done by calculating its amount according to 
 the expression for ky above, from data obtained at several 
 reversions of the instrument, and applying an adopted mean 
 value with a negative sign (see Art. 74). When the smaller 
 pivot is westward, the correction is the same with contrary sign. 
 Thus for the Cambridge instrument the correction for inequality 
 of the pivots is negative or positive according as the illumination- 
 end of the axis is eastward or westward. 
 
 78. I propose to give an example of the calculation of level- 
 error from data furnished by the spirit-level, after making the fol- 
 lowing preliminary statements. First, the Level has a small cross- 
 level attached to it near one end, with marks on the glass for 
 putting its bubble in mid-position. This is done before every 
 scale-reading of the large Level, in order that the plane of the 
 axis of its bore, if not exactly vertical, may deviate from 
 verticality always to the same amount, that thus the readings 
 may be all consistent with each other. [The same use should 
 be made of the cross-level in determining the value of the scale- 
 interval by the method described in Art. 75.] Also this cross- 
 level gives the means of indicating the position of the large 
 Level as to East and West. Again, it is usual, for greater 
 certainty, to infer the level error from several sets of double 
 levellings, and accordingly, as may be seen from the expression 
 
THE TRANSIT INSTRUMENT. 67 
 
 for h in Art. 70, if there be n such sets, the mean value h is 
 given by the equation 
 
 If we now assume, regard being had (as in Art. 57) to the effect 
 of the inclination of the axis on the time of a transit above pole, 
 that this angle is positive or negative according as the West or 
 East end of the axis is highest, the a-readings will pertain to the 
 west end of the bubble, because the bubble rises towards higher 
 parts, and the sign of h will be that proper for correcting the 
 effect of the error. After these explanations the subjoined 
 instance of calculating the correction of level error from data 
 extracted from the records of the Cambridge Observatory will be 
 easily understood. 
 
 1850, Oct. 21, 2 h . Illumination-end of the axis Eastward. 
 
 Cross Level E 
 
 East reading. 
 d. 
 9,9 
 
 West read 
 d. 
 . 11,2 
 
 W. . 
 
 8,3 . 
 
 . 13,0 
 
 
 . ... 105 
 
 . 11,3 
 
 W. . 
 
 8,1 . 
 
 . 13,0 
 
 
 .... 10,1 
 
 11,3 
 
 W 
 
 7 9 
 
 . 13,0 
 
 
 
 2. x -72,8 
 
 Hence = - x ~ 9 y = l',50 = 1",95, because l d = 1",3. 
 
 The correction is thus shewn to be positive, the sum of the West 
 readings being greater than the sum of the East readings. By 
 a reversion of the instrument on July 16, 1850, the amount 
 of correction for inequality of the pivots was found to be 
 0",408, and by a reversion on May 6, 1851, 0",491. Adopting 
 the mean of these for application to the levelling on Oct. 21, 
 1850, applying it with a negative sign because the illumination- 
 end was eastward, and including the correction 0",20 for diurnal 
 aberration, the concluded level-correction is 
 
 + 1",95 - 0",45 - 0",20 = + 1",30. 
 
Year. 
 
 1828 
 
 Correction. 
 
 rr 
 
 + 018 
 
 Year. 
 1843 
 
 Correction. 
 n 
 
 -0 331 
 
 Year. 
 1849 
 
 1829 .. 
 
 .... +0,13 
 
 1844 
 
 .. -0,28 
 
 1850 
 
 1837 
 
 -004 
 
 1845 
 
 . . -0 26 
 
 1851 
 
 1839 
 
 -0 10 
 
 1846 
 
 .. -0,22 
 
 1852 
 
 1842 .. 
 
 . -0.19 
 
 1847 
 
 .. -0,22 
 
 1854 .. 
 
 68 PRACTICAL ASTRONOMY. 
 
 79. In order further to ascertain the character of this 
 correction for inequality of the pivots, I subjoin here the values 
 obtained for it in the use of the Cambridge transit instrument 
 from 1828 to 1854. In every instance the correction is given 
 for the case of illumination-end eastward. 
 
 Correction. 
 // 
 
 -0,48 
 -0,53 
 -0,49 
 -0,68 
 
 The first two values were obtained by my predecessor Mi- 
 Airy, the others by myself. That opposite the year 1843 was 
 considered doubtful, as being the mean of three discordant 
 values. The continuous change of value is most probably to be 
 ascribed to dissimilar wear of the pivots due partly to turning 
 the instrument for observing, and partly to the application of 
 the Y's of the level to the pivots in the operation of levelling, 
 which was most commonly performed with the telescope 
 horizontal and pointing southward. By levellings taken in 
 1838 and 1839 with the telescope inclined from the Zenith 45 
 northward or southward I found that the additional correction 
 4- 0",30 was required. These circumstances all point to the 
 desirability of being able to calculate the level error by some 
 method that shall be independent of the form of the pivots, and 
 not require the use of the spirit-level. Such a method I put 
 in practice at the beginning of 1850, making use also from that 
 time of the collimating eye-piece for determinations both of 
 collimation error and level error. The spirit-level was used as 
 an auxiliary to the end of 1854, but the adopted level errors 
 virtually rested on data furnished by the collimating eye-piece. 
 After 1854 the use of the spirit-level was wholly discontinued. 
 The above-mentioned method of making the transit-observations 
 independent of the forms of the pivots cannot be fully described 
 till the meridian or azimuth error has been under consideration. 
 
THE TRANSIT INSTRUMENT. 69 
 
 It will also be previously necessary to describe the collimating 
 eye-piece, and the method of using it for determinations of 
 collimation and level errors, which I now proceed to do. 
 
 The Collimating Eye-piece. 
 
 80. This important ^auxiliary instrument, which enables 
 the observer to obtain instrumental corrections exclusively by 
 optical means, was the invention of Bohnenberger of Tubingen, 
 who has given a description of it in the Astronomische Nach- 
 richten (Band IV., 1826, col. 327 336.) My attention was first 
 called to it by Henderson, late Astronomer Koyal at the Cape 
 of Good Hope, who brought me a specimen (made apparently 
 according to the above-mentioned description), having a 
 metallic reflector with a hole at the centre, through which the 
 wires and their reflected images were looked at with a Ramsden 
 eye-piece. On trial I found this construction to be extremely 
 inconvenient on account of the limited field of view, and the 
 small interval between the eye-glass and the wires, rendering it 
 difficult to hold a lamp for throwing light upon the reflector. 
 On mentioning these circumstances to the late William Simms, 
 he constructed for me the instrument represented by Figures 
 19 and 20, in which a three-glass eye-piece is substituted for 
 the Ramsden eye-piece, and for the metallic reflector a piece of 
 plate glass, the reflection from which, as will presently be 
 explained, gives the means of seeing the wires together with their 
 reflected images with quite sufficient distinctness. By these 
 changes the above-stated inconveniences were entirely removed. 
 As far as I am aware the collimating eye-piece has since been 
 uniformly made according to this pattern. I brought it into use 
 in the Cambridge Observatory in the year 1850 ; the next year 
 it was adopted at Greenwich, when the new Transit-circle was 
 first made use of. It had already attracted the attention of 
 Bessel, Gauss, and Lament, but had not, I believe, been 
 definitively employed for exact determinations relating to 
 meridian observations with the transit instrument and mural 
 circle, before I made such application of it at the Cambridge 
 Observatory. 
 
70 PRACTICAL ASTRONOMY. 
 
 81. Figure 19 represents the collimating eye-piece detached 
 from the telescope ; d is the plate-glass reflector, seen through 
 an aperture made in the tube to allow the light from an ex- 
 ternal lamp to fall on the reflector; m and n are two small 
 milled heads, one for turning the reflector to the required angle 
 of inclination, and the other for acting by a screw to fix it when 
 put in position. Figure 20 exhibits a vertical section of the 
 telescope and attached collimating eye-piece, made by a plane 
 containing the common axis, and cutting at right angles the 
 plane of the plate-glass reflector D. IT is a lamp, the light of 
 which, being incident horizontally on the reflector adjusted to 
 an inclination of 45 to the axis of the telescope, is sent down as 
 a divergent beam to the object-glass FOF, after transmission 
 through which it is reflected at the surface of mercury in the 
 trough GG. At a is a portion of a micrometer-wire situated at 
 the geometrical focus of the object-glass, and movable by the 
 micrometer-head M and opposing spring at N. Kegarding, for 
 the moment, the telescope as a collimator furnished with a 
 wire at a, if there were another collimator pointing oppositely 
 to this in a direction parallel to its axis, a dark image of the 
 wire at a would be formed at the focus of the second collimator 
 precisely in the manner explained in Art. 54. But by the reflec- 
 tion at the mercury the rays that go to form this image are 
 turned into the contrary direction, so that the telescope becomes, 
 in fact, its own collimator, and the image is formed at the 
 position a'. being the optical centre of the object-glass, Oa 
 is the axis of the incident centrical pencil, Oa that of the 
 reflected centrical pencil, and by reason of the reflection Oa 
 and Oa make equal angles with the vertical direction through 
 0. The wire at a and the image at a' are both seen distinctly 
 with the collimating eye-piece, provided the wire be accurately 
 at the geometrical focus of the object-glass (see next Art.). 
 Figure 20 represent the form of a pencil by which a point of 
 the image is seen, its course being, first, through the plate-glass 
 Z>, then through the third lens (7, the second lens B, and the 
 first lens A, to the eye at E. This may be called a negative 
 pencil, as originating in a stoppage of light from the lamp by 
 
THE TRANSIT INSTRUMENT. 71 
 
 Fig. 19. Fig. 20. 
 
 m 
 
72 PRACTICAL ASTRONOMY. 
 
 the wire at a. At the same time this wire, by reason of the 
 incidence of the light upon it, might be expected to be seen by 
 the eye at E by positive reflected pencils. This is found to be 
 the case when the telescope is directed to the dark sky. (I 
 have, in fact, taken advantage of this circumstance, by making 
 use of the collimating eye-piece to illuminate wires in a dark 
 field for taking transits of faint objects (see Art. 37). This 
 twofold manner of viewing the wires has given occasion to 
 misconception in some published accounts of the use of this 
 instrument.) But when the telescope is directed to the trough 
 of mercury, the moment the light reflected from the surface of 
 the mercury illumines the field, the wires appear dark on a 
 bright ground, because the very small amount of light that can 
 be reflected from fine cylindrical wires is extinguished by the 
 flood of the reflected light. Thus the wire at a and the image 
 at a' are both viewed by negative pencils, and under favourable 
 circumstances I have seen them so nearly alike, as to blackness 
 and definition, that it was hard to distinguish one from the 
 other. This, however, occurred with the telescope of the mural 
 circle, but in no instance with that of the transit instrument, 
 owing probably to some fault in the optical conditions of the 
 lenses of the object-glass. 
 
 82. The optical effect produced by the collimating eye- 
 piece may now be distinctly stated. In order that the Figure 
 might not be too complicated, the negative pencil from a is 
 represented only by its axis, which is exactly analogous to that 
 of the negative pencil from a'. Both the wire at a and the 
 image at a are to be regarded as origins of pencils which by 
 means of the lens at C, and the limiting diaphragm at c, form 
 distinct images at b, where also there is a diaphragm. These 
 images are viewed by the eye at E by means of the two lenses 
 A and B of an adjustible Ramsden eye-piece. It hence appears 
 that any movement of the wire at a by the micrometer-head M t 
 and the consequent movement of the image at a', are seen by 
 the eye at E, so that by turning the micrometer the wire and 
 its image may be brought into coincidence. To effect this 
 coincidence is the express purpose of the whole arrangement. 
 
THE TRANSIT INSTRUMENT. 73 
 
 For ensuring accuracy the image should be brought into contact 
 with the wire by just excluding the line of light between them 
 an equal number of times on each side, and recording the 
 micrometer-reading for each contact. Then the mean of all the 
 readings is the reading for coincidence of the image with the 
 wire. When the micrometer-wire is set to this reading, the 
 plane passing through it and the optical centre of the object- 
 glass is as nearly vertical as by the optical means it can be 
 made to be. These explanations may suffice to shew how an 
 essential astronomical element, namely, the direction of the 
 vertical at the place of observation, formerly found by a plumb- 
 line or a spirit-level, is determined by means of the collimating 
 eye-piece. 
 
 83. As it is important that the wire and its image should 
 be seen with equal distinctness, special means are provided for 
 satisfying this condition. This is done, as already intimated by 
 the reference in Art. 36 to Figure 6, by means of apparatus 
 capable of moving, in the direction of the telescope-axis, the 
 tube in which the frames of the fixed wires and the micrometer- 
 wire are held, and fixing it in the position for which, according 
 to the judgment of the eye, a wire and its image appear equally 
 well defined. When this is the case the wires will have been 
 placed at the geometrical focus of the object-glass by a method 
 (alluded to in Art. 54) which is particularly adapted to this 
 purpose, inasmuch as the deviations of the wire and its image 
 from focus are equal in opposite directions, as is evident from 
 optical considerations, and on that account any difference of 
 definition is the more easily recognised by the eye-piece. From 
 Figure 20 it is also to be seen that this is an inverting eye- 
 piece, resembling in that respect the micrometer-microscope for 
 reading off circle graduations. 
 
 84. The errors of collimation and level may be simulta- 
 neously obtained by measures taken with the collimating eye- 
 piece in reverse positions of the instrument, as the following 
 argument will shew. In Figure 21 is the optical centre of 
 the object-glass, the telescope pointing to the Nadir, and OZ is 
 the vertical direction ascertained by making the micrometer- 
 
PKACTICAL ASTRONOMY. 
 
 wire and its image coincide by the process explained in Art. 82. 
 Oc represents the plane of collimation, and OD is drawn from 
 to the position of the middle wire D before the reversion. 
 
 Hence the ^ ZOc ( = L) is the deviation of the plane of colli- 
 mation from the vertical, and therefore the level error, and the 
 t cOD ( = (7) is the error of collimation of D. The * ZQD ( = a) 
 is measured by the difference of micrometer readings for the 
 vertical direction and for coincidence with D. After the rever- 
 sion, by which D is brought to D', the ^ ZOD' ( a) is simi- 
 larly measured. Then since the angle cOD is, by the definition 
 of collimation error, equal to the angle c OD', we have 
 
 ot = C+L and a'= C-L, 
 Hence (7= 
 
 Thus C and L are found by calculating according to these 
 formulae, the values of a and a' in micrometer revolutions being 
 first converted into arc. For stars above pole, the sign of the 
 level-correction is positive when the plane of collimation deviates 
 from the vertical eastward, or the west end of the axis is 
 highest, and, as already stated, the sign of the collimation-cor- 
 rection is positive when the pointing of the telescope is east- 
 ward from the plane of collimation. The position of the micro- 
 meter-head being given, the micrometer readings indicate the 
 relative places of Z t c, and D. 
 
THE TRANSIT INSTRUMENT. /o 
 
 85. If L be ascertained by the spirit-level, we shall have 
 <7=a-A or (7=a' + Z; so that the collimation error might 
 be determined without reversing the instrument. It is, how- 
 ever, to be remarked that these values, whether obtained by the 
 spirit-level or the collimating eye-piece, would be affected by 
 inequality of the pivots or irregularities of their forms. On 
 this account, with prospective reference to a process (hereafter 
 explained) for obviating this source of error in the Cambridge 
 transit instrument, the level error was determined at each 
 reversion by both methods, the telescope in every case of level- 
 ling being horizontal and pointing southward. As it was 
 found that the two methods did not give the same results, from 
 the differences accruing from comparisons made in a certain 
 limited number of years, two mean differences were calculated 
 from time to time, one applying to the case of illumination east, 
 and the other to that of illumination west. By applying these 
 differences to level errors found by the spirit-level, it was con- 
 sidered that the level error was virtually determined by the 
 collimating eye-piece and reversing the instrument. This 
 course was adopted because it would have been inconvenient to 
 reverse the instrument on every occasion of rinding the level 
 error; and again, relatively to the proposed correction of the 
 effects of irregularities of the pivots, it was essential that the 
 level error should be uniformly determined by the collimating 
 eye-piece. Accordingly this mode of calculating the level error 
 of the Cambridge instrument was begun in 1850, when the 
 collimating eye- piece was first used, and was continued to 
 February 20, 1855 ; from which date the calculation was made 
 on a different principle in the following manner. It was con- 
 sidered that as the collimation-correction varies but slowly, the 
 same mean value of G might be used during a considerable 
 interval, and the values of level error be calculated during each 
 such interval by the formula L = oiC, or L=Ca.', the 
 variations of the value of L being taken account of by frequent 
 measures of a, or of a', in the same interval. In consequence of 
 the adoption of the above-mentioned processes, all the correc- 
 tions of collimation and level errors which I have employed in 
 
76 PRACTICAL ASTRONOMY. 
 
 the reduction of the Cambridge observations from the begin- 
 ning of 1850 to the end of 1860 are in effect determinations by 
 the collimating eye-piece. 
 
 86. The method of finding the errors of collimation and 
 level which, as being found in practice to be the most exact, is 
 adopted in well-appointed fixed observatories, is to obtain the 
 collimation error by means of a north and a south collimator, as 
 explained in Art. 54, and then to measure a or a' by the colli- 
 mating eye-piece. The values of (7, as well as those of a or a', 
 might in this way be frequently determined without the trouble 
 of reversing the instrument, and in using the formula L = a. 
 (7, or L=Ca } instead of adopting mean values of C, the 
 individual values would be used in order to take account of any 
 sudden or irregular variations which the collimation error 
 might be liable to. This process, as not requiring the instru- 
 ment to be reversed, is constantly employed in the use of the 
 transit-circles of the Greenwich and Cambridge Observatories. 
 A method of calculating corrections for the forms of the pivots, 
 applying alike to this mode of determining collimation and level 
 errors, and to that described in Art. 85, will come under consi- 
 deration after the discussion of meridian error. 
 
 87. The foregoing methods of finding collimation and level 
 errors have not required astronomical observation, being con- 
 ducted solely by instrumental means (with the exception of the 
 process given in Art. 66 for obtaining the collimation error of a 
 portable transit instrument by transits of Polaris). There is, 
 however, a method of ascertaining the level error by observing 
 transits of Polaris at meridian passage both directly, and by 
 reflection at the surface of mercury, whereby good results are 
 procurable without previously knowing either collimation or 
 azimuth error. It is evident that this method, combined with 
 the use either of north and south collimators, or of the collima- 
 ting eye-piece, would give the level and collimation errors with- 
 out reversing the instrument. But as it involves processes of 
 observing and calculating which properly belong to Division III., 
 the account of the details of the method is deferred till the sub- 
 jects of that Division are under treatment. At present we may 
 
THE TKANSIT INSTRUMENT. 77 
 
 proceed to consider by what means meridian or azimuth error is 
 calculated. 
 
 Calculation of Azimuth Error. 
 
 88. The azimuth al error is the deviation of the plane of 
 collimation, corrected for level error, from the plane of the 
 meridian, which is the plane that passes both through the 
 Zenith of the place of observation and the Pole of the heavens. 
 Consequently this error is necessarily determined by astrono- 
 mical observation. For the purpose, however, of obtaining 
 formulae for calculating azimuth error, it suffices to take for 
 granted that the observer is able to note with precision the 
 time, according to an astronomical clock, of the transit of a 
 star, whether near or distant from the Pole, across a wire of the 
 telescope. In Figure 22, P is the pole of the heavens, Z is the 
 
 Fig. 22. 
 
 Zenith of the place of observation, Znri the plane of collimation 
 passing through the Zenith, error of level as well as that of colli- 
 mation being supposed to be corrected, and an is the path of a 
 star between the plane of collimation and the plane of the me- 
 ridian. Let Per = 8, Za- z, / crZn = x = the azimuth error in 
 seconds of space. Then 
 
 an x sin z, t <rPn = on cosec B = x sin z cosec 8. 
 
 /y* 
 
 Hence ^ o~Pn (in time) = = sin z cosec B = hx, 
 
 15 
 
 h being put for sin z cosec 8. Let t be the noted clock -time 
 
 J.O 
 
 of transit of the star, corrected (according to formulae that will 
 
78 PRACTICAL ASTKONOMY. 
 
 be obtained under Division III.) for collimation and level errors, 
 A the star's actual apparent Right Ascension, and T the excess 
 of true sidereal time above the clock-time. 
 
 Then the true time of meridian transit =t + T+hx = A 
 The true time of transit of another star = t' + T+ r + h'x = A', 
 T being the clock's loss in the interval t' t. 
 
 A'-A + t-t'-r 
 
 From these two equations x 
 
 h'-h 
 
 which is the general formula for the calculation of azimuth 
 error. In order that it may give good results the stars should 
 be selected so that the denominator h' h may have a large 
 value. Hence one at least of the stars should be near the 
 Pole ; or, if both be near the Pole, h' and h must have different 
 signs, which will be the case if one observation be taken above 
 and the other below the Pole, because for a star below pole the 
 polar distance (S) is to be considered negative. 
 
 89. The general formula for azimuthal error may be safely 
 applied either in its original form, or as modified by given con- 
 ditions, in the four following cases. 
 
 (1) If one star be near the Pole and the other not far 
 from the Equator, one of the factors h and h' will be large, and 
 much larger than the other, so that ti h will not be small. In 
 using the general formula for this case the Apparent Right 
 Ascensions A and A' will have to be obtained by interpolation 
 from catalogues of computed values, such as those for Upper 
 Transits at Greenwich of a few stars very near the north or 
 south pole, computed in the Nautical Almanac for every day of 
 the year, and those for Upper Transits of a much larger number 
 distant from the poles, computed for every tenth day. The 
 Right Ascensions in the list of the Nautical Almanac severally 
 depend on a very large number of transit observations taken at 
 the Greenwich Observatory, and may safely be employed as 
 data for calculating azimuthal error. For these reasons the ap- 
 parent R. A. made use of for the calculation should be those of 
 stars included in that list. The correction r may be inferred from 
 the clock's rate as determined by comparisons of the clock- 
 
THE TRANSIT INSTRUMENT. 79 
 
 times of transit of the same star, or stars, on two or more con- 
 secutive days. 
 
 (2) If both stars be near the Pole, and their difference of 
 Eight Ascension be nearly 12 h , the transits may be taken, one 
 above and the other below the Pole, nearly at the same time, 
 and the correction T would then be so small that it might be 
 neglected. In this case h' h is large, because h and h' are 
 both large and of contrary signs, and the general formula 
 becomes 
 
 the + or sign applying according as A is greater or less than 
 A'. The stars 51 (Hev.) Cephei and 8 Ursae Minoris are 
 favourably situated for the application of this formula, and con- 
 temporaneous transits of these stars have not unfrequently 
 been used in computing azimuth error for the reduction of 
 Cambridge transit observations. 
 
 (3) If two transits of the same star near the Pole be taken, 
 one above and the other below pole, h' h will be large, 
 because for the transit below pole the polar distance is negative, 
 so that h and h' will have different signs. Also A' A is the 
 increment (e) of the star's Right Ascension in the interval t' t. 
 Hence adding 12 h according as t' is greater or less than t, the 
 formula for this case is 
 
 which is independent of the star's Right Ascension. The star 
 being supposed to be in the list of the Nautical Almanac, the 
 change (e) of the R. A. in the interval t' t may be inferred 
 from the computed apparent R. A. of the star at the times of 
 transit. 
 
 (4) If three consecutive transits of a star near the Pole be 
 taken, there will be another equation like the first, namely, 
 
 it being assumed that the corrections e and T, for change of the 
 star's R. A. and rate of the clock, are the same for the interval 
 of 12 h between the second and third observations as those for 
 
80 PKACTICAL ASTRONOMY. 
 
 the interval of 12 h between the first and second observations. 
 Hence the two equations give 
 
 2 h'-hx=2 
 
 the H- or sign to be used according as the sum of the first and 
 third clock times is less or greater, by about 24 h , than twice the 
 second. This result is independent both of the star's E. A., and 
 of the change of its R. A., as well as of the clock's rate. 
 
 90. As these Lectures are intended to answer the twofold 
 purpose of describing the best means at present available for 
 taking exact astronomical observations, and at the same time 
 indicating the fundamental principles on which the science of 
 Practical Astronomy has been, or might be, raised, I take occasion 
 to remark here, that if it were required to construct a Cata- 
 logue of the places of stars ab imtio, the fourth of the above 
 methods of calculating the azimuth error would be employed, 
 inasmuch as it is independent of any results of antecedent 
 observations. 
 
 91. Although, as we have seen, there is no necessity for 
 a meridian mark in an Observatory furnished like that of 
 Greenwich or Cambridge, still there are circumstances in the 
 use of astronomical instruments in which a fixed distant mark 
 is serviceable, if not necessary. I propose, therefore, to shew 
 how to select a position for such a mark, and how, when the 
 mark is set up, its distance in arc from the meridian may be 
 ascertained. These determinations may be made without diffi- 
 culty now that we are able by the foregoing methods to calcu- 
 late the amount of the azimuth error. It is, in the first instance, 
 requisite to correct this error mechanically. For this purpose 
 the value in arc of one revolution of the screw by which the 
 azimuth adjustment is effected must be found. (The screw is c 
 or d in Figure 10, p. 38.) This may be done with sufficient 
 accuracy by carefully measuring with compass and scale the 
 interval, parallel to the axis of the screw, occupied by a certain 
 integral number of its threads, and then measuring the length 
 of the axis of motion intervening between the two middle 
 transverse sections of the Y's. If h be the first measure em- 
 
THE TRANSIT INSTRUMENT. 81 
 
 bracing n intervals, II the measured length of the axis, and R 
 be put for 206265", the value of one revolution in arc is p . 
 
 This quantity having been calculated, the correction of azimuth 
 error is effected by turning the micrometer-heads (c and d in 
 Figure 10) through the portion of a complete revolution corre- 
 sponding to the amount of the error, and in the direction which, 
 as indicated by its sign, is proper for correcting it. Let us 
 now suppose that the micrometer-wire has already been put in 
 the position of the mean of all the wires (see Art. 61), and that 
 the micrometer reading for that position has been recorded ; 
 also that by adjusting the wire-frame the micrometer-wire in 
 this position has been put into the plane of collimation (Arts. 
 48 50); lastly, that the level error has been mechanically cor- 
 rected either by turning the screws a and b, fig. 9, p. 87, till the 
 micrometer- wire and its image, as seen with the collimating eye- 
 piece, coincide (see Art. 82), or by using the Spirit-level for the 
 same purpose in the manner described in Art. 71. After these 
 operations, if the telescope be pointed horizontally southward, the 
 micrometer- wire will indicate the actual position of the plane of 
 the meridian, and near this plane it is required to set up a 
 firmly fixed mark, the image of which can be accurately bisected 
 by the micrometer-wire. It may not be possible till after repeated 
 trials to fix upon an appropriate mark, or place it in the most 
 suitable position. When this has been effected, the mean of 
 six or more micrometer-readings for bisection of the mark is to 
 be recorded. Then the difference between this micrometer- 
 reading and that for the position of the mean of all the wires 
 measures the distance of the mark from the plane of the meri- 
 dian. On account of uncertainties arising from atmospheric 
 disturbances the adopted measure should be the mean result of 
 several such determinations. This quantity being well ascer- 
 tained, it is always possible afterwards, assuming the mark to 
 be steady, to determine how far the plane of collimation is from 
 the mark, and, by inference, how far it is from the meridian, 
 and if the occasion required, to place it in the meridian. Also 
 c. 6 
 
82 PRACTICAL ASTRONOMY. 
 
 by bisections of the mark from time to time the variations of 
 meridian error would be ascertainable, and any abrupt or exces- 
 sive deviations from its ordinary value might be detected. 
 
 92. Having now shewn how to find the errors of collimation, 
 level, and azimuth, we may proceed to determine the time- 
 correction required to be applied to the time of transit across the 
 mean of the wires to obtain that of transit across the meridian. 
 Let a, b } c be respectively the three errors in seconds of space, 
 inclusive of their proper signs, which, it should be remembered, 
 have been given to them on the principle of taking each to be 
 positive when it causes the observed time of transit of a star 
 above pole to be earlier than meridian transit. Since by the 
 law of coexistence of small errors, the value of each is the same 
 as if the other two did not exist, the formulae of calculation will 
 "be separately deduced by means of the diagrams (1), (2), (3) of 
 Figure 23. In Fig. 23 (1), P is the pole of the heavens, Z the 
 zenith, cr the point of transit of a star across the plane of 
 collimation, which coincides with the plane of the meridian, 
 because, by supposition, there is neither level nor azimuth error. 
 The adopted line of collimation describes a small circle parallel 
 to the meridian at the distance from it measured by the small 
 equatorial arc Ee, which, therefore, is the collimation- error. 
 The z <rPn, subtended by the portion an of the small circle 
 described by the star, is the angular interval between the 
 observed transit and meridian transit. Hence since an = Ee = a, 
 if 8 be the star's polar distance, 
 
 Z crPn (in time) = =-= cosec 8, 
 
 J.O 
 
 which, consequently, is the required correction for error of 
 collimation. 
 
 In Fig. 23 (2), HZ'h is the great circle described by the line 
 of collimation supposing that there is no collimation or azimuth 
 error, H and h being the points of its intersection with the 
 meridian. The Z ZHZ', being the angular deviation of the 
 plane of collimation from the plane of the meridian, is equal to 
 
THE TRANSIT INSTRUMENT. 
 
 83 
 
 b the level-error. Hence if Per = B and Zcr = z, we have, since 
 crn = bcosz and Z crPn = an cosec 8, 
 
 (in time) = cos z cosec S, 
 
 J.D 
 
 which is the correction for error of level. 
 
 In Fig. 23 (3), HZH' is the azimuth error (c) supposing 
 that there is no collimation or level error. Hence as an = c sin z, 
 
 aPn (in time) = sin z cosec & 
 
 62 
 
84 PRACTICAL ASTRONOMY. 
 
 Let T be the whole reduction to the meridian. Then, since 
 this is equal to the sum of the separate corrections, we obtain 
 
 f , , N cosec S 
 
 T = (a + o cos z + c sin z) -^ , - . 
 
 93. This angle measures the inclination of the adopted 
 line of collimation to the plane of the meridian for any position 
 of the telescope, and might on that account be termed generally 
 the total error of collimation; because when that line coincides 
 with the plane of the meridian the instrumental error is zero. 
 But the formula for calculating T was obtained on the supposi- 
 tion that the pivots are equal and exactly cylindrical. If this 
 be not the case, we may still call the deviation of the line of 
 collimation from the plane of the meridian the total error of 
 collimation; but the use of the transit instrument is not 
 strictly logical unless we are able to calculate this error what- 
 ever be the forms of the pivots, or the changes of form they may 
 undergo, assuming only that by reason of original construction 
 they are always very approximately cylindrical and of the same 
 size. This calculation I have succeeded in effecting by means 
 of a practical solution of the following problem : 
 
 A line being drawn to the optical centre of the object-glass 
 from a selected point of the middle wire nearly mid-way be- 
 tween the two horizontal wires (called previously "the line of 
 collimation "), it is required to find, whatever may happen to be 
 the exact forms of the pivots, the small angle which this line 
 makes with the plane of the meridian for any position of the 
 telescope. 
 
 The solution rests on the principle of determining the 
 position of the line of collimation by first finding by direct 
 micrometer measures the position of another straight line fixed 
 relatively to the instrument, and then after ascertaining the 
 fixed geometrical relation between the two lines, to infer 
 therefrom the position of the line of collimation. The subsidiary 
 straight line may be conceived to be fixed relatively to the 
 instrument by passing through two dots at the extremities of 
 the pivots, and -its position relatively to the piers to be ascer- 
 
THE TRANSIT INSTRUMENT. 
 
 85 
 
 tainable by means of micrometer-microscopes pointing towards 
 the dots as exhibited in Figure 1 (page 21). This method was 
 originally proposed by the Astronomer Koyal, who applied it 
 first in testing the forms of the pivots of the Alt-azimuth 
 instrument of the Greenwich Observatory. I shall give here a 
 description in detail of the mode in which I have employed it 
 to obtain corrections for the forms of the pivots of the Cambridge 
 transit instrument. 
 
 94. Two long micrometer-microscopes of considerable mag- 
 nifying power, constructed expressly for the purpose by 
 Mr William Simms, were fastened by screws in two rectangular 
 cuttings at the tops of the piers, which were formerly occupied 
 by the lever-counterpoises (see Art. 29). These microscopes 
 were mounted so as to admit of a vertical adjustment, and of 
 horizontal adjustments perpendicular to, and in the directions 
 of, their axes, and thus could be made to point, according to the 
 judgment of the eye, along the axis of motion of the instrument, 
 and at the same time be placed at distances from the pivots 
 proper for seeing the dots distinctly. In order to have the 
 means of adjusting the positions of the dots, as well as fixing 
 them relatively to the pivots, two caps were provided, which 
 carried the dots, and fitted on to the ends of the pivots. 
 Figure 24? represents one of the caps, and the mode of its 
 
 Fig. 24. 
 
 attachment to the pivot. By the capstan-headed screw at e 
 the cap is firmly clamped to the pivot, and by the capstan-heads 
 
80 PRACTICAL ASTRONOMY. 
 
 at d, d, d, pertaining to screws which work in a fixed cylinder 
 covered by the movable cylindrical piece that carries the dot 
 (their stems passing through elongated holes in the latter), this 
 piece and the dot may be adjusted to any required position. 
 The diagram accompanying the figure of the cap is a magnified 
 representation of a circle aa described on a plate of white metal 
 on the plane face of the movable cylinder, at the centre of which 
 the dot, which is circular, is engraved, together with a smaller 
 concentric circle bb of known radius, and a broken straight line 
 cc passing through the centre of the dot. The uses of the small 
 circle and straight line will be seen from the following account of 
 the instrumental processes in which this apparatus is employed. 
 95. The first operation is to place the dots very nearly on 
 the axis of motion of the instrument. This is to be done 
 tentatively by the movements given to the dot by the screws 
 d, d, d, till, as seen with the microscope, it describes a very 
 small re-entering curve when the instrument is turned about 
 its axis through a complete revolution. Next, the micrometer- 
 wire is to be adjusted vertically. To do this a small plumb-line 
 apparatus is applied in the following manner. The cap having 
 been pushed close up to the end of the pivot and clamped, the 
 microscope is adjusted in the direction of its axis so that the 
 images of the dot and of the line cc are seen distinctly together 
 with the micrometer-wire put roughly in a vertical position. 
 The distance of the plane face of the cap from a shoulder of the 
 pivot is then gauged, the cap is temporarily removed, and the 
 plumb-line apparatus placed in such a position that the line 
 hangs down at the distance measured by the gauge, and can be 
 seen distinctly through the microscope. The micrometer- wire 
 can now be placed with sufficient exactness parallel to the 
 plumb-line by turning the microscope by hand about its axis. 
 The cap being after this restored to its former position, the line 
 cc will, by reason of the gauge, be seen distinctly, and may be 
 placed parallel to the micrometer- wire by turning the instrument 
 about its axis. When this is done the corresponding reading of 
 the setting-circle should be observed and recorded. Then by 
 turning the instrument from this position through 90, the line 
 
THE TRANSIT INSTRUMENT. 
 
 87 
 
 co will be placed in a horizontal position, and will serve to 
 adjust the micrometer- wire to horizontally. But as to do this 
 requires the microscope to be turned about its axis, a new 
 adjustment in the direction of its axis should be made in order 
 to secure that the micrometer-wire and the line cc shall both be 
 seen distinctly when the wire is finally adjusted horizontally. 
 The readings of the setting-circle for the vertical and horizontal 
 positions of cc being determined, the micrometer-wire can 
 always be adjusted either vertically or horizontally by means of 
 that line, care being taken that the wire and the image of the 
 line be both well seen in focus. If these determinations of 
 the vertical and horizontal directions be not quite accurate, no 
 sensible error will arise provided the dot be always very near 
 the axis of motion. These preliminary operations have to be 
 performed in exactly the same mariner for both dots. 
 
 96. We may now proceed to obtain a general formula for 
 collimation -error independently of the particular forms of the 
 pivots. In Figure 25, A and B are the positions of the dots, 
 
 Fig. 25. 
 
 and Ax, Ay, Az are rectangular axes, Ax being drawn parallel 
 to the plane of the meridian southward, Ay vertically upward, 
 and Az in a horizontal direction westward, making a small 
 angle with AB. The co-ordinates of A and B parallel to Ax 
 and Ay are measured by the microscope-micrometers, adjusted, 
 as described in the preceding Article, for either horizontal or 
 vertical measures. Let the telescope be directed to the Nadir 
 Point, and suppose AC in the plane yAz to be parallel to the 
 
88 PRACTICAL ASTRONOMY. 
 
 line of collimation in that position. The angle yA C (= a) is 
 measured by the collimating eye-piece (see Art. 84), and is 
 reckoned positive when A G deviates from the vertical westward. 
 Let F l Y l and F 2 F 2 4- k be the vertical ordinates of A and B, 
 Fj and F 2 being the recorded micrometer readings, F t and F z 
 factors by which the micrometer readings are converted into 
 seconds of arc, and k an unknown constant, the value of which 
 is determined when the measures Y 1 and F 2 are referred to the 
 same horizontal plane. To find the value of F lt let Z) = the 
 measured distance between the dots, h = the diameter of the 
 small circle bb (Fig. 24), as determined by the maker of the cap, 
 and M the measure of that diameter in micrometer revolu- 
 tions, which measure may be supposed to have been taken just 
 after Y l was recorded. Then if R = the number of seconds in an 
 
 arc equal to radius, -=r- is the measure, in seconds of space, of 
 the angle subtended at B by the diameter bb. Hence the angle 
 subtended by any smaller micrometer measure m is 
 
 Consequently the factor F^ is equal to - . So for the opposite 
 
 T I 75 
 
 cap, F z = f , supposing the given diameters h and h' to be 
 
 unequal. The values of F 1 and F 2 having been calculated by 
 these formulae, we shall have for the value in arc of the angle 
 BAz, F z F 2 + k F t Fj, in which k only is unknown. Hence, 
 since tyAC=u. ) if 6 be the constant angle CAB, we shall 
 have 
 
 = W-(F 2 Y 2 + k-F l Y 1 )-a ............... (1). 
 
 It will now be supposed that when the above measures 
 were taken the Illumination-end was westward, so that, as B 
 was by construction always at that end, F 2 Y z + k and F l Y l 
 are respectively the ordinates of the dots B and A for Illumination 
 West. The instrument is next to be reversed and the same 
 process repeated. Let F 2 ' and F/ be respectively the micrometer 
 readings for the bisections of the west and east dots (which are 
 now A and B), and let F^Y^ + k and F^Yf be their ordinates 
 
THE TRANSIT INSTRUMENT. 89 
 
 reckoned from the same horizontal plane as before, F{ and F 2 ' 
 being calculated by processes already indicated with respect to 
 jPj and F z . Since instead of we have now its supplement, 
 and instead of a an angle a' measuring, as before, the deviation 
 of the line of collimation westward from the meridian, the 
 analogous equation will be 
 
 i8o- e = 90 - (^T; + & - jyr/) -*' ............ ( 2 )- 
 
 The equations (1) and (2) determine the unknown quanti- 
 ties k and 0. In fact, putting, for brevity, Y w for F 2 Y 2 F^Y l 
 and Y. for J^'F,' - FY^ we obtain 
 
 (3)> 
 
 97. Reverting to Figure 25, .4' and I? are new positions of 
 the dots (before the reversion) after turning the eye- end of the 
 telescope through an angle (z) from the vertical towards the 
 south ; A'x, A'y, A'z are rectangular axes, A'y being vertical, 
 A 'x horizontal southward, and A'z westward. A' O is parallel 
 to the new direction of the line of collimation, so that the angle 
 C'A'B' = 6. Let f = the angle by which A C' deviates from the 
 plane yA'x towards A'z. Then y lt y^ being the micrometer 
 readings in the vertical direction, and x v a? 2 those in the hori- 
 zontal direction, for bisections of A' and B' respectively, let fy^ 
 and / 2 y 2 + k be the measures of the vertical ordinates, and g^ 
 and <7 2 a? 2 + k' the measures of the horizontal ordinates, the 
 factors f v / 2 , ff v g^ having to be determined, as already shewn, 
 by means of micrometer measures of the diameter bb. Hence 
 we have the following data for calculating f : 
 
 Z O A'z = 90 - /. B'A'z = e, a small angle, 
 
 Z A'y = *, nearly, z ffA'y = 90 - (fa + k -fa), 
 
 /. C r Ax = 90 - z, nearly, z B'A'x = 90 - (g^ + k' - g^). 
 But by Analytical Geometry, 
 
 cos C'A'B' = cos VA'z cos B'Az + cos C'A'y cos B'A'y 
 
 + cos CA'x cos B'A'x. 
 
SO PRACTICAL ASTRONOMY. 
 
 Hence, putting small arcs for their sines, and unity for cos e in 
 a small term, we obtain 
 
 cos 6 = f + (f z y z + h -fyj cos z + (g 2 x z + k' - g^) sin z. 
 Now by the equation (4) we also have very nearly, 
 
 and if ^be the zenith distance of the pointing of the telescope 
 reckoned southward, Z- z = 180, z = Z- 180. Hence by sub- 
 stituting in the above formula, and putting for k its value from 
 the equation (3), it will be found that 
 
 sn 
 
 The deviation of the eye-end of the telescope from the plane of 
 the meridian having been assumed (in Fig. 25) to be westward, 
 that of the object-end will be eastward, and consequently f is 
 the toted error of collimation with its proper sign. 
 
 98. Supposing now the pivots to be cylindrical and equal, 
 and the extremities of the axis of motion to be at A and B, let 
 f t be the value of f in that case. Then, having regard to the 
 expressions for which Y w and Y e were substituted in Art. 96, it 
 is evident that as the axis of motion has the same position in 
 space after as before the reversion, we shall have T w = Y e , and 
 also / 2 y 2 /^ = F w the position being fixed. Moreover, the 
 c's being reckoned positive southward, g z x z -f k' g^ will be 
 equal to the angular deviation of the west end of the axis 
 southward, and will consequently be the azimuth error with its 
 proper sign. Putting therefore c for the azimuth error, we 
 have for equal cylindrical pivots, 
 
 a + a.' 
 
 ! 
 
 .. . . 
 
 = --- cos Z+ c sin Z. 
 
 By referring to Art. 92 it will be seen to follow as a 
 corollary from this equation that - is equal to the collimation 
 
THE TRANSIT INSTRUMENT. 91 
 
 error, and - ^-^ to the level error. These values may also be 
 4 
 
 obtained independently as follows. The Z yAC (Fig. 25), 
 measured from the vertical towards Az, is equal to a, and if it 
 were wholly due to level error, would have a negative value as 
 respects the transit observations, because it would signify that 
 the west end of the axis is depressed ; but so far as it is due to 
 collimation error it has a positive value, because a westward 
 deviation of the line of collimation from the plane of collimation 
 at the eye-end implies an eastward deviation at the object-end, 
 and consequently a plus correction of the transit-time. Hence 
 if L be the level error and C the collimation error, since the 
 value of a is made up of both, we shall have a = L + (7. 
 After the reversion nothing is changed except the sign of C, so 
 that a! L C. From these two equations it follows that 
 
 By subtracting the above expression for f x from that for 
 the result is 
 
 cos 
 
 + (ff& - 9i x i + M - c) sin 
 
 Consequently in order to calculate the correction f f t for 
 any given zenith distance, we have to find the value of the 
 constant quantity k' c. Before shewing how this may be 
 done, the above formula will be modified for the purpose of 
 making it applicable both to Illumination West and to Illumi- 
 nation East. The equation in its present form is assumed to 
 apply to the case of the Illumination (or the dot B) being 
 westward. Since Y w has been put for F 2 Y 2 F^Y V if we assume 
 that the suffix w indicates that the expression applies to 
 Illumination West, we can,, by putting Y e for F 2 Y 2 f\Y lt 
 indicate that the same expression applies to Illumination East. 
 Consequently the above value of f f x is adapted to the case of 
 
92 PRACTICAL ASTRONOMY. 
 
 Illumination East by simply changing Y w into Y e and F e into 
 YV. Hence it will be seen that 
 
 " 
 
 - c sn 
 
 according as the Illumination is West or East. The value of 
 k' c is found in the following manner. 
 
 99. In the case of equal cylindrical pivots the corrected 
 plane of colliraation coincides with the plane of the meridian, 
 and the corrected line of collimation consequently passes through 
 the pole. Hence if we introduce the condition that f = f t at 
 the pole, the corrected line of collimation in the case of unequal 
 and non-cylindrical pivots will be made to pass through the 
 pole, and the satisfying of that condition gives the means of 
 determining k' c. For, after putting F and X Q to represent 
 the values of fjf^f^ and g<p z ~9\ x i respectively for the 
 position of the pole, and I for Z, I being the co-latitude, we 
 obtain 
 
 or, k' - c = - 5 cosec 
 
 After calculating the value of Tc c from this formula, and 
 substituting it in the general expression for f f t , this quantity 
 can be calculated for any given zenith distance for loth positions 
 of the instrument, and the time-correction for the forms of the 
 pivots may then be obtained from the equation 
 
 _ fc 
 
 Correction = * gl cosec N. P. D. 
 JLo 
 
 100. I proceed to give an example of the calculation of the 
 constants in the formula for f % v for the purpose of afterwards 
 shewing how to obtain the time-correction for forms of the 
 pivots in any given instance of a transit observation. We have, 
 as fixed data, the length D of the straight line joining the dots 
 
THE TRANSIT INSTRUMENT. 93 
 
 = 49 i>l -,80, the diameter of the circle II at the Illumination-end 
 = IM -,0465, and the diameter of that at the other end = O in -,045S. 
 [These values of the diameters were carefully determined by 
 Mr Simms : the angles which the diameters subtend at the 
 opposite dots will be found from the above data to be 192",60 
 and 189' / ,70.] The data for calculating the constants in the 
 general expression for f f l will be extracted from a complete 
 series of measures contained in pages Ixxviii. and Ixxix. of 
 Volume XX. of the Cambridge Observations, which were taken 
 on September 29, 1858, for the purpose of obtaining the pivot- 
 correction for any given N. P. D. It is to be understood that 
 each adopted measure of the co-ordinate of a dot is the mean of 
 three recorded measures, two for contact of the micrometer-wire 
 with the circumference of the dot on opposite sides, and the third 
 for an estimated bisection of the centre of the dot by the axis of 
 the wire. So also the adopted measures of the diameters bb are 
 in each instance means of three trials. The measures for the Pole 
 and the Nadir Point, being fundamental quantities, are all means 
 of six trials. The measures of diameters are designated by the 
 letter M with the suffix 1 or 2 attached. According as this 
 letter, or any other letter used in the subjoined example, has 
 the suffix 1 or 2, it pertains to the east or west end of the axis. 
 In order that the micrometer measures may be expressed in low 
 numbers, the recorded eastern vertical measures are all diminish- 
 ed by 10 r ,000, and the western vertical measures by 9 r ,000. For 
 the same reason constants are subtracted from the horizontal 
 measures, as will be indicated in the following account of the 
 calculations. 
 
 Illumination West. 
 
 Y l = 10,336 - 10,000 = 0,336, M t = 12,537, \ = 0,0458, 
 F 2 = 9,072- 9,000 = 0,072, M 2 = 12,516, fc a = 0,0465, 
 
 = [1,18718], 
 
 J^ = 5",084, ^F, = 1",108, Y w = v = _ r',988. 
 
94 PRACTICAL ASTRONOMY. 
 
 Illumination East. 
 
 r. r. r. r. in. 
 
 F; = 10,390 - 10,000 = 0,390, M^ = 12,711, \ = 0,0465, 
 F 2 '= 9,028- 9,000 = 0,028, Jf a = 12,340, \ = 0,0458, 
 
 /F; = 5",909, *y F; = o",4so, Y 6 = 
 
 For obtaining the values of / 2 i/ 2 /^ and g<P*~~g\ x \ 
 sponding to the position of the pole, the data and the calculation 
 are as follows : 
 
 Illumination West, N. P. D. 0. 
 
 r. r. r. r. in. 
 
 ^ = 10,597-10,000= 0,597, M 1 = 12,532, ^ = 0,0458, 
 y ^ 8,667- 9,000 = -0,333, Jf 2 = 12,531, h 2 = 0,0465, 
 
 2 y 2 -/ iyi = - 14",155 = F , 
 
 r. r. r. r. in. 
 
 , = 10,993 - 10,900 = 0,093, M^ = 12,551, ^ = 0,0458, 
 ^ a = 9,416- 9,200 = 0,216, Jf a = 12,569, & 2 = 0,0465, 
 
 g^i = 1",406, g& = 3",310, g z x z - g^ = 1",904 = X . 
 
 F F / F + F\ 
 Hence Tc c = -^ i cosec ^ + f F ~ " ) cot ^ ~ -^o 
 
 = + 0",752 cosec I - 9", 427 cot i - 1",904 
 = - 12 // ,836, I being = 37 . 47' . 8". 
 
 Illumination East, K P. D. 0. 
 
 r. r. r. r. in. 
 
 yl = 10,250 - 10,000 = 0,250, M t = 12,640, \ = 0,0465, 
 y ; = 8,745 - - 9,000 = - 0,255, M 9 = 12,388, h z = 0,0458, 
 
THE TRANSIT INSTRUMENT. 95 
 
 - 7",7i4 = 
 
 < = 10,945 - 10,700 = 0,245, ^ = 12,658, A x == 0,0465, 
 ^ 2 '= 9,174- 9,300 = -0,126, M z = 12,449, 2 = 0,0458, 
 
 ^' = - = [1)18228]j ^ 
 
 Hence 
 A;' - c = - FFe cosec I + F - e cot Z - 
 
 = - 0",752 cosec Z - 2",986 cot I + 5",648 = + 0^,569. 
 Consequently the final expressions for f f t are, 
 
 Illumination West, 
 f - f , = + 0",75 + (/^, -/ l2/1 + 4",73) cos (8-0 
 
 Illumination East, 
 
 / l2/1 + 4",73) cos (8 - Z) 
 9& + 0",57) sin (3 - Z). 
 101. The measures on September 29, 1858, were taken 
 for every tenth degree of N. P. D. above pole, and for the polar 
 distances 5, 0, + 5, these three being included for the 
 purpose of obtaining the values of f f t for Polaris and 8 Ursse 
 Minoris by interpolation. The order of proceeding was as 
 follows: the Illumination being eastward, the micrometer- 
 wires were first adjusted vertically, and all the horizontal 
 measures were taken, and after adjusting the wires horizontally 
 and to distinct vision of the dots, all the vertical measures were 
 taken, ending with those for Nadir Point. The instrument 
 was then reversed, and without altering the horizontal adjust- 
 ments of the wires, all the vertical measures were again taken, 
 beginning with those for Nadir Point ; and lastly, after adjusting 
 
96 PRACTICAL ASTRONOMY. 
 
 the wires vertically and to distinct vision, all the horizontal 
 measures were again taken. In other instances, as in that of 
 September 30, 1858 (given in the same pages of Volume xx.), 
 the measures were taken for every fifth degree of ISL P. D. 
 above pole, and for every tenth degree below pole. As the 
 interval between the Y's is somewhat greater than that between 
 the shoulders of the transverse axis which are contiguous to 
 
 G 
 
 them, the axis is liable to a small displacement whereby the 
 images of the dots are thrown out of focus. This fault was 
 corrected by turning the instrument backwards and forwards 
 about its axis, and at the same time pushing it moderately in the 
 direction proper for the correction, till both dots could be seen 
 distinctly for the same position of the instrument. This pre- 
 caution having been attended to, it was not thought necessary 
 to measure the diameters of the circles on each occasion of 
 measuring the ordinates of the dots. On September 29 the 
 measures of the diameters were taken only for 0, 30, 60, 90, 
 120 of N. P. D., and for the Nadir Point. After giving these 
 explanations I proceed to adduce an example of calculating 
 the time-correction for forms of the pivots, both for Illumi- 
 nation West and Illumination East, for a given N. P. D., select- 
 ing that of 60. 
 
 Illumination West, N. P. D. 60. 
 
 y l = 10,406 - 10,000 = 0,406, 3/ t = 12^512, h, = 0^458, 
 y a = 9,169- 9,000 = 0,169, M 2 = 12,514, h^ 0,0465, 
 
 fa -f l!fi = -3",555, 
 
 r. r. r. r. in. 
 
 10,969 - 10,900 = 0,069, M t = 12,524, ^ = 0,0458, 
 9,143- 9,200 = -0,057, M z = 12,551, A 2 = 0,0465, 
 
 &i = - 0",S75, g^ - g l x l = - l r/ 5 920. 
 
THE TRANSIT INSTRUMENT. 97 
 
 Illumination East, N. P. D. 60. 
 
 y l = 10,223 - 10*000 = 0*223, M, = 12^679, \ = 0,0465, 
 y t = 9,354- 9,000 = 0,354, M 2 = 12,382, ^ = 0,0458, 
 
 = [1,18528], 
 
 
 / 2 y 2 =5",432, /&-/& = 2",045, 
 
 x l = 10,758 - 10,700 = 0,058, M^ = 12^665, h, = 0,0465, 
 ar a = 9,059- 9,300 = - 0,241, l/ 2 = 12,428, 7i 2 = 0,0458, 
 
 a 
 
 
 ^^ = 0",S82, g^ = - 3",679, ^ -^^ = - 4" 5 561. 
 Consequently, Illumination West, 
 
 f - ft = + 0",75 + (- 3",56 + 4",73) cos 22 . 12',9 
 
 + ( - 1",92 - 12 // ,84) sin 22 . 12',9 
 = + 0",75 + l /r ,09 - 5^,58 = - 3 // ,74, 
 
 Illumination East, 
 
 ? - ft = - 0",75 + (2",04 + 4 r/ ,73) cos 22 . 12',9 
 
 + ( - 4",56 + 0",57) sin 22 . 12',9 
 = - 0",75 + 6",27 - 1",51 = + 4",01. 
 
 Hence the time-correction, ^ ** cosec 60, is *- 8 ,288 for 
 
 lo 
 
 Illumination West, and + 0*,309 for Illumination East. These 
 values, obtained by independent calculation, agree with those 
 on page Ixxviii of Volume xx. of the Cambridge Observations. 
 
 102. The pivot-corrections for all the particular N. P. D. for 
 which micrometer-measures were taken having been calculated 
 according to the foregoing example, and arranged in tabular 
 form with N. P. D. for argument, the correction for any 
 assigned N. P. D. might be obtained by simple interpolation. 
 c. 7 
 
98 PRACTICAL ASTRONOMY. 
 
 As, however, it was found that the corrections varied gradually 
 in amount from year to year, and the measures were subject 
 to incidental irregularities, the table of values for any year was 
 combined with those of two other years, one shortly preceding 
 and the other shortly following, and from the three a mean table 
 was formed, from which the correction for any given N. P. D. 
 for the intermediate year was derived by simple interpolation. 
 This pivot-correction may be immediately applied to the clock- 
 time of transit across the mean of the wires (obtained as shewn 
 in Division ///.), and then the corrections for the errors of 
 collimation, level, and azimuth may be obtained and applied 
 just as if the pivots were exactly equal and cylindrical. The 
 reasoning I have now gone through, although long and some- 
 what complicated, was yet necessary for shewing how transit 
 observations may be freed from errors due to irregularities and 
 changes of the forms of the pivots, and how, consequently, to 
 make a completely logical use of the transit instrument 1 . 
 
 103. Before proceeding to treat of the methods of taking 
 and reducing transit observations, some account will here be 
 given of the Astronomical Clock, and of the conditions it is 
 required to satisfy when used in an Observatory. For details 
 respecting the parts of chief importance in the construction of 
 
 1 The method of obtaining the pivot-corrections which I put in practice with 
 the means I had at command, is adopted in principle in the use of the Cam- 
 bridge Transit -circle, but with various simplifications and improvements as to 
 details which I shall take this occasion to point out. (1) The dots were engraved 
 on the pivots themselves, and as nearly as possible on the axis of motion, so 
 that no subsequent adjustment of their position was required, such as that I 
 applied to the dots on caps (Art. 95). (2) A spring acting on the end of one 
 of the pivots keeps them always to the same bearing, and thus the inconvenience 
 I experienced from the images of the dots being put out of focus by displace- 
 ments of the axis (Art. 101), is obviated. (3) The micrometer-wire of the 
 microscope is adjusted horizontally by a small spirit-level, movable with the 
 micrometer-apparatus about the axis of the microscope, and so adjusted by the 
 instrument-maker that when the bubble is in mid-position the micrometer-wire 
 is sufficiently horizontal, considering the small movement of the dot, for mea- 
 suring its vertical ordinate. This expedient supersedes the use of the small 
 plumb-line which I had recourse to (Art. 95). (4) As the Transit-circle is not 
 reversed, the pivot-corrections are required for only one position of the in- 
 strument. 
 
THE TRANSIT INSTRUMENT. 99 
 
 clocks for astronomical purposes, the student will do well to 
 read Chapter III. of Godfray's " Treatise on Astronomy." As 
 far as the Lectures are concerned it will suffice to refer only to 
 the clock with Graham's mercurial pendulum, this being the 
 form of pendulum which is found by experience to be best 
 adapted for obtaining accurate astronomical measures of time. 
 The particular advantage of this construction is, that it allows 
 the astronomer himself both to alter the clock's rate and to 
 correct imperfect compensation. These adjustments are effected 
 by means which it is the object of the following statements to 
 describe. 
 
 104. First, for altering the clock's rate, the mercury is put 
 in a glass cylinder attached to the end of the pendulum-rod, 
 and by turning a screw the cylinder and contained mercury 
 may be raised or lowered relatively to the rod. As the weight 
 of the mercury constitutes the chief part of the weight of the 
 pendulum, this operation changes the position of the centre of 
 oscillation, and, by consequence, the time of an oscillation of the 
 pendulum. A small graduated circular disk, attached to the 
 stem of the screw and turning with it, is. read off by an index 
 fixed to the rod. It being understood that the clock's rate is 
 its gain or loss upon 24 hours in the sidereal interval between 
 consecutive meridian transits of the same star, let the index be 
 set to a certain reading, and the clock be rated, by transits of 
 stars taken ad libitum, in the manner stated at the end of 
 Art. 89 (1). After altering the index-reading by a certain 
 integral number of divisions, let the clock be rated in the same 
 manner again. The difference of the index-readings and the 
 corresponding difference of the rates being noted, the index- 
 reading which would correspond to a certain desired rate might 
 then be found by a simple proportion, and the length of the 
 pendulum might be. altered accordingly. For convenience as 
 respects the reduction of transit observations it is usual to 
 maintain a small losing rate, the minute-hand being put for- 
 ward when the clock's loss exceeds one minute. 
 
 105. Compensation for changes of the length of the pendu- 
 lum produced by changes of temperature is effected by the 
 
 72 
 
100 PRACTICAL ASTRONOMY. 
 
 mercurial pendulum in the following manner. The cylindrical 
 column of mercury, being uncovered at the top, is free to 
 expand upwards upon an increment of temperature, which, 
 therefore, has the effect of raising the centre of gravity of the 
 mercury, and, by consequence, the centre of oscillation of the 
 pendulum. At the same time the pendulum-rod, by expanding 
 downwards from the point of suspension, produces the opposite 
 effect as respects the position of the centre of oscillation. When 
 these two results of the change of temperature just counteract 
 each other, the required compensation is effected, the distance 
 of the centre of oscillation from the point of suspension, and 
 consequently the time of an oscillation of the pendulum, being 
 thereby made invariable. Analogous reasoning applies to the 
 effects of a decrement of temperature. Now the astronomer is 
 in a better position than the clock-maker for telling whether or 
 not the compensation be good, because for doing this completely 
 exact transit observations carried on through the hottest and 
 coldest parts of at least one year are required. It is the 
 special advantage of the mercurial pendulum that it gives the 
 astronomer the means of correcting the compensation if found 
 thus to be imperfect. Supposing, for instance, the observations 
 to shew that the losing -rate comes to a maximum in the hot 
 months, this will indicate that the length of the pendulum is at 
 that time too great, and consequently that the effect of the 
 expansion of the rod is greater than that of the expansion of 
 the mercury. The clock is in that case under-compensated, and 
 requires an addition to be made to the quantity of mercury ; or, 
 if it be found that the losing rate diminishes in the cold months 
 till it attains a minimum, the pendulum is then too short, and 
 the contraction of the pendulum-rod by the cold is not compen- 
 sated for by the contraction of the mercury ; that is, as before, 
 the clock is under-compensated. But if the clock goes faster 
 in the summer months, or slower in the winter months, than 
 in Spring or Autumn, it is over-compensated, and requires that 
 some mercury should be taken out of the cylinder. 
 
 106. The simplest practical method of adjusting the quantity 
 of the mercury in the cylinder so as to effect complete compen- 
 
THE TRANSIT INSTRUMENT. 101 
 
 sation seems to be the following. Suppose that after giving to 
 the clock a small losing rate by the process explained in Art. 
 104, the clock were rated by transit observations during a whole 
 year (in the manner that will be explained under Division IV.}. 
 Then, from what has been said in Art. 105, according as the 
 summer losing rates are found to be greater or less than those 
 of winter, the compensation requires mercury to be added to or 
 subtracted from the quantity (Q) that was in the cylinder when 
 the rate was adjusted. To calculate how much, let R the 
 maximum losing rate, and r the minimum losing rate, in the 
 course of the twelve months, be extracted from the reduced 
 transit observations, and suppose that the contemporaneous 
 temperatures are known to be T and t. At the end of that 
 period let the quantity of mercury be altered to Q q, and the 
 observations be continued through another twelve months, and 
 let R', /, T' y t' be the new values corresponding to R, r, T, t. 
 Then to reduce the two sets of values to the same conditions as 
 respects temperature, let R" r" be the value that R' r 
 would have had if the difference of temperature T' t had 
 been equal to T t. Hence, as small differences of rate may be 
 assumed to be proportional to the differences of temperature, 
 
 T ' t 
 
 we shall have R" r" = (R f r') -7 7, . Hence the additional 
 
 JL & 
 
 T t 
 
 mercury q has produced the change R r (R f r) -^ -, 
 
 J- t 
 
 in the difference between the maximum and minimum rates. 
 But there is still to be corrected the difference R' r. Let q' 
 be the unknown quantity of mercury required for this purpose. 
 
 T t 
 
 Then we may assume that q is to q as R r (R f r) 7 -7 
 
 JL t 
 
 is to R' r\ From which it follows that 
 
 - j (r - o - (R'-r) (T- t) * 
 
 This formula gives in terms of known quantities the amount 
 of mercury to be put in or taken out of the cylinder to complete 
 the compensation, at least, approximately. If after this has 
 
102 PRACTICAL ASTRONOMY. 
 
 been done, further correction appears to be needed, it may be 
 calculated by means of a like formula, obtained by combining 
 the transit observations of a third twelve months with those of 
 the second. Should there still be residual variations of rate, 
 whether from defect of compensation or other causes, they will 
 have no appreciable effect, so long as they are neither large nor 
 abrupt, on the final determinations of Right Ascension, sup- 
 posing transit observations of fundamental stars to be carried on 
 in the Observatory night and day during the year. 
 
 III. METHODS OF TAKING TRANSIT OBSERVATIONS, AND 
 CALCULATIONS OF CORRECTIONS. 
 
 107. It will be proper to begin this Division of our subject 
 by stating in what manner a setting-circle is used in preparing 
 to take a transit observation. The construction and uses of 
 setting-circles have been described in Art. 34, but it was not 
 shewn how, before employing the setting-circle to point the 
 telescope to a celestial object intended for observation, its 
 spirit-level has to be adjusted. This may readily be done, if a 
 collimating eye-piece be adaptable to the telescope, and the 
 telescope be thereby pointed so that the reflected image of each 
 of the two horizontal wires is coincident with the other wire ; in 
 which case the path across the field midway between the 
 horizontal wires, which the object observed is usually made to 
 traverse, will be exactly in the nadir direction. Under these 
 circumstances the index which will always be used for settings 
 should be made to point to an angle below pole equal to the 
 latitude of the Observatory, for instance to 52. 13' for the 
 Cambridge Observatory. After this adjustment has been made 
 by the tangent-screw and clamp (Art. 35), the bubble of the 
 level is to be put in mid-position by turning the screws x and y 
 (Figure 7 in p. 28). Then if the index be set to any given 
 circle-reading, the telescope, after being moved till the bubble 
 is in mid-position, will point to the north polar distance signified 
 by that reading. It should, however, be noticed that the adjust- 
 ment of the level was made independently of any effect of 
 refraction, on which account in setting for a star or other object, 
 
THE TRANSIT INSTRUMENT. 103 
 
 the N. P. D. as altered by refraction should be set for. For 
 doing this readily the observer should have at hand a small 
 table of approximate refractions with N. P. D. for argument. If 
 the use of a collimating eye-piece be not available, the adjust- 
 ment of the level might be performed by first setting the index 
 to the apparent N. P. D., as altered by refraction, of a bright 
 star, such as Arcturus, and after catching it as it enters into the 
 field of view (which it is easy to do), to turn the telescope till the 
 path of the star's image is mid- way between the two horizontal 
 wires. Then, the telescope retaining its position, the bubble is 
 to be put in mid-position by turning the screws x and y pro- 
 vided for this adjustment. 
 
 108. Proceeding, in the next place, to give an account of the 
 methods of taking transit observations of the different kinds of 
 celestial bodies, I shall assume that in every instance the Eight 
 Ascension and Polar Distance of the object to be observed, as 
 also the error of the clock on true sidereal time, are, at least 
 approximately, known. (By what antecedent operations and 
 observations the clock's error, and the places both of stars and 
 of the moving bodies, may have been approximated to, or how 
 by beginning de novo they might be sufficiently well determined 
 for being used to obtain places of higher degrees of accuracy, 
 will be unfolded in the sequel of the Lectures, more especially 
 in the treatment of what relates to meridional transit and circle 
 observations under Division IV.) The polar distance being 
 approximately known, we can set for the object in the manner 
 described in Art. 107, and the clock's error being sufficiently 
 ascertained we shall know when to look for the entrance of the 
 object into the field of view. When fairly in the field, the 
 image has to be brought, by shifting the telescope, into a 
 position intermediate, as nearly as may be, between the two 
 horizontal wires. Suppose, first, the image to be that of a star, 
 which may be considered to be a point of light (see Art. 19). 
 The observer, as it approaches the first wire, takes the number 
 of a seconds' beat from the clock-face, and then goes on counting 
 the beats mentally, looking at the same time intently at the 
 central point of the image in order to note its position just at 
 
104 PRACTICAL ASTRONOMY. 
 
 the beat which next precedes the transit across the wire, and 
 again its position at the beat next after the transit. Also to the 
 best of his judgment he estimates the ratio of the space between 
 the first position and the wire to that between the two positions 
 in tenths of the interval 1 , and the counted second of the first 
 position with this proportional part of a second added is 
 recorded as the clock-time of the transit. This operation is 
 repeated at each of the wires, the intervals between the wires 
 allowing time for recording the several transits. After the 
 transit across the last wire, the clock-face is looked at to see 
 whether the counting has been correct, and the hour and 
 minute of the last transit are set down. This mode of taking 
 transits is called for distinction the eye-and-ear observation. 
 
 109. The same method of observing to the tenth of a 
 second is employed when the object is either the First or Second 
 Limb of the Sun, care being taken in the setting to ensure that 
 the Sun's centre shall pass as nearly as may be mid-way 
 between the horizontal wires. On account of the Sun's heat 
 the eye of the observer generally requires to be protected by 
 darkened glasses; but occasionally satisfactory observations 
 can be taken when the Sun is seen through cloud or mist. 
 The Cambridge transit-telescope had a small wheel attached at the 
 eye-end, carrying near its periphery a set of glasses of different 
 degrees of opacity, which, on turning the wheel by hand, could 
 be placed, as the occasion required, opposite the aperture 
 through which the field of view was seen. Also to prevent any 
 disturbing effect being produced by the incidence of the Sun's 
 rays on parts of the instrument, a movable dark screen has to 
 be provided, by shifting which the solar rays may either be 
 wholly cut off, or be allowed to pass through a circular opening 
 in the screen so as to fall on the object-glass during the interval 
 
 1 For a long time astronomers were satisfied with noting the time of transit 
 to the nearest second. Bradley in some cases estimated to the half or the third 
 of a second. Maskelyne at the beginning of his career divided the second into 
 eight parts; but eventually (in September, 1772) he began the practice of 
 recording the times of transit to tenths of a second. (See Grant's History of 
 Physical Astronomy, p. 489.) 
 
THE TRANSIT INSTRUMENT. 105 
 
 occupied by taking the observation. When both limbs have 
 been observed, the mean of the transit-times for the two limbs 
 is the time of transit of the Sun's centre, and their difference is 
 a measure of the Sun's diameter, the adopted value of which, 
 as being an important astronomical element, should be the 
 mean result of a large number of observations. 
 
 110. The full moon would be observed in the same manner 
 as the Sun, and when, also, it is not quite full, the transits of 
 both limbs would still be taken, because, after applying a 
 correction for a small defect of illumination (the mode of calcu- 
 lating which will be subsequently explained), the transit-time 
 of the Moon's centre, and a measure of the diameter, may be 
 inferred from the times of transit of the limbs. As the Moon's 
 diameter is an important datum, and the opportunities for 
 measuring it are comparatively rare, advantage should be taken 
 of all that are available. 
 
 111. Transits of Jupiter and of Mars are observed as 
 follows. The number of wires being supposed to be seven, 
 transits of the first limb are taken at the first, third, fifth, and 
 seventh wires, and transits of the second limb at the second, 
 fourth, and sixth wires. The mean of all these transit-times is 
 strictly the observed time of transit of the planet's centre across 
 the mean of all the wires. The excess of the mean of the 
 transit-times of the second limb above the mean of those of the 
 first limb is considered to be a measure of the planet's diameter. 
 (More generally and strictly the diameter is measured by the 
 excess of the mean of the times of transit of the second limb 
 across three, or four, of the wires, reduced, as hereafter shewn, 
 to time of transit across the mean of all the wires, above the 
 mean of the times of transit of the first limb across the remaining 
 four, or three, wires, similarly reduced.) The ansse of Saturn's 
 Ring (not the body of the Planet, excepting when the Ring 
 disappears) are observed in the same manner, for the purpose of 
 obtaining the transit-time of the planet's centre and a measure 
 of the external diameter of the Ring. The observed times of 
 transit of Mercury and Venus apply only to their illumined 
 limbs. In any case of the observation of a single limb of a 
 
106 PRACTICAL ASTRONOMY. 
 
 planet, the reduction to sidereal time of transit of centre is 
 made by adding or subtracting the sidereal interval occupied by 
 transit of semidiameter, as derivable from those calculated in 
 the Nautical Almanac for every second day "at transit at 
 Greenwich." The assumed values in arc of the semidiameters 
 at the Earth's mean distance from the Sun from which the 
 calculations are made, are given in page V. of the Preface. 
 When both limbs are taken and the planet is sensibly gibbous, 
 a correction for defect of illumination, calculated by a process 
 that will be given afterwards, is applied. When planets are 
 very small, transits of their estimated centres are taken. 
 
 112. In observations of Polaris and 8 Ursaa Minoris, or 
 other stars very near the pole, it is not possible to judge with 
 any certainty of the instant of transit to the fraction of a 
 second. As, however, the instant at which the star is exactly 
 bisected, or is struggling, as it were, to cross the wire, may 
 sometimes be noted with great precision, in every case an 
 estimate is made of the transit-time to the tenth of a second. 
 Transits of stars near the pole are required to be taken from 
 time to time at all the wires, because, as stated in Art. 60, by 
 means of these observations the equatorial intervals of the 
 several wires from the mean of all are calculated, and these 
 intervals it is necessary to know for a reason which will be 
 stated in the next Article. But because of the slow motion of 
 stars near the pole, each set of such observations takes a long 
 time ; on which account, in general, a shorter way of obtaining 
 their transits across the mean of the wires is adopted, by 
 employing the micrometer- wire in the following manner. First, 
 just before or after the observation, the coincidence reading of 
 the micrometer-wire with the middle wire D is found by the 
 process indicated in Art. 51. Then the micrometer-wire is 
 placed successively at the distances of three revolutions, two 
 revolutions, and one revolution from D on the west side, and at 
 the distances of one, two, and three revolutions on the east side, 
 and transits of the star are taken at each of the six positions 
 and at the middle wire, the intervals between the positions 
 allowing of time for recording the seven transits. The mean of 
 
THE TRANSIT INSTRUMENT. 
 
 107 
 
 the transit-times is the time of transit across the wire D, and 
 the interval of this wire from the mean of all being known by 
 the method explained in Art. 60, the transit-time across the 
 mean of the wires may be inferred. As the Cambridge and 
 Greenwich Transit-circles have no micrometer-wire, additional 
 sets of fixed wires, the intervals between which are small, are 
 inserted, one on each side of the middle wire, by means of 
 which the abbreviated observation may be made in the manner 
 above described ; and in order that the mid- wire may be readily 
 distinguished, the intervals between it and the two adjacent 
 wires are larger than the others. 
 
 113. When transits of both limbs of the Moon, or a Planet, 
 are taken at the same meridian passage, generally a correction 
 is required on account of defect of illumination of one of the 
 limbs, in order to obtain the exact time of transit of centre, 
 and an exact measure of the diameter. The object of the 
 following investigation is to give the means of calculating such 
 corrections. Figure 26, which applies more expressly to cor- 
 
 (5) 
 
 rections of measures of vertical diameters for defect of illumi- 
 nation, may be made use of for the present purpose. In (1) of 
 Fig. 26, P is the pole of the heavens, S the Sun's place, and V 
 
108 PRACTICAL ASTRONOMY. 
 
 that of a planet at the time of taking the measures. The polar 
 distance PS of the Sun and PV that of the planet, and the 
 difference of their Right Ascensions SPY, will be supposed 
 to be deduced from the Nautical Almanac, and from these 
 data the z. PVS and the arc SV may be calculated. As transits 
 of cusps are usually not taken on account of the large probable 
 error of such observations, I shall suppose the Moon or planet to 
 be gibbous, and to deviate little from a circular form. In that 
 case the boundary of illumination opposite to the Sun, being 
 the projection of a circle on a plane perpendicular to the line of 
 vision, is an ellipse, as represented by the curve Aba in Fig. 
 26, (3). The angle of projection is equal to the inclination of 
 the line of vision to the line drawn from the Sun's centre to the 
 centre of the Moon or planet. In Fig. 26, (4) EVS is this 
 angle, E, 8, V being respectively the positions of Earth, Sun, 
 and Planet in space, and according as the angle EV8 is greater 
 or less than 90, the illumined body is horned or gibbous. Let 
 A a, the axis major of the ellipse, be equal to 2a; then if 
 the semi-axis minor = b, we shall have b = a cos EVS, and 
 
 sin EVS = (l - ^V = e. Suppose SE( = R) and SV( = r) to 
 be obtained from the Nautical Almanac. Then 
 
 e = sin EVS= - sin SJEV= - sin SV, 
 r r 
 
 by which expression e can be calculated, SV being already 
 known. Since DVn in (3) coincides with the circle of polar 
 distance PV through V in (1), and AVa is at right angles to 
 SV, it follows that the tAVn(= 6} is equal to 90- / PVS, 
 and is therefore known. Hence the angle which a parallel of 
 declination makes with VA is 6 + 90. Now by Conic Sections, 
 if & be calculated from the equality 
 
 sin & = e sin (0 + 90) = e cos 0, 
 
 the length of the perpendicular from V on a tangent to the 
 ellipse parallel to DVn is a cos 6'. Hence if I be the observed 
 interval occupied by the transit of the diameter, and 2/ be the 
 
THE TRANSIT INSTRUMENT. 109 
 
 & 
 
 true interval, I = 7 + J cos & = 2 J cos 2 -^ , assuming the obser- 
 
 T Q' 
 
 vation to be exact. Hence I = ^ sec 2 -^ . From this equation 
 
 2i Z 
 
 the interval of transit of semi-diameter, as given by the obser- 
 vation, may be calculated, and the value be compared with that 
 in the Nautical Almanac. From a collection of such com- 
 parisons a correction of the tabular diameter in arc might be 
 obtained, but not as accurately as from the micrometer measures 
 of vertical diameters which will have to be considered in treat- 
 ing of the Mural Circle. When the transit-time of centre is 
 inferred from the transit-times of both limbs, the correction 
 
 required for defect of illumination is plainly + 1 7 -= \ , accord- 
 ing as the first or second limb is defective, and is therefore 
 
 I ff 
 
 equal to + ^ tan 2 . 
 
 114. The foregoing expressions may be applied in obser- 
 vations of Venus, Mars, and Jupiter, when the disks are 
 approximately circular, and transits of both limbs are taken. 
 For Jupiter the corrections will in any case be excessively 
 small, and for the other planets they will often be far less than 
 unavoidable errors of observation. It is, however, right in 
 principle to correct errors however small, when they are due to 
 known causes, and can be accurately calculated. 
 
 115. For the Moon the calculation of & may be simplified 
 as follows. As the correction can be applied only when the 
 Moon is nearly full, let in Fig. 26 (5) $' be the point opposite 
 the Sun's centre, and M be the place of the Moon's centre ; so 
 that S'M is a small arc. But in this case sin $F= sin S'M= S'M 
 
 7? 7? 
 
 nearly. Also since = 1 nearly, and by Art. 113 e = sin SV 9 
 
 it follows that sin & or e cos 6 S'M cos 6 S'N nearly. Or & 
 is nearly equal to S'N, which is the difference of R. A. of the 
 Moon and the point opposite the Sun, multiplied by the cosine 
 of the Sun's declination. After thus finding & with sufficient 
 approximation, the corrections of the measure of diameter and 
 
110 PRACTICAL ASTRONOMY. 
 
 of transit-time of centre may be calculated by the formulae 
 obtained in Art. 113. 
 
 116. It will frequently happen that by the intervention of 
 clouds, or from other causes, a transit observation is not taken 
 at all the wires, in which case the mean of the observed transit- 
 times is reduced (by formulae which will presently be given) to 
 time of transit across the mean of all the wires, and the weight 
 of the observation is considered to have to that of a complete 
 observation, the ratio of the number of the recorded times to 
 the whole number of wires. According to Cambridge practice, 
 the observation was inserted in the computing books, but not 
 reduced, if it was taken at less than three wires. For the 
 above-mentioned reduction of broken observations, it is neces- 
 sary to be able to calculate the interval of any wire from the 
 mean of all for any given polar distance. The method of 
 obtaining these intervals for an equatorial star has already been 
 indicated in Arts. 59 61. In the use of the formula, 
 
 sin 15a = sin B sin lot, 
 
 for this purpose (Art. 60), the polar distance S was supposed to 
 be taken from the Nautical Almanac (see Art. 89) ; but it 
 should be noticed that this datum might be independently 
 obtained by a contemporaneous observation with the Mural 
 Circle, or one observer might determine S and t simultaneously 
 by means of a Transit-circle. We have next to ascertain how 
 the intervals are to be calculated for any polar distance, and 
 for any object, whether it be a star or a moving body. 
 
 117. For a star not very near the Pole, if 8 = the star's 
 polar distance, h = a, given equatorial interval, and e? = the 
 required interval for the polar distance 3, we shall have, with 
 sufficient approximation, d h cosec S. 
 
 118. For a star near the Pole, sin h = sin S sin d. Suppose 
 that in any year a considerable number of complete transit 
 observations of Polaris, either above or below pole, are taken, and 
 let A represent a polar distance consisting only of degrees 
 and minutes, chosen so as to be little different from the mean 
 of the polar distances of Polaris in the period during which the 
 
THE TRANSIT INSTRUMENT. Ill 
 
 calculated intervals are to be used. It is first required to 
 deduce from all the observations a mean value of the interval 
 of any given wire from the mean of all for the polar distance A . 
 Let d t be this interval, and let S and d be the given polar 
 distance and observed interval pertaining to any one of the 
 observations. Then since sin h sin 8 sin d sin A sin d v we 
 
 have sin d^ = cosec A sin 8 sin d. 
 
 From the last formula, after converting d into arc, the value of 
 d^ in arc can be calculated. Each observation furnishes in this 
 manner a value of d v and the mean of the several values, which 
 mean will be called d oy is the adopted interval corresponding 
 to the polar distance A . The same process has to be gone 
 through for each wire. 
 
 119. Now let A + tt represent any polar distance of 
 Polaris, n being expressed in seconds, and let d + q be the 
 corresponding interval. Then since sin (A + n) sin (d Q + q) is 
 equal to sin A sin d , it will be found, if only first powers of the 
 small quantities n and q be retained, that q = n cot A tan d . 
 The value of n may be taken from the pages of the Nautical 
 Almanac containing the apparent places of Polaris ; but as 
 declinations are there given in place of polar distances, it will 
 be convenient to substitute in the above expression 90 D Q 
 for A and n for n. We shall then have for the interval 
 d Q + q corresponding to the declination D + n the general 
 expression 
 
 cZ (in time) + tan D tan d Q , 
 
 id 
 
 n being now taken immediately from the Nautical Almanac. 
 
 1 20. The same formula is applicable to any other circum- 
 polar star in the Nautical Almanac, as S Ursse Minoris, or 51 
 (Hev.) Cephei ; but in these cases we have the choice of 
 determining D by independent observations, or inferring its 
 value from that adopted for Polaris. The latter method is 
 perhaps to be preferred, the value for Polaris admitting of 
 superior exactness of determination on account of the proximity 
 of this star to the Pole. If d Q ' and D Q ' be corresponding values 
 
112 
 
 PRACTICAL ASTRONOMY. 
 
 of wire-interval and declination for the second star, those for 
 Polaris being d and D Q , we have 
 
 sin h = cos D sin d cos D ' sin d Q ' ; 
 or sin d ' = sec D ' cos D sin d Q . 
 
 Hence d ', corresponding to a selected declination D ', may be 
 calculated, and then the interval corresponding to any other 
 declination D ' + n may be obtained by means of a formula 
 analogous to that for d Q + q given above. 
 
 121. The subjoined Table of the several intervals of the 
 wires from the mean of all for the case of an equatorial star, 
 and for given declinations of Polaris and 8 Ursse Minoris, was 
 formed by calculating according to the rules indicated in Arts. 
 116, 118 120. The data for the calculation consisted of the 
 transit-times of eleven complete observations of Polaris taken 
 from October 3, 1853, to April 13, 1854, as contained in Volume 
 XIX. of the Cambridge Observations. 
 
 
 T 
 
 
 
 Wire. 
 
 for an 
 Equatorial 
 Star. 
 
 Interval for Polaris. De- 
 clination = 88 32' + TO. 
 
 Interval for S Ursae 
 Minoris. Declination 
 = S636' + n. 
 
 
 S. 
 
 m. s. s. 
 
 m. s. s. 
 
 A 
 
 - 40,352 
 
 -26. 20,03-0,30171 
 
 -11. 20,39-0,05671 
 
 B 
 
 -26,867 
 
 -17. 30,72 -0,1997i 
 
 - 7. 33,03-0,03771 
 
 C 
 
 - 13,577 
 
 - 8. 50,59-0,10171 
 
 - 3. 48,93-0,019w 
 
 T> 
 
 + 0,068 
 
 + 2,68 
 
 + 1,16 
 
 E 
 
 + 13,521 
 
 + 8. 48,37 + 0,10071 
 
 + 3. 47,98 + 0,019w 
 
 F 
 
 + 26,908 
 
 + 17. 32,33 + 0,20071 
 
 + 7. 33,72 + 0,037 
 
 G 
 
 + 40,310 
 
 + 26. 18,35 + 0,30071 
 
 + 11. 19,69 + 0,056^ 
 
 The intervals (d ) for Polaris corresponding to the declina- 
 tion 88 32' are each the mean of the values derived by the 
 process stated in Art. 118, from the eleven intervals given by 
 the observations, and the intervals (e ') for 8 Ursse Minoris cor- 
 responding to the declination 86 36' were calculated from the 
 values of d ' by the formula in Art. 120. The above Table 
 was used from September 21, 1853, to the end of 1854, on 
 
THE TRANSIT INSTRUMENT. 113 
 
 which account the selected values of D and D^ are 88 32' and 
 86 36'. The equatorial intervals were calculated from the 
 same values of d (see Art. 60). [The above Table was formed 
 independently of that in page xxxiv of Volume XIX., from which 
 it slightly differs, owing to differences of the data and details of 
 the calculation.] 
 
 122. For observing transits of faint objects which admitted 
 of very little illumination of the field, I found that dark bars 
 of moderate breadth could be used with good effect. A set of 
 such bars, two on each side of the system of wires, was inserted 
 in the Cambridge transit-telescope between August 29 and 
 September 21, 1853, and after that date transits of faint objects, 
 such as the asteroids, were often taken at the bars, and 
 generally at the first edges, the observations of disappearance 
 being thought to be more certain than those of re-appearance 
 at the second edges. These bar-transits are all reduced to 
 transits across the mean of the wires, the intervals of the bar- 
 edges from this mean being determined as follows. In all the 
 instances, from September 26, 1853, to July 24, 1854, in which 
 for this purpose transits of Polaris were taken at the wires and 
 at the bars in the same meridian passage, the interval between 
 each bar-transit and the mean of the wire-transits reduced in 
 the usual way to the mean of all the wires, was determined. 
 To each such interval for a given edge, and the corresponding 
 declination of Polaris, a weight was attached proportional to 
 the number of wire-transits, and from all the intervals and 
 declinations, account being taken of the weights, a mean 
 interval (t') corresponding to a mean declination (D'\ was 
 deduced. Then the equatorial interval (a) for the given edge 
 was obtained by means of the formula, sin I5a = cos D' sin I5t'. 
 The same computation was gone through for all the edges, and 
 the eight intervals were arranged in the order of the transits 
 of a star above pole for Illumination East. Then, for facility of 
 calculation, a list was formed of the reductions to the mean of 
 the wires for an equatorial star in all the cases that could occur 
 of transits, broken and complete, at first edges, both for Illumi- 
 nation East and Illumination West, so that the reduction in 
 C. 8 
 
PRACTICAL ASTRONOMY. 
 
 any instance of a bar-transit was obtainable by simply mul- 
 tiplying the proper equatorial reduction by the cosecant of 
 polar distance. (See Cambridge Observations, Vol. XX., pages 
 ii and iii). 
 
 123. In the case of the Sun, or any body of the Solar 
 System, the diurnal motion is diminished by the apparent 
 increment of Right Ascension due to the motion of the body 
 in its orbit. If, therefore, I be the horary increment of its 
 R.A., as given in the Nautical Almanac, or by other means 
 obtainable, any interval for the moving body is greater than 
 that for a star of the same polar distance in the ratio of 3600 
 to 36007. Hence for the Sun, or a Planet, or a Comet, 
 
 , 3600 
 
 3600^7 
 
 124. For the Moon another factor is required, in order to 
 take account of parallax, the effect of which in this case is of 
 sensible amount. In Figure 27, A is the place of the observer 
 
 L C 
 
 on the meridian AE } G the Earth's centre, CE the radius of the 
 Equator, CAZ the direction of Geocentric Zenith, L Az that of 
 Astronomical Zenith, and M the position of the Moon in its 
 orbit, so that the angle AMG is the parallax (p). The effect of 
 this parallax on the Moon's apparent motion may be determined 
 as follows. If we suppose an angular motion about the Earth's 
 axis, equal and opposite to the Earth's rotation, to be impressed 
 in common on all bodies, the Earth will be, as it appears to be, 
 at rest, and the motions of the other bodies will be just what 
 
.'> 
 
 THE TRANSIT INSTRUMENT. 115 
 
 they appear to be. But in that case if co be the impressed 
 angular motion and D the distance of the Moon from the 
 Earth's axis, the motion impressed on the Moon will be Deo ;' :/ ., 
 and if w be the Moon's orbital motion resolved parallel to the 
 Equator, the whole motion in space about the Earth's axis is 
 Deo + w. This quantity, divided by AM the distance of the 
 spectator from the Moon, gives the Moon's local angular motion, 
 or that which is measured by a transit instrument at the place. 
 The angular motion relative to C the Earth's centre is the same 
 quantity divided by CM. Hence the above interval d is short- 
 ened by the parallax in the ratio of the distances, or the 
 
 AM 
 
 additional factor for the Moon is -- . Now we have 
 
 AM sin t ACM sin Z- 
 
 , A sin ^ AMG sin p AC CE 
 
 also ->.i>= - A 7ir = - ; and .'. sin p = -^^ . -T^T T . sin Z.. 
 CM sin ^ ZA M sm Z^ ' CE CM 
 
 Suppose that simultaneously with the transit observation the 
 Moon's apparent zenith distance zAM was found by a Mural 
 Circle or a Transit Circle to be Z, and that the local angle of 
 
 A C 
 the vertex ( = e), and ratio -^ ( = p\ are known. Then 
 
 Z^Ze; and if P be the Moon's Equatorial Horizontal 
 
 s~i TJI 
 Parallax, j>= sin P. Hence we have 
 
 sin p - p sin Psin (Z - e), 
 
 by which formula, after deriving P by interpolation from the 
 Nautical Almanac, p may be calculated. Then the required 
 factor, being equal to cosec (^- e) sin (^ e j9), is readily 
 obtained. In this calculation the arcs p and P should not be 
 substituted for their sines. For the latitude of the Cambridge 
 Observatory p = [9,9990916], and e = 11' 12", the ratio of the 
 Earth's axes being assumed to be that of 297 to 298. 
 
 If the Moon's apparent zenith distance be not determined 
 by observation, the ratio of AM to CM may be calculated 
 
 82 
 
116 PRACTICAL ASTRONOMY. 
 
 as follows, If X = ^ ALE, the latitude of the observatory, 
 tACE = \-e' y and if & = *MCE, the Moon's Geocentric 
 Decimation, to be obtained by interpolation from the Nautical 
 Almanac, the / A CM = X e A. 
 
 AM _$m*ACM _ sin (X-e-A) 
 ~~~ 
 
 Also sin p = p sin P sin (\ .e A 
 
 p sin Psin (X e A) 
 
 or tanp = -- . v -^T- 
 
 1 /> sm Pcos (X e A) 
 
 It will generally be sufficiently accurate to omit the small term 
 in the denominator. After obtaining p from this equation, the 
 ratio of A M to CM is obtainable from the formula above. (For 
 the Cambridge Observatory the val tie of X e to the nearest 
 second is 52 1' 40".) Putting Ffor this factor, we have for the 
 Moon 
 
 3600 
 d= 
 
 8600 
 
 / being taken immediately from the column of 'Variation of 
 the Moon's K. A. in one hour of longitude ' under the head of 
 Moon-culminating Stars in the Nautical Almanac. 
 
 125. I take occasion to remark here that although for the 
 Moon the interval of any wire from the mean of the wires is 
 partly dependent on parallax, the measurement of the Moon's 
 diameter by transits, spoken of in Art. 110, is not thereby 
 sensibly affected. The reason of this is that the Moon's apparent 
 diameter is greater to the spectator at A than to the spectator 
 at (7, in the same proportion that the apparent motion is greater, 
 so that the duration of the transit of diameter is independent 
 of the place of the observer. This is true also of transit 
 measures of the Moon's diameter taken out of the meridian 
 with an Equatorial Instrument. 
 
 126. Having shewn how to calculate the interval of the 
 transit of any celestial body across any wire from the mean of 
 the transit-times of a complete observation, we are now able to 
 obtain a rule for applying a correction for reducing to that 
 
THE TRANSIT INSTRUMENT, 117 
 
 mean the mean of the observed times of a broken observation. 
 The rule may be inferred from a particular instance, such as 
 the following. The number of wires being seven, suppose the 
 times to be observed at four wires and to be omitted at three; 
 and let m be the required reduced time ef transit across the 
 mean of all the wires. Then if m + a, m -f &> m + c, m + d be 
 the observed times, and m -f &, m +f, m + g the- omitted 1 times, 
 we shall have 
 
 _ (m -f a) -f (m + 5) + (m + c) -f (m + d) e +f'+g 
 ~1~~ ~~^' 
 
 supposing the observed times to be correct, because in that case 
 a+b+c+d+e +f+g = 0. The signs of a,, &,, c, &c. are 
 positive or negative according as they designate intervals after 
 or before mid-transit. (See Table in p. 112.) Hence the 
 following rule : Obtain the mean of the observed times, and 
 add to it the sum of the calculated intervals of the lost wires 
 from mid-transit divided by the number of the observed times. 
 
 127. The process of reducing an incomplete observation of 
 any star to the time of mean transit having been indicated, we 
 may now proceed to take into consideration the method of 
 obtaining Level error by transits of Polaris which was mentioned 
 in Art. 87. Suppose, first, that there is neither collimation 
 error, nor azimuth error, and on that supposition let HZ'hN' in 
 
 Figure 28 represent the great circle described by the line of 
 collimation, cutting the great circle of the meridian at H and h 
 
118 PRACTICAL ASTRONOMY. 
 
 in the horizon. Then the inclination of the plane of collimation 
 to the plane of the meridian is the z ZHZ ', or the small arc ZZ 
 drawn through the zenith at right angles to the two planes. A 
 transit of Polaris is taken at four wires by reflection at the 
 surface of mercury, and at the remaining three by direct obser- 
 vation, the slow motion of the star allowing time, after securing 
 the reflection observation, to set for the other. The mean 
 of the noted times is in each case reduced to time of transit 
 across the mean of the wires by the method explained in 
 Art. 126, a table of intervals being employed such as that for 
 Polaris in page 112. Let t v and 2 be the reduced times for the 
 reflection and direct observations respectively. Then since by 
 the principle of reflection / cOh s. a Oh, the small arcs an and 
 an are equal, and the time in which the star describes either 
 
 is 5 ft - O. Let PZ the colatitude of the observatory = I, 
 
 2t ' ^ 
 
 and Pn the star's polar distance = 5, so that Zn = l for 
 Polaris above Pole. Then <rn, which is equal to Z2Tco&(l $), 
 
 15 
 
 is also equal to - - (t ~ 2 ) sin 5. 
 
 ft 
 
 The correction for Level error is positive if the West end of the 
 axis be highest (Art. 78), and in that case Polaris crosses the 
 plane of collimation before passing the meridian in the direct 
 observation, and after passing the same in the reflection 
 observation. Hence t t is greater than 2 , and we have 
 
 1^ 
 
 ZZ' = an sec (1-$) = *^- ft - Q sin 8 sec (I - 8). 
 
 If Polaris be below Pole, 2 is greater than ^ for a positive 
 azimuth error, and the arc Zn is I + 8. Hence the same formula 
 applies by merely changing the sign of 8. Or, taking 8 always 
 positive, and the respective times for the reflection and direct 
 observations to be always ^ and t# the general formula is 
 
 Level-correction = -^-ft 2 ) sin 8 sec (I + 8), 
 according: as Polaris is above or below Pole. 
 
THE TRANSIT INSTRUMENT. 119 
 
 The effect of collimation error, if it existed alone, would be 
 to make the line of collimation describe a small circle of the 
 sphere parallel to the plane of the meridian, and consequently, 
 as the reduced transit-time would be the same for the direct, 
 as for the reflection, observation, ^ t 2 would be zero. Also if 
 there were only azimuth error, each reduced time would be the 
 transit-time across the same vertical plane inclined at a small 
 angle to the plane of the meridian, and hence as far as depends 
 on azimuth error t 2 will again be zero. Thus, since each 
 small error is independent of the amounts of the other two, the 
 determination of Level error by this method is not affected by 
 actual small errors of collimation and azimuth. 
 
 128. When the Level error has been found either by the 
 Spirit-level, or by the foregoing method, we have the choice 
 between two ways of determining azimuth error. For we may 
 either obtain the sum of the collimation and level errors by 
 the collimating eye-piece (Art. 84), whence, the level error 
 being known, the collimation error may be inferred ; or we may 
 find the collimation error directly by means of collimators 
 (Arts. 53 56). The collimation and level errors being thus 
 in either case both ascertained, the azimuth error may be 
 calculated by the methods indicated in Arts. 88 and 89. It 
 will be seen that these two ways of finding the three errors do 
 not require the instrument to be reversed, and are, therefore, 
 applicable when a Transit- Circle is used without reversion. It 
 is, however, to be remarked, that as the determinations ought to 
 be free from any effects of irregularities of the forms of the 
 pivots, the correction of such effects should be provided for by 
 special measures of the co-ordinates of the dots (Arts. 94 96) 
 for all the positions of the telescope that are employed for 
 calculating the three errors. Upon the whole, for a non- 
 reversed Transit-Circle, the best method seems to be to 
 determine the collimation and level errors by combining with 
 the use of collimators that of the collimating eye-piece. 
 
 129. To complete what has to be said relative to the 
 Transit Instrument under the Third Division, it remains to 
 
120 PRACTICAL ASTRONOMY. 
 
 give an account of the Chronographic 1 method of registering 
 transits by means of a galvanic circuit. The apparatus em- 
 ployed at Greenwich for this purpose is described, with 
 illustrations by plates, in the Appendix to the Volume of 
 Greenwich Observations for 1856, to which " Description " 
 recourse must be had by those who desire to be fully acquainted 
 with all the details of this method. As far as regards these 
 Lectures it will suffice to give a general account of the chief 
 parts of the apparatus, such as that contained in the Intro- 
 ductions to Volumes of the Greenwich Observations (in the 
 Volume, for instance, for 1874, pp. x and xi), which for the 
 most part I have followed in the subjoined statements. 
 
 ISO. A clock being mounted in a suitable position, its 
 movement is governed by the conical rotation of a sidereal 
 seconds' pendulum, and to ensure regularity of the movement, 
 which is an essential condition, apparatus is provided whereby, 
 as the radius of the conical arc increases, a revolving plate is 
 
 1 This is also called the American method,, having been, both as to principle 
 and working, invented solely by physicists and astronomers of the United 
 States. I was first made acquainted with it by Professor Mitchel of Cincinnati, 
 Borne time in 1848 or 1849, on the occasion of his visiting the Cambridge 
 Observatory, when, if I remember rightly, he had come from America for the 
 purpose of executing a commission to procure in Europe a large telescope for 
 the Cincinnati Observatory. The method he described to me involved the 
 principle of automatic registration of seconds by dots impressed on a revolving 
 circular disk by a point acted upon by a galvanic current in connection with the 
 seconds' pendulum of an astronomical clock ; a radial shifting of the recording 
 point after each revolution of the disk; and means of impressing dots by 
 another point acted upon by the same galvanic current at the will of the 
 observer. By this arrangement, which comprised all the more essential parts 
 of the invention, the seconds' dots radiated from the centre of the disk, and the 
 seconds' intervals were, therefore, measured by arcs increasing with the radius. 
 Mr W. C. Bond, Director of the Harvard College Observatory, introduced a 
 great improvement by making the records on a cylinder, covered with paper, 
 which was caused by appropriate mechanism to revolve uniformly about its 
 axis, and at the same time to have a uniform slow motion in the direction of 
 the axis. By these means the seconds' dots are separated by equal intervals, 
 and are arranged without confusion on the turns of a spiral. (For a history of 
 the successive steps of this important invention, see Annals of the Harvard 
 Observatory, Vol. L, Part I., pp. xxiv xxviii.) 
 
THE TRANSIT INSTRUMENT. 121 
 
 dipped deeper in an annular trough of glycerine, thus supplying 
 increased resistance without affecting the centrifugal force upon 
 the pendulum. A spindle of the clock gives a revolving motion 
 to a brass barrel, about 20 inches long and 12 inches in 
 diameter; and as the clock with conical pendulum moves 
 without jerks, the barrel revolves with a motion sensibly uni- 
 form. A point on the cylindrical surface moves through 0,3 
 inch nearly in a second of time. The surface is completely 
 covered with a sheet of paper, which, when filled by the registra- 
 tion, can be easily removed, to be replaced by another sheet. 
 A different spindle of the clock turns two long screws, which 
 are both parallel to the axis of the barrel, and cause a travelling 
 frame to traverse its whole length. In one revolution of the 
 barrel the frame moves through 0,1 inch. The travelling 
 frame carries two levers, each armed at one end with a pricking 
 point, and so mounted that when the opposite end is pulled 
 away from the barrel, the pricking end is impressed on the 
 barrel, and makes a permanent puncture on the paper. The 
 prickers are mounted in such manner that when their points 
 have entered the paper, they yield laterally to the motion of the 
 revolving barrel, and do not scratch the paper. [In Bond's 
 apparatus (see Note p. 120) a pen traces a continuous spiral 
 line on the sheet of paper, and the records are indicated by 
 small offsets, those for the observations being distinguished from 
 the others by being somewhat longer.] Two galvanic magnets 
 are fixed on the travelling frame, so as to attract the lever ends 
 opposite the pricking points, and thus in order to produce 
 punctures on the paper by these points, it is only required to 
 send galvanic currents through the magnets. 
 
 One of the prickers registers the seconds of the Sidereal 
 Standard Clock; for which purpose the wires of its magnet, after 
 passing through a galvanic battery, are led to the Sidereal 
 Standard Relay, which completes the circuit, and produces a 
 puncture at each second, excepting the first second of each 
 minute. The omission of the corresponding puncture serves to 
 indicate the commencement of each minute. The circuit can 
 be broken at pleasure, and the movement of the travelling- 
 
122 PRACTICAL ASTRONOMY. 
 
 frame be suspended when not required for recording obser- 
 vations. 
 
 The other pricker is used for registering times of observation. 
 The wires of its galvanic magnet, after passing through a battery, 
 are led to the Transit-Circle-pier, and to insulated brass rings 
 on the transverse axis of the telescope, whence conducting wires 
 pass within the telescope-tube to a contact-piece near the eye- 
 end, where the observer by a touch of the finger can complete 
 the circuit, and thus make a puncture on the revolving barrel 
 at the instant he judges the object observed to be on a transit- 
 wire. The same arrangement is made available for transit 
 observations with three other instruments, the Altazimuth, the 
 Great Equatorial, and the Sheepshanks Equatorial. Also as the 
 clock Hardy is still employed for eye-and-ear observations of 
 stars near the pole, its contact springs are occasionally thrown 
 into circuit with the magnet of the observations-pricker, so as 
 to give the means of obtaining from the chronograph-register 
 the difference between its time-indications and those of the 
 Sidereal Standard Clock. 
 
 131. The chronographic method of recording transits has 
 two special advantages; (1) it allows of taking the times of 
 transit of an object across wires separated by small intervals, 
 and, consequently, of observing several objects in quick succes- 
 sion, at the same time that each is taken at a sufficient number 
 of wires ; (2) the estimation of the fraction of a second can be 
 made with greater certainty by this method than by the eye- 
 and-ear observation, inasmuch as the position of the observations- 
 dot relative to the antecedent seconds-dot may be estimated 
 from the register at leisure by sight in tenths of the seconds- 
 interval, or if great precision be required it may be read off by 
 applying to the interval a scale of ten equal parts. Moreover, 
 the discordance between transits taken by different observers, 
 due to what is termed personal equation, is found to be less 
 when the time of transit is recorded by means of the senses of 
 sight and touch, than when the transits are observed by eye and 
 ear. The mode of eliminating the influence of personal equa- 
 tion on the results of transit observations will come under con- 
 
THE TRANSIT INSTRUMENT. 123 
 
 sideration in the next Division. I propose to add here, as an 
 appropriate conclusion of Division III., some remarks on the 
 character of this peculiar error, and its amount under different 
 circumstances. 
 
 132. In Volume XLIX. of the Astronomische Nachrichten, 
 cols. 1 30, the editor, Professor C. A. F. Peters, has given an 
 elaborate and interesting account of facts relating to personal 
 equation, together with a discussion of its influence in chrono- 
 graphic registration of transits, as compared with that in the 
 ordinary method of observing transits. The following particu- 
 lars have been extracted from that article. Maskelyne, in his 
 account of the Greenwich Observations of 1795, made the remark 
 that although the transits of his assistant Kinnebrook agreed 
 with his own at the beginning of that year, in August they were 
 O s ,5 later, which difference increased in 1796 to 8 ,8. Bessel, 
 induced by that remark, subjected the facts at his command to 
 a full enquiry, from the results of which (given in Part 8 of the 
 Konigsberg Observations) it appeared that Walbeck observed 
 later than himself by 1 8 ,02, and Argelander by 1*,22, and that, 
 as shewn by direct comparisons, Struve's transits were later by 
 0*,04 in 1814, and by 8 ,77 in 1834. In the same Article 
 mention is made of other differences of personal equation, not 
 quite so remarkable : Wolfers observed earlier than Nehus by 
 8 ,73, Gerling later than Nicolai by 0*,78 ; Rogerson at 
 Greenwich observed in 1854 earlier than Main by s , 70, 
 although in 1840 he observed later by O 8 , 15, whereas in the 
 whole of the twelve years 1841 1853, between Main and 
 Henry there was scarcely any difference. These facts shew 
 that personal equation is very different in different individuals, 
 and in the same individual is not a constant quantity. Professor 
 Peters has also stated that although Bessel's observations of 
 transits were earlier than Argelander's by more than a second, 
 his observations of sudden phenomena, as occultations of stars 
 by the Moon, were not earlier by more than s , 28. 
 
 133. For judging of the comparative values of transits 
 observed by the chronographic register and by the usual method 
 Professor Peters has given the results of five sets of observations, 
 
124 PKACTICAL ASTEONOMY. 
 
 made by five combinations of two out of four observers, each 
 set of observations consisting of transits of stars taken by eye and 
 ear on one day, and on the day next before or after by galvanic 
 registration. The stars observed by the two methods were for 
 the most part the same in the same set, but different stars were 
 observed in the different sets. Each observer took transits of 
 each star at two or at three wires by the usual method, and at 
 three or at four wires by the register-apparatus. The general 
 result of 53 comparisons may be stated to be, that personal 
 equation is less subject to variation, and the probable error of 
 the transit-time of a star across any wire is considerably (erhe- 
 blich) less, by the new method than by the old. It appears, in 
 fact, from all the comparisons that the probable error of a 
 single transit is 9 ,082 by the usual method, and 8 ,060 by the 
 register-apparatus, according to one computation, and s , 087 
 and 0",055, according to another. It may also be mentioned, 
 as farther indication of the superiority of the latter method, 
 that it agrees better than the other with observations, such as 
 those of Polaris, which are not affected by personal equation. 
 From the foregoing considerations it is evident that differ- 
 ences of personal equation have to be determined, whether the 
 transits be observed by the galvanic or the ordinary method, 
 as is done, for instance, very fully from year to year for 
 the galvanic method in the Introductions to the Greenwich 
 Observations. (See the Introduction to the volume for 1873, 
 pp. xliv liii.) 
 
 134. From the experience I had as Director of the 
 Cambridge Observatory I am able to add the following particu- 
 lars respecting personal equation in eye-and-ear observations. 
 From reduced transits taken in the year 1841, nearly contem- 
 poraneously, by myself (C) and one of my assistants, Mr John 
 Glaisher (G), I was led to infer that our observations were 
 "very nearly alike/' and from like contemporaneous transits 
 taken in the same year by G and my other assistant, Mr Baldrey 
 (B), it appeared that B observed earlier than G by 8 ,28. 
 (See Cambridge Observations, Vol. XIII. for 1840 and 1841, 
 p. xxxviii.) These observations were not made with any 
 
THE TKANSIT INSTRUMENT. 125 
 
 reference to personal equation. In 1843, July 24, 12 ft 13*, 
 transits were specially taken for determining the difference of 
 the personal equations of C and G. Eight stars of various polar 
 distances were selected, one observer took the transit of the 
 star at the first four wires, and the other at the remaining three, 
 and the successive observations of the transits were commenced 
 fey the two observers alternately. The mean result of the eight 
 comparisons was, that C observed earlier than G by 0*,004, 
 which is so small a quantity that the difference of the personal 
 equations may be taken to be :zero. The particular results, 
 however, indicated a small difference depending on the polar 
 distance of the star, C observing the slower moving stars later, 
 and the quick moving stars earlier, than G. (See Vol. XV. for 
 1843, p. xii.) In the Introductions of the subsequent Volumes 
 of Cambridge Observations, determinations of differences of 
 personal equations are given in a large number of instances, 
 in consequence of frequent changes of the observers having 
 occurred. All these determinations were made by reducing 
 independently the transits of each observer to the same epoch, 
 0* of sidereal time, the clock's rate being in no case inferred 
 from the transits of different observers. This method I consider 
 to be preferable to that spoken of above as having been 
 employed on July 24, 1843. The amounts of difference of 
 personal equation range generally between 8 ,2 and O s ,4, but 
 in several instances are much smaller, not exceeding the 
 twentieth of a second. I found, for instance, that the transits 
 of Mr Charles Todd, Mr Carrington, my son J. L. Challis, and, 
 as is stated above, those of Mr John Glaisher, agreed closely 
 with my own. The last-named observer agreed with Mr James 
 Glaisher of the Greenwich Observatory, who, as I have under- 
 stood, agreed with the Astronomer Eoyal, as also did Mr Main, 
 and it has already been mentioned that there was very little 
 difference between Mr Main and Mr Henry. Here then are 
 nine observers, under various circumstances, whose transits were 
 found to be in accordance ; from which it would seem allowable 
 to infer that there exists a faculty, although not possessed by all 
 observers, of taking transits nearly, or wholly, unaffected by 
 
126 PRACTICAL ASTRONOMY. 
 
 personal equation ; that, in short, there is a right way and a 
 wrong way of observing transits by eye and ear. 
 
 135. Another peculiarity, which is of the nature of personal 
 equation, has been detected in transit observations of the 
 circular discs of the Sun, Moon, and Planets, some observers 
 recording transit-times of the two limbs which are neither 
 consistent with those of stars nor with each other. In the 
 Introduction to Volume XIII. of Cambridge Observations, for 
 1840 and 1841, pp. xxii xxiii, I have adduced a special in- 
 vestigation of corrections to be applied on this account to 
 Mr Baldrey's observations of transits of the Sun's limbs, and to 
 the inferred Right Ascension of the Sun's centre. Like cor- 
 rections are also obtained, but with less certainty, for transits of 
 the limbs of the Moon and of several Planets. The discrepancy 
 appears to arise chiefly from discordant observations of second 
 limbs. It may here also be mentioned that I detected a con- 
 stant difference between my own microscopic bisections of a 
 graduation-line of the Mural Circle and those of one of my 
 assistants ; which, probably, was to be attributed to certain phy- 
 siological conditions of the organs of sight, differing in that 
 respect from personal equation, which seems to have relation to 
 mental conditions. 
 
 IV. CALCULATION OF APPARENT RIGHT ASCENSIONS FROM 
 
 THE TRANSIT OBSERVATIONS, AND OF THE MEAN RIGHT 
 
 ASCENSIONS OF STARS. 
 
 136. Before indicating the processes which in the existing 
 state of Practical Astronomy are employed for inferring ap- 
 parent Right Ascensions from reduced meridian transits, and 
 obtaining concluded Mean Right Ascensions of stars, I propose, 
 with the view of distinctly exhibiting the principles involved in 
 those processes, to state historically how they have actually 
 been discovered, and to shew also how they might be logically 
 arrived at by beginning de now. The requisites for freeing the 
 transit-instrument from the effects of faulty mechanical con- 
 ditions, and placing it exactly in the plane of the meridian, 
 
THE TRANSIT INSTRUMENT. 127 
 
 were very imperfectly understood by astronomers who preceded 
 Flamsteed, and even the methods which he employed for these 
 purposes fell far short of attainable accuracy. It appears from 
 what is said in Division IL, that it is possible to effect all the 
 instrumental adjustments without knowing the Right Ascension 
 of any star (see Art. 90). This having been done according to 
 rule, an astronomer provided with a clock going pretty re- 
 gularly, might, although his transit-instrument were imperfectly 
 adjusted, ascertain the clock's rate with much accuracy by 
 transits from day to day of the same unknown star, or stars, 
 inasmuch as the transits of a given star might be supposed, for 
 a few consecutive days, to be equally affected by defect of 
 instrumental adjustment. The clock's rate, which is the differ- 
 ence between the clock- times of transit of the same star on 
 consecutive days and 24 hours, being thus sufficiently deter- 
 mined, he might proceed to calculate the differences of sidereal 
 intervals between the transits of different stars, and after fixing 
 upon one of them as a point of reference, the sidereal intervals 
 between the transit of that one and the transits of all the 
 others on the same day might be inferred. These intervals, 
 which would be affected by imperfect instrumental adjustment 
 and other sources of error, might, however, be called pro- 
 visionally Right Ascensions of the different stars, such deter- 
 minations of Right Ascension having been actually made by 
 the older astronomers. Flamsteed, having undertaken to 
 observe Sun, Moon, and Planets, as well as stars, and per- 
 ceiving the advantage of choosing a point of reference not 
 defined by the place of a star, selected for his purpose the point 
 of intersection of the Equator with the Ecliptic, which is named 
 the first point of Aries. There is, besides, a particular reason 
 why a star should not be chosen for point of reference, arising 
 out of the well ascertained fact that stars have proper motions, 
 in consequence of which the determinations of Right Ascension 
 might be affected to an unknown amount, and in different 
 degrees, by the differences of the proper motions of the selected 
 star and those of the other stars. The adoption of the equi- 
 noctial point for the origin of arcs of Right Ascension, gets rid 
 
128 PEACTICAL ASTRONOMY. 
 
 of this source of error, and at the same time allows of in- 
 dependent determinations of the proper motions of stars in R A. 
 Accordingly I proceed now to describe a method of determining 
 the position of the first point of Aries, which, although differing 
 in details from that employed by Flamsteed, is the same in 
 principle. 
 
 137. It is not possible to determine the place of the 
 Equinox except by combining transit observations of the Sun 
 with simultaneous observations of its declination by a Mural 
 Circle, or a Transit-Circle. Hence, although the mode of taking 
 observations of declination has not yet been under consideration, 
 I shall anticipate so far as to assume that transits of the Sun's 
 centre and observations of its declination are taken at the same 
 time at several meridian passages both preceding and following 
 the Vernal Equinox. (As respects the logic of the present 
 enquiry it may be remarked, that the supposed observations of 
 the Sun's declination are simply obtained by instrumental 
 means, and are in no way dependent on determinations of 
 places of stars, or of their right ascensions.) Supposing, in 
 accordance with what is stated in Art. 136, the errors of colli- 
 mation, level, and azimuth, of the Transit Instrument to be 
 ascertained prior to any determination of the right ascensions 
 of stars, and the clock to be rated by transits of unknown stars, 
 let t lt t 2 , t a , &c. be the clock-times of the several transits of the 
 Sun's centre observed near the Equinox, corrected both for 
 instrumental errors and for clock-rate, so that t 2 t lt t s t v &c. 
 are true sidereal intervals between the transits. Then if 
 d^ d 2 , d z , &c. be the simultaneously observed declinations of 
 the Sun's centre, and d the declination at any arbitrary time t, 
 and if we take, for the sake of simplicity, only three transits, 
 we shall have by a known formula of interpolation 
 
 , , -- j , 
 ' - 2 
 
 ft - ft - ' ft - .) ft - f s) 2 ft - ,) ft - 3 ' 
 
 Suppose now that d= 0, or that the Sun is at the Equinox. 
 Then for determining the corresponding clock -time t Q , we have 
 an equation of the form = A + Bt + Ct*, in which A, B, G 
 
THE TRANSIT INSTRUMENT. 129 
 
 are given functions of t v 2 , t a and d v d 2 , d z . Hence, after ob- 
 taining a positive value of t from the solution of this equation, 
 if we subtract this value from each of the times t lt t z , t a we 
 obtain sidereal intervals reckoned from the instant at which 
 the Sun passed from the South to the North side of the Equator. 
 (Similarly, sidereal intervals reckoned from the instant the Sun 
 passed from the North to the South side of the Equator might 
 be obtained by observations made near the Autumnal Equinox.) 
 We have now to find the clock-error when the Equinox is on 
 the meridian of the place of observation, which is equivalent to 
 making the clock indicate local sidereal time. 
 
 138, The rate of change of the Sun's right ascension about 
 the time of the Equinox may be inferred as follows from the above 
 mentioned Transit observations. Let, for instance, a be the 
 increment of the Sun's R. A. in the interval t n t n _^ between two 
 meridian transits, which, for the sake of simplicity, will be sup- 
 posed to be the one next before, and the one next after, the passage 
 of the Sun through the equinoctial point. Now this increment 
 takes place in 24 h -f a of sidereal time, so that t n t n _^ = 24 h + a ; 
 or a = t n t n _ t 24 h . Hence the increment in the interval 
 
 t n ~ t is (t n - t n ^ - 24 h ) tn ~ to , which must be the Sun's actual 
 
 l n ~ Vi 
 
 right ascension at the epoch t n , because at the beginning of that 
 interval the R. A. was zero. Hence the correction of the clock- 
 error at the time t n is the excess of that R A. above t n , which is 
 
 equal to t Q - 24 h n . So the correction of clock-error at 
 
 f "~ t 
 
 the time t n _^ is - t Q + 24 h ~ n ~ l . (The first or second of these 
 
 St ~~ \-i 
 
 expressions should be preferred according as t n 1 is less or 
 greater than t - t n _^ When the clock has been actually, or 
 virtually, corrected according to either expression, the seconds' 
 hand is made to point to O h ,0 when the Equinox is on the 
 meridian, and consequently the clock will indicate local sidereal 
 time. The error of the clock at a given epoch being thus cor- 
 rected, and its rate being supposed to be continuously deter- 
 mined, its error at the clock-time of the meridian transit of any 
 c. 9 
 
130 PRACTICAL ASTRONOMY. 
 
 object can be deduced, and accordingly the clock and transit- 
 instrument be employed to recerd the R. A. of any number 
 of objects as referred to the Equinox of the given epoch. The 
 rate of regression of the Equinox being supposed to be uniform 
 and of known amount, the origin of these R A. might be 
 transferred to the position of the Equinox at any other given 
 epoch ; and it is evident that the R. A. of a given star, as deter- 
 mined by transits on different days, might also be referred to 
 this origin, and the mean of several determinations be adopted 
 as the value at the new epoch. 
 
 139. Flam steed formed his catalogued right ascensions in 
 the Historia Ccelestis for the most part in the above manner, 
 making use of a very accurate value of the precession of the 
 Equinoxes. But his results were affected by two sources of error 
 with which he was unacquainted. Assuming the regression of 
 the Equinoxes, as well as the earth's rotation about its axis, to 
 be uniform, astronomers at that time inferred that the arc of 
 the Equator between the Equinox and the Circle of Declination 
 of a star might be exactly measured by a correctly regulated 
 clock. Bradley discovered that for two reasons this is not the 
 case. First, because the actual regression of the Equinox is 
 not uniform, but oscillates about a mean regression, and again, 
 because, by reason of the aberration of light, the apparent place 
 of a star undergoes variations depending both on its position in 
 the heavens and on the season of the year. Bradley, after 
 allowing for regular precession, distinguished by his observations 
 between these two kinds of irregularity, and not only deter- 
 mined the quantities and laws of both, but ascribed them to 
 true causes, the variation of the rate of precession to nutation 
 of the earth's axis, and aberration to a relation between the 
 velocity of the earth in its orbit and the velocity of the pro- 
 pagation of light. The former cause has been confirmed by the 
 theory of gravitation, Laplace having found that the attractions 
 of the Sun and Moon on the protuberant equatorial parts of the 
 earth produce a nutation of its axis agreeing in law and amount 
 with the results of Bradley's observations. Respecting what I 
 conceive to be the rationale of the observed effects of aberration 
 
THE TRANSIT INSTRUMENT. 131 
 
 in telescopic observations I propose to introduce a Note in a 
 subsequent part of this work. At present we may proceed to 
 consider how transit observations are freed from the con- 
 sequences of these two sources of error. 
 
 140. Admitting it to be proved by the theory of gravity that 
 the uniformity of the earth's diurnal rotation is not affected by 
 the nutation of its axis, it is evident that the difference of the 
 meridian-transits of two stars, as shewn by a well regulated 
 clock, does not exactly measure the difference of their R. A., 
 inasmuch as the pointing of the telescope towards any star will 
 be altered in a certain manner by such nutation. At the same 
 time it is altered by the apparent displacement of the star by 
 aberration. Bradley's discoveries gave the means of calculating 
 precisely the amounts by which the visual direction of the star 
 is changed in R. A. by each of these causes, and consequently of 
 correcting the instrumental measurement of the apparent differ- 
 ence of R,. A. of the stars so as to reduce it to the true difference, 
 as measured by a clock regulated according to the rate of the 
 earth's rotation. It should also here be noticed that the theory 
 of gravity has detected another cause of error due to the 
 circumstance that the attraction of the earth by the Planets 
 changes the inclination of the plane of the earth's orbit to the 
 plane of its Equator, thereby producing a small progression of 
 the first point of Aries along the Equator, the law and amount 
 of which have been determined by the same theory. This is. 
 Planetary Precession, as distinguished from Lunisolar Preces- 
 sion, the two together constituting General Precession. When 
 the fluctuations of value arising from these several causes have 
 been eliminated from the instrumental measures of the differ- 
 ence of R. A. of two stars, the resulting mean measures are com- 
 parable with each other, and from a large number of such deter- 
 minations an exact measure of the difference of the mean R. A. 
 of the stars at the epoch of the mean of the times of the 
 measures may be inferred. The different measures, for greater 
 certainty, may be extended over a considerable number of years. 
 Maskelyne selected for such observations 36 stars suitably dis- 
 tributed over the heavens, and bright enough to be seen in day- 
 
 92 
 
132 PRACTICAL ASTRONOMY. 
 
 time, and compared each in R A. a great number of times with 
 one of them, a Aquilae. The R A. of this one was referred, by 
 the intervention of observations of the Sun, to the mean place of 
 the Equinox at a certain epoch, the process being mainly the 
 same as that of Flamsteed, which has already been described in 
 Arts. 137 and 138. After determining the absolute R. A. of 
 the 36 stars in this manner, Maskelyne named them Fzmda- 
 mental Stars 1 . The Apparent and Mean Places of fundamental 
 stars form an essential part of the contents of the Nautical 
 Almanac 2 . At the present time this work contains the mean 
 places of 149 standard stars for the beginning of the year, the 
 apparent places of 144 for every tenth day of the year, and the 
 apparent places of the remaining five, which are near the North 
 or South pole, for every day. The mean places of 136 were 
 derived from the Seven-year Catalogue forming Appendix I. of 
 the volume of Greenwich Observations for 1862. 
 
 141. In the determination of the place of the Equinox by 
 Flamsteed's method, only observations of the Sun taken near 
 the times of the Equinoxes are made use of. In the Volumes 
 of the Greenwich Observations, as also in those of the Cambridge 
 Observations, the place of the first point of Aries and the 
 obliquity of the Ecliptic, together with the mean error of the 
 assumed R. A. of the fundamental stars, are calculated from all 
 the meridian observations of the Sun taken in the course of 
 any one year, beginning at January 1. The process by which 
 this is done cannot be logically indicated till the reduction of 
 Circle observations of the Sun's decimation has been treated of; 
 at present we may go on to enquire in what way the apparent 
 R. A. of objects are deduced when their transits have been 
 taken contemporaneously with those of fundamental stars. 
 
 142. The formulae now generally employed for calculating 
 apparent places of stars from mean places, or, reversely, mean 
 places from apparent, depend upon astronomical researches of 
 
 1 See Grant's History of Physical Astronomy, p. 489. Respecting the history 
 of Practical Astronomy generally, much valuable and interesting information is 
 given in Chapter xviu. of that work. 
 
 3 The original publication of the Nautical Almanac, which was due to Mas- 
 kelyne, took place in 1767. 
 
THE TRANSIT INSTRUMENT. 133 
 
 Bessel contained in his Fundamenta Astronomice published in 
 1818, and his Tabulce Regiomontance published in 1830. The 
 former work consists principally of two catalogues of all the 
 observations of stars made by Bradley at Greenwich in the 
 years 1750 1762, reduced to the beginning of 1755, together 
 with discussions relative to the constant of aberration and the 
 Precession of the Equinoxes. In both works Bessel commences 
 his investigations relative to Precession and the Nutation of 
 the earth's axis with adopting Laplace's gravitational formulae 
 given in Tom. in. of the Mfoanique Celeste, but without 
 accepting the numerical values of the principal constants. 
 These he undertook to correct, first, by adopting a more recent 
 determination of the mass of Venus, and then by calculating 
 general precession exclusively from comparisons of Bradley's 
 places of stars 1755,0 with places in Piazzi's Catalogue 
 of 1800,0, after applying a general correction to the latter 
 deduced from absolute Right Ascensions of Maskelyne's 
 36 fundamental stars inferred from his own observations. 
 By this means he finally obtained in the Tabulce Regio- 
 montance, p. v., for the indefinite epoch 1750 + 1, the subjoined 
 formulas, in which ty is the lunisolar precession, or movement, 
 in the interval t, of the point of intersection of the Equator 
 with a fixed great circle with which the Ecliptic coincided 
 at the beginning of 1750, ijr l is the general precession or 
 movement in the same interval of the intersection of the 
 Equator with the actual position of the Ecliptic, co is the arc 
 between the Pole of the Equator and the Pole of the fixed 
 Ecliptic at the epoch 1750 -f t, and ta l that between the Poles 
 of the Equator and the actual Ecliptic at the same epoch : 
 
 ^ = 50",37572 - 0",0001217945 2 , 
 ^ = 50",21129* + 0",0001221483 2 , 
 to = 23 28' 18",0 + 0",OOQ00984233 2 , 
 6), = 23 28' 18",0 - 0",48368 - 0",00000272295* 2 . 
 
 According to these formulas the constant of general precession 
 at 1750,0 is 50",21129, and the annual diminution of the 
 obliquity of the Ecliptic is 0' / ,48368. For constant of annual 
 
134 PRACTICAL ASTRONOMY. 
 
 aberration Bessel adopts Delambre's value, 20",255, as being 
 nearly the same as that given by Bradley's observations of 
 7 Draconis. Also taking the Sun's parallax to be 8",5776 
 as found by Encke, he obtains for the constant of diurnal 
 aberration 0" 30847 (Tab. Reg. p. xxn). The values of 
 Solar Parallax and constant of annual aberration which 
 I have employed in Art. 63 for calculating local diurnal 
 aberration, give a value of that constant larger than 0", 30847 
 by 0",01576. 
 
 143. Since the publication of the Tabulae Eegiomontance, 
 astronomers have bestowed much labour on investigations for 
 correcting as accurately as possible Bessel's values of the 
 principal constants. This may be pre-eminently said with 
 respect to three dissertations contained in the Memoirs of the 
 Imperial Academy of Sciences of St Petersbourg, Tom. in. 
 (1844), pp. 17285, the first of which, by Otto Struve, is a 
 discussion on the constant of Precession, the second, by Professor 
 C. A. F. Peters, treats of the numerical constant of Nutation, 
 and the third, by W. Struve, is on the constant coefficient of 
 the Aberration of fixed stars. It will be proper to give here 
 some account of the processes and results of each of these 
 important contributions to Practical Astronomy. 
 
 (1) The value 50",21129 of general precession was obtained, 
 as stated in Art. 142, by comparing Bradley's R. A. of 1755,0 
 with Piazzi's of 1800,0, the interval being 45 years. 0. Struve 
 compared Bradley's R. A. of the same date with R. A. observed 
 at Dorpat reduced to the epoch 1825,0, and thereby found that 
 in the interval of 70 years the movement of the Equinox was 
 greater by 0",97 than that given by Bessel's constant. Ac- 
 cordingly this constant requires to be increased by one-seventieth 
 part of the difference, or 0",01386. To calculate the value of 
 Bessel's constant at the epoch 1780,0, which is the mean 
 between 1755,0 and 1825,0, we have 
 
 - = 50",21129 + 0",0001221483 x 80 - 50",22106. 
 
 (Mi 
 
 The addition of 0",01386 to this result gives 50",23492 for the 
 
THE TRANSIT INSTRUMENT. 135 
 
 corrected value of annual precession at the epocli 1790,0. It 
 should be farther stated that the correction + 0",97 was obtained 
 by O. Struve after eliminating, with all the accuracy attainable, 
 from the R. A. given by observation, errors due to the motion of 
 the Solar System in space. 
 
 (2) Dr Peters in his investigations relative to Nutation, 
 adopting formulae derived from the theoretical researches of 
 Laplace and Poisson, calculated the numerical values of the 
 coefficients from 603 observations of transits of Polaris taken 
 above and below Pole at the Dorpat Observatory from 1822 
 Nov. 4 to 1838 Oct. 15, of which number 249 were observed by 
 W. Struve, and the remaining 354 by his assistant W. Preuss. 
 After applying corrections for difference of the personal equa- 
 tions of the observers, and taking into account 0. Struve's 
 determination of the annual general precession, Peters found 
 for the coefficient of the principal term of the obliquity of the 
 Ecliptic a value which, combined with two other values deduced 
 by independent computations of MM. Busch and Lundahl, gave 
 as the mean result 9",22305 for the epoch 1800,0. 
 
 (3) For determining the constant of aberration W. Struve 
 observed transits of seven stars with a transit-instrument set up 
 in the Prime Vertical at the Imperial Observatory of Poulkova, 
 the zenith distances of the stars at meridian passage in no 
 instance exceeding 1 30'. Having carefully got rid of instru- 
 mental errors, and taken into account Peters' value of the 
 constant of nutation, he estimated at 0",0111 the total amount 
 of probable error due to observation, to annual parallax and 
 proper motion of the stars, and to nutation, and finally 
 obtained for the constant of aberration 20'',4451. 
 
 144. To complete this account of the fundamental quan- 
 tities pertaining to observational astronomy, the following 
 particulars are added relative to determinations of the Obliquity 
 of the Ecliptic, and the amount of its annual variation. Bessel 
 did not accept the values of the obliquity and of its annual 
 diminution (0",48368) given by the theoretical expression for co t 
 in Art. 142, but availing himself of obliquities deduced from 
 observations made by different astronomers in the interval from 
 
136 PRACTICAL ASTRONOMY. 
 
 1755 to 1815 (see Fundamenta Astronomice, p. 61), he obtained 
 for the epoch 1800 + t the formula, 
 
 Obliquity = 23 27' 54",80 - 0",4570. 
 
 Peters, adopting the apparent obliquities deduced by Bessel 
 from Bradley's observations, as given in page 58 of the Fund. 
 Astron., calculated therefrom by his own formulae a mean 
 obliquity 23 18' 14",06 for 1757,3, by comparison of which with 
 a value deduced by W. Struve from observations at Dorpat 
 (Vol. VI. of Dorpat observations, p. LXIX.), namely, 23 27' 42",61 
 for 1825,0 he found for 1750,0 the value 23 28' 17",44, and for 
 1800,0, 23 27' 54",22, giving a mean annual diminution of 
 0",4645 (see pp. 189 and 190 of the Memoirs cited in Art. 143). 
 Leverrier, according to results exhibited in page 203 of Tom. IV. 
 of the Annals of the Paris Observatory, gives for the epoch 
 1850 + 1, Obliquity = 23 27' 31 /; ,83 - 0",47594f. 
 
 The details of observational means by which the constants 
 treated of in the foregoing Articles may be determined or 
 corrected, will come under consideration in the sequel of the 
 Lectures. 
 
 145. I propose in the next place to give an account of the 
 derivation of formulae proper for calculating the Annual Varia- 
 tions of the mean R. A. and Decl. of any star at any epoch, these 
 formulae being auxiliary to those required for calculating the 
 differences in R. A. and Decl. between the apparent and mean 
 places of a star at the same epoch. Bessel deduced his formulae 
 in p. x. of the Tab. Regiomont. for the annual variations in R. A. 
 and Decl. from the expressions for ty, -fy v o>, &>j already adduced 
 in Art. 142. By a like process Peters obtained formulae 
 differing from Bessel's only as to the numerical values of the 
 coefficients, the new values having been deduced from 0. Struve's 
 corrected general precession, and, as respects terms containing 
 t z , by taking advantage of M. Leverrier's fuller development of 
 the gravitational theory, and introducing corrected values of 
 the masses of the planets. Putting, in accordance with Bessel's 
 notation, ^ for lunisolar precession from 1800 to 1800 + 
 referred to the Ecliptic and Equinox of 1800, for the angle at 
 
THE TRANSIT INSTRUMENT. 137 
 
 the epoch 1800 + t between the earth's Equator and the 
 Ecliptic of 1800, and X for the angle subtended at the Pole of 
 the Equator by the arc joining the Poles of the Ecliptics of 
 1800 and 1800 + 1, we have the following results of Peters's 
 researches : 
 
 ^ = 50",3798* - 0",0001084ft 
 w = 23 27' 54" + 0,00000735ft 
 X = 0,15119* - 0,00024186ft 
 
 m = ^ cos w - ^ = 46",0623 + 0,0002849ft 
 at at 
 
 n = ^ sin o> = 20",0607 - 0,0000863* ; 
 
 at 
 
 and the mean R A. and Decl. of the star being a and 8, 
 
 Annual Precession in R. A. = m .+ n tan 8 sin a, 
 Annual Precession in Decl. = n cos a. 
 
 Also putting ^ for the general precession from 1800 to 1800 + 1, 
 and Wj for the mean obliquity of the Ecliptic at 1800 + 1, the 
 same researches gave 
 
 ^ = 50",241U + 0,0001134ft 
 
 w, = 23 27' 54" - 0",4738* - / ',0000014* 2 ; 
 
 and consequently the annual diminution of the obliquity is by 
 this process 0",4738. (See Memoirs, &c., before cited, pp. 
 193195.) 
 
 146. After the foregoing discussion of the values of the 
 principal constants involved in the Formulas of Reduction which 
 are required for the calculation of Apparent places of Stars from 
 their Mean places, it remains to adduce these formulae (given 
 originally by Bessel in pages xxix and xxx of the Tab. 
 Regiomont.), with corrected coefficients, and to indicate suitable 
 modes of applying them. The subjoined equations, which are 
 adapted to these purposes, were taken from the Nautical 
 Almanac of 1879, and it may be assumed that their numerical 
 coefficients will be sufficiently accurate for at least ten years 
 
138 PRACTICAL ASTRONOMY. 
 
 from the present date (1877). For the sake of convenience in 
 making use of the formulae,- the sets of equations for calculating 
 Apparent Declination are exhibited together with those for 
 calculating Apparent Right Ascensions, although the former 
 have more immediate reference to observations with the Mural 
 Circle. The values of the coefficients are all adopted from 
 formulse given by Professor Peters in pages 71 76 of hisNume- 
 rus Constans Nutationis (= pages 195 200 of the Memoir before 
 cited), in which his own constant of Nutation, and the constants 
 of Precession and Aberration, determined, as already stated, by 
 0. and W. Struve, are introduced. 
 
 The Nutation of Obliquity being represented by A&>, and 
 that of Longitude by A, the formulae for computing these 
 quantities, given in page v of the Nautical Almanac of 1879, 
 are 
 
 Aw - 9",2237 cos & - 0",0895 cos 2 Q, + 0",5507 cos 20, 
 AZ = - 17,2526 sin Q. + 0,2073 sin 2Q - 1,2693 sin 20, 
 
 where $ is the mean Longitude of the Moon's ascending Node, 
 and the true Longitude of the Sun. 
 
 The Apparent Right Ascension of a star being a' in arc, 
 and its Apparent Declination S', at the time t reckoned each 
 year in fractional parts of a tropical year from the instant at 
 which the Sun's mean Longitude is 280 (which for 1879 is 
 Jan. + d ,227), the star's mean Right Ascension in arc, and 
 its Mean Declination, being respectively a and S, for the begin- 
 ning of the year, and Ac, Ac' its annual proper motions in R. A. 
 and Decl., the equations for calculating a.' a. and 8' 8 were 
 reduced by Bessel to the following convenient forms : 
 
 a' - a = Aa + Bb + Cc + Dd + Ac, 
 8' - S = A a + Bb ' + Cc ' + Dd' + 1 A c', 
 
 in which the factors A, B, C, D are the same at the same time 
 for all stars, a, b, c, d and a, b', c', d r are quantities depending 
 on the place of each star. The following equations, derived 
 from the foregoing expressions for Ao> and A.L, combined with 
 the formulas for the mean precessions in Right Ascension and 
 
THE TRANSIT INSTRUMENT. 139 
 
 Declination (Art. 145), and the known law and amount of 
 aberration, are employed for calculating the twelve quantities 
 A, B, &c. above specified. (See Nautical Almanac for 1879, 
 pp. 295 and 512j : 
 
 A = - 20",4451 cos w cos 0, B = - 20"4551 sin 0, 
 
 C=t- 0,02519 sin 20 - 0,34243 sin Q + 0,00410 sin ft, 
 D = _ 0",5507 cos 20 - 9",2237 cos Q + 0",0895 cos 2&, 
 a cos a sec 8, a = tan o> . cos B sin a sin 8, 
 
 6 = sin a sec 8, V cos a sin 8, 
 
 c = 46",0848 + 20",0539 sin a tan 8, c' = 20" 5 0539 cos a, 
 6? = cos a tan 8, d' = sin a, 
 
 where is the Sun's true Longitude, Q, the mean Longitude of 
 the Moon's node, and o> the obliquity of the Ecliptic, each for 
 the time t. The values of log A, log B, log C, log D are given 
 in the Nautical Almanac for every day of the year. 
 
 147. The Astronomer Koyal has obtained the following 
 formulae for calculating the excess of Apparent above Mean 
 
 R. A. in time ! = ^ (a' - a) L and the excess of Apparent 
 
 above Mean North polar distance {= (8' 8)} ; 
 
 jL( a '_ a ) + 300,00 =Ee + Ff+Gg + H1i + L + 1 
 
 JLQ 
 
 - (S' - 8) + 300,00 = Ee' + Ff + Qg +Hh' + L+ I'. 
 
 The particular advantage of these forms is, that all the terms 
 on the right-hand sides of the equations have positive numerical 
 values (excepting when the first equation is applied in the case 
 of a star less than 3 . 10' from the Pole), the relations between 
 the new quantities E, e, F, f, &c. and the quantities A, a, B, b, 
 '&c. of Bessel's formulae, being given by the following simple 
 equations : 
 
 #=,4 + 25, .F= + 25, =(7+1,2, # 
 L = 210 - 1,2 E-l^F-ZoG- 1,2 H, 
 
140 PRACTICAL ASTRONOMY. 
 
 r = 210 - 25 e' - 25/ - 1,2 g - 25 A'. 
 
 To facilitate this method of calculation, the values of 
 log F t log G, log H and L, which are independent of the 
 particular star, are given in the Nautical Almanac for every 
 day of the year. (See pp. (x) and (xi) of the Appendix to the 
 Greenwich Observations of 1847, containing the Twelve-year 
 Catalogue of 1836 1847.) Applications of both methods to a 
 particular instance are exhibited in pages 512 and 513 of the 
 Nautical Almanac of 1879. 
 
 148. The means of calculating the clock-time of transit of 
 any celestial object across the meridian having been indicated 
 in Division III., it is now required to shew how the true 
 sidereal time of transit is inferred from the clock-time by 
 correcting for the clock-error. For this purpose, it is first 
 necessary to form a catalogue of stars arranged according to 
 their mean R. A. at a given epoch, and to determine their R. A. 
 by a large number of observations, in order that they may be 
 used as fiducial points of reference, or fundamental stars. These 
 stars should be bright enough to be seen in day-time, at 
 moderate distances from the Equator, and their distribution in 
 Right Ascension should be pretty uniform. In the course of 
 the foregoing discussions (Arts. 136 147), it was shewn how 
 a provisional catalogue of R. A. might be formed ab initio, and 
 how, after the effects of precession, nutation, and aberration on 
 the apparent places of stars had been ascertained, the transit- 
 observations of a given star extending over a large number of 
 years might be all made to contribute to an accurate determi- 
 nation of its mean R. A. at a given epoch. By such means the 
 R. A. in the catalogues of mean places of stars produced in 
 successive Nautical Almanacs have been derived from numerous 
 transit observations taken at the Greenwich Observatory during 
 the superintendence of Maskelyne, Pond, and the present 
 Astronomer Royal. In the more recent catalogues ascertained 
 
THE TRANSIT INSTRUMENT. 141 
 
 changes of places of the stars due to proper motion 1 have been 
 taken into account. For these reasons if the stars selected for 
 use as fundamental stars be contained in one of the lists of the 
 Nautical Almanac, their Mean and Apparent R. A., as there 
 calculated, might be adopted without alteration. The process, 
 however, I am about to describe, which has been employed at 
 the Cambridge Observatory, although it takes advantage of 
 the Mean and Apparent R. A. of the fundamental stars as 
 calculated in the Nautical Almanac, really makes use of values 
 that are independently determined by observations made at the 
 Observatory. 
 
 149. The first list of mean R. A. of fundamental stars used 
 at the Cambridge Observatory was adopted by Professor Airy, 
 without alteration, from the Naut. Aim. of 1829, and the num- 
 ber of Stars was 34 inclusive of Polaris and 8 Ursae Minoris. 
 For the next year the adopted fundamental R. A. were derived 
 from the R. A. resulting from the observations of 1829 by add- 
 ing the Annual Variations in R. A.; and so on through successive 
 years to 1835 inclusive, at which date the number of the stars 
 was 31. On taking the superintendence of the Observatory in 
 1836, I followed the same rule, deriving the fundamental R. A. 
 for that year from the results of the transits of 1835 by adding 
 the annual variations, and, with the exception of omitting 
 Sirius, I adopted the same list of stars. The R. A. for 1837 were 
 obtained in the same manner ; but a new point of departure 
 was established by the list of fundamental R A. for 1838, 
 which were calculated on the following principles. The differ- 
 ence between the mean R. A. of every two of the stars taken 
 consecutively, was assumed to be the same as a mean difference 
 derived from all the transit observations of the two stars in the 
 eight years 1830 1837, by giving to the difference as deter- 
 mined by the observations of any one year a weight proportional 
 to the product of the numbers of the observations of the two 
 stars in that year. The epochs of the concluded mean differ- 
 
 1 The proper motions now adopted in the Nautical Almanac are those de- 
 termined by Main and Stone, contained in Vols. xix., xxvm. and xxxin. of the 
 Memoirs of the Eoyal Astronomical Society. 
 
142 PRACTICAL ASTRONOMY. 
 
 ences of R. A. thus obtained would be different for different 
 stars, and would generally be nearer to 1834,0 than to 1838,0. 
 As, however, the variations of the differences between the two 
 epochs would in all cases be extremely small, it was thought 
 sufficiently exact to refer all the concluded mean differences to 
 the common epoch 1838,0, after applying corrections for relative 
 proper motions in the instances in which the absolute proper 
 motions obtained from the most reliable sources indicated that 
 the difference of R. A. might thereby be sensibly affected. This 
 
 being understood, let a v 2 , a 3 , a n be the adopted mean 
 
 R. A. 1838,0 in order of magnitude, and A t , A 2 , A^.A^the 
 successive differences determined as above said. Then 
 
 a z = a i + AI> 3 = 2 + A 2 = a x + A x 4- A 2 , and so on. 
 
 Consequently a t + a 2 + + a n = na^ + Q, 
 
 Q being a given function of A t , A 2 ,...A M _ r If now we introduce 
 the condition that the sum of the excesses of the adopted R. A. 
 above the corresponding R. A. of the Naut. Aim. shall be zero, 
 the left-hand side of the above equation will be equal to the 
 sum of the Naut. Aim. R. A. Hence putting S for this sum, 
 we have S na v + Q, from which equation a t may be calculated, 
 and then in succession a 2 , a z ...a n . It resulted from this calcula- 
 tion that the mean of the adopted R. A. was less by O s ,083 than 
 the mean of those which would have been obtained from the 
 observations of 1837 by the former process. Consequently a 
 change in the origin of the R. A. was made at the beginning of 
 1838. As the true position of the Ecliptic relatively to the 
 assumed origin is determined from year to year by the 
 Circle observations of the Sun, this change is of no importance. 
 The adopted R. A. of Polaris and 8 Ursse Minoris were those 
 given by the observations of 1837 diminished by O s ,08. 
 
 150. Subsequently to 1838 the assumed Mean R. A. of the 
 fundamental Stars for any year were derived in general from 
 the R. A. resulting from the observations of the preceding year 
 by merely adding the annual variations of the Naut. Aim. 
 When, however, the number of the observations was small, the 
 assumed R. A. were deduced from the results of the observations 
 
THE TRANSIT INSTRUMENT. 143 
 
 of two preceding years by applying the annual variations, the 
 weight given to each year's result being proportional to the 
 number of the observations by which it was obtained. Even- 
 tually the assumed R. A., beginning with those of 1844, were 
 calculated by the following rule. If A be the Star's assumed 
 R. A. for any year, and A' the R. A. resulting from a number 
 (ri) of observations in that year less than 20, the assumed R. A. 
 
 77 
 
 for the next year is A -f (A' A) ^- plus the annual variation. 
 
 Having found it desirable to employ a larger number of clock- 
 stars, I added nine in the fundamental Catalogue of 1844, 
 the assumed R. A. of which were adopted from the Naut. Aim. 
 with the correction + O s ,02 applied, excepting that for two stars 
 observed in 1843 the corrections were + O s ,03 and 8 ,00. As 
 the mean excess of the R.A. of the fundamental stars above 
 the R. A. of the Naut. Aim., resulting from the observations of 
 1843, was + 8 ,020, it will be seen that the place of the Equinox 
 would not be sensibly altered by employing the adopted R. A. 
 of the additional stars. 
 
 151. We are now prepared to consider how the clock's daily 
 rate and the clock-error at any given time may be calculated. 
 It has been stated (Arts. 149 and 150) that after assuming 
 (provisionally) approximate Mean R. A. of the clock-stars for 
 the beginning of a certain year, the adopted Mean R. A. for the 
 beginnings of successive years are inferred from the results of 
 the yearly observations by the applications of Annual Variations 
 of Mean R. A. Formula? for calculating the Annual Variations 
 for any epoch are given in Art. 145. The Apparent R. A. of 
 any clock-star for any year, at the time of its meridian transit 
 on any day of the year, is derived from its adopted Mean R. A. 
 at the beginning of the year by applying a quantity termed the 
 Correction, the formulae for calculating which are those con- 
 tained in Art. 146. As already mentioned (Art. 146) both sets 
 of formula are the same as those used for the computations of 
 the Naut. Aim. Consequently, after applying to the Apparent 
 R. A. of the Naut. Aim. the excesses of the adopted Mean R. A. 
 for the beginning of the year above the corresponding Mean 
 
144 PRACTICAL ASTRONOMY. 
 
 R. A. of that work, we may regard the results as Apparent R. A. 
 independently determined by observations at the Observatory. 
 A clock regulated to sidereal time ought to indicate these 
 Apparent R. A. when the stars to which they pertain are on the 
 meridian. The difference between the clock-time of the transit 
 of a fundamental star and its calculated Apparent R. A. is 
 presumed to be the error of the clock. It is a great part of the 
 work of a well-appointed Observatory to determine such clock- 
 errors by observations carried on night and day. If a clock-star 
 has proper motion, it will plainly give an erroneous clock-error 
 if this motion is not taken into account in calculating the 
 Apparent R. A. For this reason the Annual Variations of the 
 Naut. Aim., which include ascertainable proper motions, should 
 be adopted. 
 
 152. For calculating the correction for clock-error to be 
 applied to the clock-time of meridian transit, the observations 
 are divided into groups severally containing observations of 
 stars proper for giving clock-errors. These groups are frequently 
 separated by intervals during which no observations were taken, 
 but, as often as may be, they consist of the night and day 
 observations made in consecutive intervals of 24 hours. The 
 mean of the clock-errors in each group is considered to pertain 
 to the mean of the times of transit of the stars which furnish 
 them. From a comparison of this mean error with errors 
 similarly derived from the next preceding and following groups, 
 a preceding and a following rate are calculated, each expressed 
 as proportionate loss of the clock in 24 hours, and applying to 
 the mean between the epochs of the errors from which it is 
 calculated. From the two rates a rate is inferred which is 
 assumed to be uniform throughout the middle group. No 
 definite rule can be given for deducing the adopted rate ; atten- 
 tion should be paid to the probable accuracy of the rates on 
 which it depends, and also to the proportion of the intervals 
 separating the preceding and following mean clock-errors from 
 the intermediate one. The rule which I have most frequently 
 adopted was obtained in the following manner. Let e, E, e' be 
 the three mean clock-errors, t-r,t,t + T f the corresponding mean 
 
THE TRANSIT INSTRUMENT. 145 
 
 24 
 clock- times, and r, r the computed rates. Then r = (Ee) 
 
 = the rate at t ~, and r = ->- (e / ~.E r )=the rate at t + ^ . 
 
 Zi T 
 
 Hence, supposing R to be the rate at the time t, and the rate 
 
 T" T" 
 
 to change uniformly in the interval from t ~ to + -^ , we 
 
 shall have 
 
 R-r r rr' + rV 
 
 ; = . . and. .*. .LI = 7- . 
 
 r r T + T T + T 
 
 The error E and rate R at the clock-time being thus known, 
 the clock-error at the next preceding O h of sidereal time may 
 be thereby calculated. Then the correction of clock-error for 
 the observation of any object whose approximate R. A. is A, 
 is the clock-error at O h , together with the proportional part of 
 the rate for the interval A, which is additive supposing the 
 clock, as is usually the case, to have a losing rate. 
 
 153. It should be noticed that if the observations from 
 which any mean clock-error is calculated were made by two 
 observers, the calculation takes account of the difference of 
 their personal equations, and thus virtually gives the error 
 as determined by the observations of one of the observers. 
 Also, if in calculating the clock-rates r and r' y the mean 
 clock-errors e, E, e are derived from groups of observations 
 made by different observers, they are reduced to errors as 
 obtained by one observer by applying corrections for differences 
 of personal equations (see Art. 134). 
 
 154. From the Apparent R. A. of a star thus found, its 
 mean R. A. for the beginning of the year may be obtained by 
 applying with opposite sign the correction calculated by the 
 formulae given in Arts. 146 and 147. If the star be in the list 
 of the Naut. Aim., the correction with the proper sign may be 
 obtained by subtracting its mean R. A. given in the Table of 
 "Mean Places of Stars," from its Apparent R. A. obtained by 
 interpolation from the Table of "Apparent Places of Stars." 
 The Apparent R. A. of stars are reduced to Mean R. A. for the 
 purpose of forming a Catalogue of their Mean R. A., the results 
 of all the observations of the same star in any year being thus 
 
 c. 10 
 
146 PRACTICAL ASTRONOMY. 
 
 made available for deducing a mean result. Also by applying 
 the annual variations the results obtained for the same star in 
 several years may be brought to the same epoch, and thus the 
 concluded R. A. of the Catalogue be made to depend on a large 
 number of observations. The same process of reduction to Mean 
 R. A. is applied to clock-stars for the purpose of farther correc- 
 tion of the assumed R. A. ; but, in Cambridge practice, this is 
 not done unless three of them at least are in the group. It is 
 from the mean of all the Mean R. A. of a clock-star obtained by 
 this process in the course of a year, that the adopted Mean 
 R. A. for the next year is deduced by applying the annual 
 variation (see Arts. 149 and 150). The Apparent R. A. of 
 Polaris and 8 Ursse Minoris are not independent determinations 
 except they are inferred from two or more alternate transits of 
 the star above and below Pole, and if this be not the case the 
 Mean R. A. are not calculated l . 
 
 1 Chapter xix. of Grant's History of Physical Astronomy contains an historical 
 account of Catalogues of the Fixed Stars, beginning with the earliest, and ex- 
 tending to those of recent date. I thought it might be useful to give the following 
 particulars extracted from this account. The several catalogues of Ptolemy 
 (A.D. 138), Ulugh Beigh (1437), Tycho Brahe (1600), Hevelius (1660), and 
 Halley's Catalogue of Southern stars (1677), after being subjected to a careful 
 revisal by Francis Baily, were reprinted, and now form Vol. xm. of the Memoirs 
 of the Royal Astronomical Society. Flamsteed's British Catalogue was published 
 in 1725, in Vol. in. of the Histnria Coelestis, and republished, with additions, by 
 Baily at the end of his "Account of the Life and Correspondence of Flamsteed." 
 Bradley's observations were published at Oxford in 1798, and from those made 
 between 1750 and 1762 Bessel formed his Fundamenta Astronomice, containing 
 3112 stars reduced to the epoch 1750,0. Then follow in order of time, Lacaille's 
 three catalogues, the principal of which, consisting of southern stars, was reduced 
 by Henderson to 1750,0, Mayer's Catalogue of 998 zodiacal stars, published in 
 1800, Maskelyne's fundamental catalogue of 36 stars, Wollaston's "Fasciculus" 
 of circumpolar stars (1800), the Histoire Celeste Fran^aise, published, unreduced, 
 in 1801, and in a reduced form by the British Association in 1837, Piazzi's cata- 
 logue of 7646 stars for the epoch 1800,0, Bessel's zone observations, reduced by 
 Weisse to 1825,0, and like observations of Santini, reduced to 1840,0, and pub- 
 lished in vol. xn. of Mem. of the Astron. Soc. Of subsequent catalogues, the 
 principal are, Taylor's Madras Catalogue of 11,015 stars formed from observations 
 in 1822-43, Pond's catalogue of 1112 stars reduced to 1830,0, the British Asso- 
 ciation catalogue of 8377 stars reduced to 1850,0 and the Greenwich twelve -year 
 catalogue (1840,0 and 1845,0) and seven-year catalogues (I. 1860,0, II. 1864,0) 
 formed by the present Astronomer Royal. Interesting details respecting the 
 above, and various other catalogues too numerous to mention here, are given in 
 Grant's History, as above cited. 
 
THE TRANSIT INSTRUMENT. 147 
 
 155. The Apparent RA. of bodies of the solar system, as 
 usually calculated in the Tables, are immediately comparable 
 with the R A. determined observationally by the processes 
 above indicated, excepting that the Tables and the observations 
 may not reckon the R A. from exactly the same position of the 
 equinox. The consideration of this point must be reserved till 
 observations with the Mural Circle have been under treatment, 
 the Transit Instrument being incapable of itself of determining 
 absolute Eight Ascensions. 
 
 156. An example of the complete Reduction of the transit- 
 observation of a star, extracted from one of the Calculation- 
 books of the Cambridge Observatory, is here subjoined : 
 
 Transit of a Aquilse, 1851 January 27, 23 h mean time. 
 Illumination-end of axis East. 
 
 N. P. D. of star = 81 . 31'. The line of collimation pointed 
 0",7 westward, and the west end of the axis was 4",1 high and 
 0",9 southward. Wires A and B lost. 
 
 h. m. s. s. 
 
 Wire I f Interval of wire II . 
 
 H , from mean 4U ' d4 
 
 III 57,1 Wire II -26,892 
 
 IV 10 8 Sum =-67, 236 
 
 V 24,7 
 
 f VI .. 38j0 5=810.31' ,=430.44' 
 
 VII 19.43.51,7 ^cosecS=0,067 
 
 T V cos z cosec 8=0,049 
 mean of the times 19.43.24,46 T V sin 2 cosec 5 =0,047 
 
 67 236 
 Corr n for lost wires -13,59 = ^ x 1,101 
 
 10,87 
 
 Corr n for form of pivots + 0,33 
 
 collimation error 0,05 = - 0",7 x 0,067 
 
 level error + 0,20 = +4",lx 0,049 
 
 ,, azimuth error +0,04 = + 0",9xO, 047 
 
 Clock-time of transit 19.43.11,39 
 
 Clock slow at O h 16,65 
 
 Corr n for losing rate 1", 17 ... +0,96 
 
 Apparent R. A 19.43.29,00 
 
 Corr u to Mean R. A +1,84 
 
 Mean R. A. 1851,0 19.43.30,84 
 
 102 
 
148 PRACTICAL ASTRONOMY. 
 
 Plate I. is a representation in outline of the Cambridge 
 Transit Instrument, copied from that attached to Woodhouse's 
 Description in the Philosophical Transactions for 1825, referred 
 to in page 5. A is the position of the apparatus for adjusting 
 the axis of motion, a full description of which, illustrated by the 
 Figures 9 and 10, is given in pages 37 39. B is the eye-end 
 of the telescope, of which the Figure 6 in pages 27 and 31 is an 
 enlarged representation, accompanied by a description in detail 
 of the different parts. At C apparatus was provided for regu- 
 lating the amount of surface of the object-glass exposed for 
 observations of the Sun, but in practice it was not found to be 
 required, the use of dark glasses of different degrees of opacity 
 at the eye-end being sufficient for all purposes (see Art. 109). 
 
 THE MURAL CIRCLE. 
 
 157. A Mural Circle is constructed for the primary pur- 
 pose of measuring the angular elevation of a celestial object 
 above a horizontal plane at the instant of its passing the 
 meridian of the place of observation. Transit-observations 
 are required for obtaining one of the spherical co-ordinates 
 by which the position of an object in the heavens is deter- 
 mined, and observations with a graduated Quadrant or Circle 
 are proper for obtaining the other. The way in which this 
 ultimate object is reached by the use of a Mural Circle 
 expressly constructed for the purpose just mentioned, and the 
 processes employed for ensuring accuracy in the determination 
 of the co-ordinate, will form the principal subject matter of the 
 portion of the Lectures now entered upon. As in the case of 
 the Transit instrument, the Circle will be treated of under 
 four Divisions (see Art. 25). 1 
 
 1 The Mural Circle is the simpler of the tvro instruments, inasmuch as it is 
 only concerned with measurement of space, whereas the other measures space 
 by the intervention of time. I have followed usage in treating of the Transit 
 instrument first, and apart from the Circle; but as respects astronomical 
 purposes neither instrument is complete by itself, on which account in the 
 treatment of the Transit instrument, there was occasion several times to refer 
 by anticipation to what is effected by the Mural Circle. The use of the 
 Transit-Circle, which is a complete astronomical instrument, is capable of 
 being explained in a perfectly logical manner. 
 
THE CAMBRIDGE TRANSIT INSTRUMENT. 
 
 PLATE I. 
 
THE MURAL CIRCLE. 151 
 
 I. DESCRIPTION OF THE PARTS AND MOUNTING OF 
 THE INSTRUMENT. 
 
 158. In describing the Mural Circle (so called because 
 when mounted it is usually contiguous to the vertical face of 
 a massive wall) I shall refer mainly to that of the Cambridge 
 Observatory, respecting which, after long use of it, I am pre- 
 pared to say that in respect to efficiency and accuracy it has 
 probably not been surpassed by any other instrument of the 
 same kind. It was constructed by Troughton and Simms, and 
 mounted in October 1832. The divisions, having been approxi- 
 mately marked, were finally cut by Mr Simms in November of 
 the same year, it being thought advisable, regard being had to 
 the weight and size of the instrument, to perform this operation 
 after it was mounted. It came into use at the beginning of 
 1833, and in the Introduction to the Volume of Cambridge 
 Observations for that year Professor Airy gave a description 
 of it, which for the most part I have followed in that which 
 is produced here. The circle is eight feet in diameter, but in 
 all other respects the Cambridge instrument is similar to the 
 mural circles of the Greenwich Observatory. The limb is con- 
 nected with the centre piece by sixteen spokes, or hollow cones, 
 and these are joined together at about the middle of their 
 lengths by consecutive bars forming a circle whose diameter 
 is about half that of the exterior circle. The limb was cast in 
 several pieces, which were afterwards united by a process 
 technically called burning together, which, when skilfully per- 
 formed, makes a connection as perfect as if the whole had 
 been cast at once. The axis on which the circle turns is a 
 hollow cone, 4J feet long, passing through a very massive stone 
 pier, the circle being on its eastern face, and both bearings of 
 the axis on the same side of the circle. At each bearing the 
 axis is armed with a steel ring which turns within a steel 
 cylinder. A considerable part of the weight at the bearing of 
 the larger ring, that near the circle, is supported by lever 
 counterpoises. The divisions of the limb are on the external 
 edge, so that the microscopes by which they are read are in the 
 
152 PRACTICAL ASTKONOMY. 
 
 same plane as the circle. The microscopes, six in number and 
 separated by equal intervals, are attached to the stone pier by 
 brass supports, the height of each of which above the point of 
 attachment is intended to supply a thermometrical expansion 
 nearly equal to the upward expansion of the central support of 
 the instrument. The microscopes are fitted with micrometers, 
 by which the interval of 5' between the divisions is subdivided, 
 and the circle-reading can be taken to integral seconds, and 
 by estimation to tenths of a second. The telescope is carried 
 by a steel rod passing through the hollow conical axis, and 
 its position on the circle is secured by frames which are 
 clamped to the limb at both ends of the telescope, one being 
 as near as possible to the object-glass cell, and the other as near 
 as possible to the wire-frame. The instrument is fixed, or 
 moved slowly, by clamps which act immediately on the limb 
 by tangent-screws, and are so disposed about the circle that 
 they can be severally used according to the convenience of 
 the observer. Originally five tangent-screws were attached to 
 the wall, but after it was found that three were sufficient, 
 one at the nadir point and the other two about 30 below the 
 horizontal diameter, the two uppermost were taken off. 
 
 159. To prevent any injurious effect from the Sun's heat 
 in mid-day observations, a wooden screen is provided, by which 
 all the parts of the limb on which the Sun's 'rays might fall are 
 shaded. That part of the screen which corresponds to the 
 interval between the tropics, consists of a sliding board with a 
 circular hole in it, a little larger than the aperture of the 
 telescope, to admit of the incidence of rays from the Sun and 
 planets on the object-glass when the screen is let down. 
 
 160. The principle of the tangent-screw and clamp may be 
 gathered from what has been said in Art. 35. I propose to give 
 here the description of one specially constructed for application 
 to the Mural Circle. Figure 29 represents the form of the 
 clamps of the Cambridge Circle. The screws a, a fasten to the 
 wall a brass plate which carries the clamp ; e, e are milled 
 heads, which are fixed to the ends of a straight rigid stem 
 for turning it ; 6, b are supports of the stem fastened to the 
 
THE MURAL CIRCLE. 
 
 153 
 
 brass plate ; the stem passes through two cylinders c, c, which 
 are fixed by being rigidly connected with the supports b, b ; the 
 left-hand cylinder is tapped for the screw portion of the stem 
 
 Fig. 29. 
 
 to work in; the right-hand cylinder contains a strong spiral 
 spring, which surrounds the stem, and is compressed between 
 a stop in the cylinder and the end of an interior movable 
 cylinder. The latter projects from the other, and abuts on a 
 transverse part of the clamp opposite the end of the screw, so 
 that the screw and spring in action are antagonistic to each 
 other. When by turning either milled head, the screw is 
 drawn out, the clamping pieces and the transverse screw d are 
 moved in the direction from left to right according to the figure, 
 and on turning the milled head in the opposite direction they 
 are moved by the force of the spring from right to left. The 
 limb of the Circle is nipped in the space ff between two 
 clamping-pieces by turning the screw d } which draws the 
 farther clamping-piece close up to the other. On turning that 
 screw back to release the hold, the Circle is undamped, and the 
 farther piece is pushed out by two small springs to allow of free 
 passage of the limb when the Circle is moved, that piece being 
 at the same time kept in place by two screw-heads seen at o 
 and o. When the Circle is clamped it partakes of the motion 
 given to the clamping-pieces by turning one or the other of the 
 milled heads either way. But for the final motion in taking an 
 observation, the milled head is always to be turned in the 
 direction which draws out the screw from the cylinder c, and 
 
154 PEACTICAL ASTRONOMY. 
 
 therefore produces motion of the clamp from left-hand to right- 
 hand. The reason for this is, that the bearing on the threads 
 of the screw may be presumed to be more complete and 
 uniform when the motion is opposed to the action of the spring 
 than in the contrary case 1 . That the observer may be able to 
 decide at once which way he is to turn the milled head, half of 
 its edge is cut (as shewn in the Figure) in such manner as to 
 be felt by the finger passing over it rougher in one direction 
 than in the other. The direction of the turning is required to 
 be that in which the greater roughness is felt. 
 
 161. It is particularly to be noticed, as being the principal 
 advantage of this form of clamp, that although the stem is rigid, 
 and its supports b, b are rigidly connected with the brass plate, 
 the clamping apparatus is so arranged that it is capable of 
 receiving after the clamping, together with the motion from 
 left to right, a small motion of rotation, in consequence of which 
 no strain is produced by the circular movement of the Circle. 
 For this purpose the apparatus is made to have a sliding motion 
 on the brass plate, and the stem passes through an elongated 
 hole in the transverse brass support of the screw d. [The 
 clamping apparatus of the Greenwich mural circles, which dif- 
 fered from this, will be described subsequently.] 
 
 162. It is now required to take into consideration the con- 
 struction and use of the micrometer-microscope, by means of 
 which, as stated in Art. 158, the graduation intervals are sub- 
 divided, and the angular indications of the Circle are read off. 
 Figure 30 represents a Troughton reading microscope, the con- 
 struction of which in all essential points is the same as that of 
 the microscopes of the Cambridge Circle, although it differs 
 from them in form. The arrangement of parts in the interior 
 
 1 In order that the Circle may be easily turned about its axis, the pivots and 
 their collars are from time to time greased by a composition, in certain 
 proportions, of oil, tallow, and bees' wax, to diminish the friction. It some- 
 times happened with the Cambridge Circle, when the composition required 
 renewing, that the force of the spring was insufficient of itself to overcome 
 the friction. This is another reason for not relying on the action of the 
 spring in moving the Circle to make a bisection. Oil alone is a better lubricant. 
 
THE MURAL CIRCLE. 
 
 155 
 
 of the box a of the micrometer, which is transverse to the axis 
 of the microscope, may be understood by referring to Figure 4 
 in page 24, so far as relates to the movement of the micrometer- 
 
 Fig. 30. 
 
 (1) 
 
 (2) 
 
 wire by turning the micrometer-head ; but in the present case, 
 instead of a single transverse wire, the turn of the micrometer- 
 head moves two wires crossing each other, so as to be suitable 
 for bisecting a line graduation in the manner shewn in Figure 
 30 (2). The graduated rim r is divided into 60 equal parts for 
 reading off to seconds of space by means of the contiguous fixed 
 index q. The tube of the microscopes consists of three parts: 
 (1) the part e is a cylindrical tube, screwed to the box of the 
 microscope, and carrying a Bamsden eye-piece /, which slides 
 in it with friction, so as to be adjustible for seeing the cross 
 wires distinctly; (2) the cylindrical tube bb, which is screwed 
 fast to the plate of the micrometer-box, has on its outside a 
 coarse screw for action with the tapped nuts d, d r with milled 
 heads, and also an interior fine screw from end to end ; a por- 
 tion of the same tube has, besides, at the end a fine screw on 
 the outside for action with a third circular nut * with milled 
 
156 PRACTICAL ASTRONOMY. 
 
 head; (3) the part g, having the form of a frustum of a cone, 
 carries at one end the cell h of the object-glass, and terminates 
 at the other in a short cylindrical part, which is inserted in the 
 tube bb, and has a fine screw on the outside for action with the 
 fine interior screw of that tube, this action giving the means of 
 adjusting the distance of the object-glass from the cross wires. 
 When the adjustment has been made, the nut i is screwed up 
 to a shoulder, and the position of the object-glass is thus fixed. 
 The dotted figure c has reference to the mode of support of the 
 microscope, and the connection of the support with the wall. 
 The upright part of the support c has a hole for the cylindrical 
 part bb to pass through, so that when the nuts d, d' are 
 screwed up, the attachment of the microscope to the support, 
 and by consequence to the wall, is effected. 
 
 Figure 30 (2) represents the field of view of the microscope, 
 shewing the image of a portion of the graduation of the Circle, 
 the bisection of one of the lines of graduation by the cross wires, 
 and the toothed comb for reading off the integral number of 
 micrometer-revolutions reckoned from the zero position marked 
 by a circular hole in the comb. The rules for reading off will 
 be given in Division II. 
 
 163. The field of view of the telescope, the apparatus at 
 the eye-end for adjusting the wire-frame, and the mode of con- 
 necting the telescope with the limb of the Circle, are exhibited 
 by Figure 31. In the field of view there are five vertical wires 
 and a horizontal wire all fixed on the same frame, and an ad- 
 ditional micrometer- wire movable by turning the micrometer- 
 head M. The fixed horizontal wire is adjustible to horizontality 
 by the apparatus at C consisting of two screws supported by a 
 frame and acting oppositely on a projection e rigidly connected 
 with the wire-frame, by which means a rotational motion about 
 the axis can be given to the wire. The parallelism of the 
 movable wire to the fixed wire depends wholly on mechanical 
 construction. The micrometer-head M has on its rim a scale of 
 100 equal parts, pointing to which there is a fixed index for 
 reading off; and by turning M the micrometer-wire may be 
 turned either way by intervention of the antagonistic spring at 
 
THE MURAL CIRCLE. 
 
 157 
 
 in. For a reason already mentioned (end of Art. 160) the 
 turning of M is always in the direction that makes the micro- 
 meter-wire approach the micrometer-head, and the nearer it is 
 
 Fig. 31. 
 
 JML 
 
 to the micrometer-head the greater is the reading of the micro- 
 meter-index. The comb on the left-hand side of the field, with 
 its indentures and circle for zero-indication, serves for reading 
 off the integral number of micrometer revolutions. Hh is a 
 brass ring, rigidly joined to the fr&mefff, and capable of being 
 clamped to the Telescope-tube by the capstan-headed screw at 
 D. That frame is clamped to the limb of the Circle by the two 
 clamps c and d, each of which consists of two rectangular clamp- 
 ing pieces drawn together by two capstan-headed screws, and 
 nipping between them the frame / and the limb LI of the 
 Circle. The attachment of the Telescope to the limb at the 
 object- end is exactly the same. After releasing the four clamps 
 the Telescope may be shifted to a new position on the Circle. 
 The screws for adjusting the wire-frame for correction of colli- 
 mation-error (not seen in the Figure), have the same position 
 as those of the Telescope of the Transit-instrument. At A and 
 B are two screws by means of which the distance of the wire- 
 
158 PRACTICAL ASTRONOMY. 
 
 frame from the object-glass may be adjusted and fixed. For 
 this purpose they pass through two small plates, fastened by 
 screws to the tube which carries the eye-end apparatus, and are 
 worked either way in the frame Hh by capstan levers, the tube 
 being finally held in position by two screws and circular plates, 
 as shewn in the Figure. The reason for having the means of 
 making this adjustment is the same as that already given in 
 Art. 83 with reference to the apparatus applied to the Telescope 
 of the Transit-instrument for effecting the same adjustment. 
 
 164. The Telescope of the Mural Circle is the usual astro- 
 nomical telescope constructed to fulfil the optical conditions 
 stated in Arts. 11 13, and is generally equal in length to the 
 diameter of the Circle, that, for instance, of the Cambridge 
 Circle having the focal length of eight feet. In case a stage be 
 placed in front of the Circle for convenience in taking reflection- 
 observations, the illumination of the field is best effected by 
 putting a lamp on the stage opposite an aperture at the middle 
 of the Telescope-tube for the incidence of rays from the flame 
 on a plane reflector inclined at an angle of 45 to the axis of the 
 Telescope. This is the arrangement in the case of the Cam- 
 bridge Circle, and the degree of illumination is regulated by 
 turning a milled head, which acts by rack-work on two rods 
 connected with two angular plates forming a square opening, 
 the size of which is by this means under command. A single 
 setting -circle, constructed like the setting-circles of the Transit 
 instrument, and attached to the east side of the Telescope-tube 
 near the eye-end, serves for setting the Telescope according to 
 the approximate N. P. D. of an object to be observed. In Cam- 
 bridge practice, however, the setting-circle was seldom made use 
 of, advantage being taken of the graduation of the Circle itself 
 to perform the operation by means of a pointer, which is fixed 
 to the wall, and points in the direction of the lines of gra- 
 duation. These lines are inscribed on a gold band (white metal 
 would have been better) at intervals of 5', and every alternate 
 line is distinguished by circular dots to enable the eye readily 
 to catch the divisions of a degree into intervals of 10'. Near 
 the gold band there is a band of composite whitish metal, on 
 
THE MURAL CIRCLE. 159 
 
 which the integral degrees are numbered correspondingly to the 
 graduation-lines, so that by means of the two bands the indica- 
 tion of the pointer, to the accuracy of about a minute, may 
 easily be seen, and the Circle, consequently, be quickly set to that 
 degree of accuracy 1 . This process implies that the observer has 
 approximately inferred the setting -angle from the polar distance 
 of the object by previous calculation. The method of obtaining 
 formulae for this calculation will be explained in Division III. 
 
 165. It has already been stated (Art. 158) that the pivots 
 of the Cambridge instrument turn in two steel collars, a larger 
 one close to the Circle, and a smaller one at the end of the 
 conical axis. The axis of motion may be considered to be the 
 common axis of these two pivots. The weight of the instru- 
 ment is principally sustained by the larger collar, on which 
 account the pressure on this collar is taken off in great degree 
 by lever counterpoises applied in the small interval between the 
 collar and the Circle. The counterpoise action is effected by 
 the upward pressure of two friction- wheels suspended by iron 
 bars from the ends of two levers whose fulcra ms are on the 
 pier, while from their other ends heavy weights hang down in 
 cylindrical holes made in the pier large enough for free suspen- 
 sion of the weights. The smaller collar is held in position 
 within a fixed frame by being pressed in the vertical and hori- 
 zontal directions by two screws, working in the frame, and by 
 two counteracting springs. Turning the vertical and horizontal 
 screws produces respectively vertical and horizontal angular 
 movements of the axis of motion, and thus gives the means of 
 correcting mechanically errors of level and azimuth. 
 
 The parts and appendages of the Mural Circle having now 
 been sufficiently described, I proceed to treat of it under the 
 Second Division. 
 
 1 In the mode of setting practised with the Cambridge Transit-Circle, an 
 additional microscope, directed for this purpose to the graduation, is made use 
 of, and by this means the setting, as often as the occasion requires, is per- 
 formed with a degree of certainty and exactness not attainable by the ordinary 
 setting-circles. This method will be particularly described in treating of the 
 use of the Transit-Circle. 
 
160 PRACTICAL AST11ONOMY. 
 
 II. THE ADJUSTMENTS OF A MURAL CIRCLE, AND CORREC- 
 TIONS OF INSTRUMENTAL ERRORS. 
 
 166. The Mural Circle being intended for measuring the 
 altitudes of objects at their meridian-transits, the line of colli- 
 mation of its Telescope is required to be adjusted to the plane 
 of the meridian. But since the Circle is only used for measures 
 of arcs of the meridian, and as these measures are not sensibly 
 altered by small deviations of the line of collimation from the 
 meridian-plane, the errors of collimation, level, and azimuth 
 may generally be corrected with sufficient accuracy by mechan- 
 ical means, without calculating their exact amounts. Also 
 these corrections may be supposed to be made as if the pivots 
 were perfectly cylindrical, and the axis of motion had a fixed 
 position in space, because the effects of non-fulfilment of these 
 conditions on the measures of arc are taken account of by the 
 microscope-readings, as will afterwards be shewn. This being 
 understood we may proceed to the consideration of methods of 
 adjusting the line of collimation so that it shall move in the 
 plane of the meridian. 
 
 167. In order to explain the requisite processes ab initio 
 (as was done in treating of the transit-instrument), I shall begin 
 with shewing how to select the place of the wall, or pier, which 
 is to support the Circle, and how, after the wall is fixed in its 
 position, it is prepared for the mounting of the instrument. 
 First, a spot having been chosen near the transit-room, east- 
 ward or westward, a meridian line has to be drawn to mark the 
 direction of the face of the wall. This may be done with suffi- 
 cient accuracy either by means of a compass, the magnetic decli- 
 nation at the locality being known, or by the method described 
 in Art. 10. Then the face of the wall may be made vertical, 
 and the perforation for receiving the axis of the instrument be 
 executed at the proper height and in the required horizontal 
 direction, by ordinary mechanical means. Greater precision 
 is required for fixing within the perforation, in sufficiently 
 approximate positions, the supports of the rings, or collars, in 
 
THE MURAL CIRCLE. 159 
 
 which the integral degrees are numbered correspondingly to the 
 graduation-lines, so that by means of the two bands the indica- 
 tion of the pointer, to the accuracy of about a minute, may 
 easily be seen, and the Circle, consequently, be quickly set to that 
 degree of accuracy 1 . This process implies that the observer has 
 approximately inferred the setting-angle from the polar distance 
 of the object by previous calculation. The method of obtaining 
 formulae for this calculation will be explained in Division III. 
 
 165. It has already been stated (Art. 158) that the pivots 
 of the Cambridge instrument turn in two steel collars, a larger 
 one close to the Circle, and a smaller one at the end of the 
 conical axis. The axis of motion may be considered to be the 
 common axis of these two pivots. The weight of the instru- 
 ment is principally sustained by the larger collar, on which 
 account the pressure on this collar is taken off in great degree 
 by lever counterpoises applied in the small interval between the 
 collar and the Circle. The counterpoise action is effected by 
 the upward pressure of two friction-wheels suspended by iron 
 bars from the ends of two levers whose fulcrums are on the 
 pier, while from their other ends heavy weights hang down in 
 cylindrical holes made in the pier large enough for free suspen- 
 sion of the weights. The smaller collar is held in position 
 within a fixed frame by being pressed in the vertical and hori- 
 zontal directions by two screws, working in the frame, and by 
 two counteracting springs. Turning the vertical and horizontal 
 screws produces respectively vertical and horizontal angular 
 movements of the axis of motion, and thus gives the means of 
 correcting mechanically errors of level and azimuth. 
 
 The parts and appendages of the Mural Circle having now 
 been sufficiently described, I proceed to treat of it under the 
 Second Division. 
 
 1 In the mode of setting practised with the Cambridge Transit-Circle, an 
 additional microscope, directed for this purpose to the graduation, is made use 
 of, and by this means the setting, as often as the occasion requires, is per- 
 formed with a degree of certainty and exactness not attainable by the ordinary 
 setting-circles. This method will be particularly described in treating of the 
 use of the Transit-Circle. 
 
162 PRACTICAL ASTRONOMY. 
 
 represented by the adjacent figure (7, consists of a circular 
 pearl disk, on which is fixed, eccentrically, a small brass ring, 
 the interior diameter of which is extremely small. At the other 
 end of A is a lens g by which the pearl is illumined, so that the 
 interior of the ring appears as a small bright circle at e. An 
 image of this circle is formed at e by means of the lens at L, 
 A being first pushed close up to the end M of the fixed tube. 
 By means of the projection n, movable in a slit, the lens L 
 (with which is connected a diaphragm _D for contracting the 
 pencils) may be placed in a position proper for forming the 
 image e where the plumb-line can conveniently be brought 
 into coincidence with it. The other movable part B carries at 
 its inserted end a single lens I, with which the image e' is to be 
 viewed by the eye at the other end, when the position of the 
 lens for distinctly seeing e' has been tentatively ascertained. 
 After the foregoing account it is, farther, to be mentioned that 
 for the purpose of levelling the axis of rotation of the Circle, two 
 ghost apparatus's, exactly alike, are fixed to the telescope-tube, 
 one at a small distance from the eye-end, and the other at the 
 same distance from the object-end. Before shewing how they 
 are both made use of for effecting the levelling, it is necessary 
 to explain the construction of apparatus specially designed for 
 adjusting the position of the plumb-line. 
 
 169. The parts of this apparatus and the mode of using it 
 may be indicated by means of Figure 33. At one end is a 
 transverse brass plate AA y which, being let into a vertical socket 
 fastened to the top of the wall, determines the horizontal direc- 
 tion of the intermediate part connecting this plate with another 
 at the other end, to which the plumb-line is attached by the 
 hook e. By turning the screw B, the tube to which the latter 
 plate is fastened, is made, by rack-work within an exterior tube, 
 to move through large spaces, for the purpose of placing the 
 plumb-line at the distance from the wall proper for being viewed 
 with the ghost apparatus, or removing it farther off to any 
 required distance. By this means, and by the tangent- screw 
 movement of the Circle, the plumb-line is brought nearly into 
 coincidence with the ghost, and the position is fixed by the 
 
THE MURAL CIRCLE. 
 
 
 163 
 
 clamping-screw C\ after which an exact adjustment is made by 
 the following means. The plumb-line efg passes at / between 
 threads of a fine horizontal screw, which is attached by brackets 
 to a second plate fastened to the other near the top, but capable 
 
 Fig. 33. 
 
 of a small separation from it at the lower part. This separation 
 is effected by turning the milled head dd, which, by the inter- 
 vention of a fine screw, gives slow motion to the plate and 
 plumb-line in the horizontal direction perpendicular to the face 
 of the wall. Then, since the line passes between threads of the 
 screw, by turning either of the milled heads hh pertaining to 
 the screw, the line is moved horizontally in a direction parallel 
 to the wall. By the two rectangular movements the plumb- 
 line may be brought into exact coincidence with the image. 
 Supposing this to have been done with the upper apparatus, 
 since by the previous arrangements the axis of the instrument 
 is approximately horizontal, and, when the axis of the telescope 
 is vertical, one ghost apparatus is exactly below the other, the 
 line may be expected to be seen in the lower apparatus not far 
 from the image. If, however, it be found to be inconveniently 
 distant, the line is to be brought half -way towards the image 
 by turning the vertical adjusting screw at the back of the wall, 
 and the operation of bisecting the upper image must be repeated. 
 The line, as seen with the lower apparatus, may always by such 
 means be brought sufficiently near the lower image to admit of 
 a coincidence being effected, in consequence of the eccentric 
 
 112 
 
164 PRACTICAL ASTRONOMY. 
 
 position of the point e,by simply turning the cylindrical piece A 
 about its axis (see Fig. 32). By this process the plumb-line is 
 made to pass through two points rigidly connected with the Circle. 
 We have next to consider how by having the means of satisfy- 
 ing this condition, the axis of motion of the Circle (as also that 
 of any mass to which the same apparatus could be applied) 
 may be horizontally adjusted. 
 
 170. The plummet is usually a brass weight of cylindrical 
 form, immersed in water, and largely indented at its vertical 
 surface, for the purpose of putting a stop to oscillatory and 
 rotatory motions. In Figure 34, let a and F . 
 
 b be the two bisected points joined by the 
 vertical line a Cb, which may be supposed 
 to pass through a point C on the axis of 
 rotation. Let CA be the direction of this 
 axis, and draw CH in the horizontal direc- 
 tion, so that A CH is the angle to be cor- 
 rected. After effecting the first bisections 
 of the two points, the screw C (Fig. 33) 
 is to be turned for unclamping, and then 
 
 by turning the screw B the plumb-line can be moved far enough 
 from the wall to be free from the ghost apparatus's, so as to allow 
 of turning the telescope through 180. By this motion let a 
 come to a and b to b', so that ad and bb' are perpendicular to A C, 
 and the angle Cad is equal to the angle ACH. Next the plumb- 
 line is to be restored to its place by turning the screw B, and 
 the observer may then proceed to bisect b' by precisely the 
 same process as that by which a was bisected in the first in- 
 stance. This having been done, let the plumb-line hang down 
 in the direction b'p. If the axis had been horizontal p would 
 have coincided with a', instead of being seen with the lens I 
 (Fig. 32) at the distance pa. Now let the dotted line Ch bisect 
 the angle b Ca ; then will z hCa = z Cad = z A CH, the angle 
 to be corrected. But pa' = twice ba = four times ha. Hence that 
 angle will be corrected by turning the adjusting screw at the 
 back of the pier till the point a' is moved in the direction from 
 a' towards p through one-fourth the interval pa, because by 
 
THE MURAL CIRCLE. 165 
 
 this motion the straight line Ca will be moved through the 
 angle a Ck, and therefore CA through the angle A CH. The 
 level-error of the axis will thus be corrected ; but because no 
 micrometer is used, and the judgment by the eye of one-fourth 
 that interval is uncertain, the correction can only be considered 
 as roughly approximate. 
 
 171. The use of the ghost apparatus is perhaps the best 
 method of effecting a first levelling of the axis of a Mural 
 Circle, and by repeating the process a sufficiently accurate final 
 adjustment might be made. But it would be preferable, after 
 the first approximation, to correct the result by means of reflec- 
 tion and direct observations of Polaris obtained in the manner 
 stated in Art. 127. Since, however, this method gives the 
 level-error in angular measure, in order to correct it mechani- 
 cally (as proposed in Art. 166), the following process might be 
 adopted. The angular error in seconds being e, the length of 
 the axis in inches I, and R being the number of seconds in an 
 arc equal to radius, the length of the small line which at the 
 position of the screw for level adjustment subtends the angle e 
 
 is -^ . Suppose a pair of compasses to be applied to the screw 
 
 so as to include in the direction parallel to its axis a number 
 n of threads, and let the opening of the compasses measured on 
 
 a scale be h in inches. Then will - be the common interval 
 
 n 
 
 between the threads, or the motion in the vertical direction 
 produced by one revolution of the screw. The required correc- 
 tion will consequently be made by turning the screw, in the 
 
 7? / ( 
 
 proper direction, through the fractional part ^ of a revolution, 
 
 which might be done by estimation, or, for greater certainty, by 
 appropriate scale measure. 
 
 172. After this correction of level-error, the error of colli- 
 mation may be mechanically corrected by using the collimating 
 eye-piece, first, for placing the telescope in a vertical position by 
 bringing the fixed wire and its reflected image into coincidence 
 by the tangent-screw motion (see Art. 84), and then for making 
 
166 PRACTICAL ASTRONOMY. 
 
 the mid-wire and its reflected image coincide by means of the 
 screws for adjusting the wire-frame horizontally. Again, by the 
 use of two collimators 1 in the manner described in Arts. 53 55, 
 we might begin with correcting mechanically the collimation- 
 error, by adjusting the wire-frame till the mid-wire is made to 
 collimate with each of two wires, one in the North collimator, 
 and the other in the South collimator, these wires having been 
 previously made to collimate with each other through an aper- 
 ture formed at the middle of the telescope-tube for this purpose. 
 The level-error can then be corrected by means of the collima- 
 ting eye-piece, by turning the screw for level adjustment till 
 the mid-wire coincides with its image. After the mechanical 
 corrections of the collimation and level-errors have been made 
 by either of these methods, the azimuth error may be calculated 
 from transit observations taken under any of the conditions 
 mentioned in Arts. 88 and 89, an astronomical clock being 
 placed in the Circle-room for taking such observations. The 
 error being thus obtained in angular measure, the movement of 
 the azimuth-adjustment screw for correcting it may be ascer- 
 tained just as in the case of the mechanical correction of level- 
 error, the account of which is given in detail in Art. 171. 
 
 173. Although the line of collimation may by the mechani- 
 cal means above indicated be made to move in the plane of the 
 the meridian with sufficient exactness for observations of zenith 
 distances, it is not possible under the usual arrangements per- 
 taining to the Mural Circle to obtain by calculation the amount 
 of the collimation -error, for the two reasons that the Circle does 
 not admit of azimuthal reversion, and is not provided with a 
 vertical micrometer 2 . However, during a portion of June, 1858, 
 
 1 The Cambridge Observatory was furnished, at my instance, with two colli- 
 mators which, not being permanently fixed, could be temporarily set up for the 
 purpose here mentioned. They were originally procured to be employed for 
 measuring the effect of flexure on the indications of the Circle for any position 
 of the telescope, by a process which there will be occasion to speak of in 
 Division in. 
 
 2 The large mural quadrant of eight feet radius, constructed by Bird, which 
 for a long time was used at the Greenwich Observatory, was too ponderous for 
 azimuthal reversion, and, consequently, could not furnish data for calculating 
 
THE MURAL CIRCLE. 167 
 
 while repairs were going on in the Transit-room, being desirous 
 of keeping up the determinations of clock-rate and clock-error, 
 I made use of the Circle and the clock in the Circle-room for 
 taking transits, and to ensure accuracy calculated the values of 
 the three corrections by the following means. The intervals of 
 the five wires from the mean of all were first obtained by 
 transits of Polaris ; then the level-error was found by reflection 
 and direct transits of the same star ; after which the telescope 
 was pointed to the nadir, and the distance of the micrometer- 
 wire from the fixed horizontal wire was adjusted so that, by means 
 of the collimating eye-piece, the two wires, with the mid-wire 
 and its reflected image, could be seen to form, according to the 
 judgment of the eye, a small square. The interval in arc 
 between the fixed and micrometer-wires being ascertained (by 
 means that will be subsequently indicated), that between the 
 mid- wire and its image became known, the half of which mea- 
 sured the angular distance of the mid-wire from the vertical 
 plane through the optical centre of the object-glass. As this 
 distance was made up of the level and collimation-errors, and 
 the level-error was already calculated, the amount of collimation- 
 error was inferred. The azimuth error was then calculated in 
 the usual way. It is plain that such an operation would not be 
 required in the case of an irreversible Transit-circle, because by 
 means of its vertical micrometer, and the use of the collimating 
 eye-piece, or of the two collimators, the error of collimation of 
 the mid-wire, and, consequently, that of the mean of the wires, 
 
 its own error of collimation ; on which account a reversible Zenith Sector was 
 employed as a subsidiary means of effecting the calculation. This was, in fact, 
 an altitude and azimuth instrument, limited as to range of altitude to small 
 distances from the zenith, and having a radius considerably larger than that of 
 the quadrant in order to diminish the probable error of comparisons of obser- 
 vations of the same star with the two instruments. In Woodhouse's Astronomy 
 (1821), Vol. I., Part i., p. 68, an example is given of the calculation of collima- 
 tion-error from a comparison of the means of the zenith distances of y Draconia 
 derived from observations made in the year 1812 with the quadrant and zenith 
 sector of the Greenwich Observatory. In the erection of the Cambridge Obser- 
 vatory a room was prepared for the reception of a zenith sector of 25 feet 
 radius, but the instrument was not procured, an exact calculation of the amount 
 of the collimation-error of the Mural Circle not being required. 
 
168 PRACTICAL ASTRONOMY. 
 
 might always be calculated. (See Cambridge Observations, Vol. 
 XX. for 1855 1860, p. Ixxxv.) 
 
 174. In order to test the position of the Circle relatively to 
 the plane of the meridian, it was usual at Cambridge to take 
 transits of known stars of various polar distances from time to 
 time with the Circle-telescope and clock (Molyneux) in the 
 Circle-room, and at the same time to compare the clock (Hardy) 
 in the Transit-room with Molyneux. The error of Hardy being 
 known, and that of Molyneux inferred, the sidereal times of the 
 transits were computed, and by comparing them with the known 
 apparent R.A. of the stars, the errors in time due to deviation 
 of the pointing of the telescope from the plane of the meridian 
 were obtained. By these it would be seen whether the adjust- 
 ment of the plane of collimation to the plane of the meridian 
 was sufficiently exact. 
 
 The Microscope-reading. 
 
 175. The Circle being supposed to be put in position, we 
 have next to consider the adjustments required to prepare the 
 micrometer-microscopes for reading oif the graduation, their 
 parts and mounting being already described in Art. 162. It is, 
 first, to be stated that the graduation proceeds in the direction 
 from the South side of the Circle over the upper part to the 
 North side, in order that the Circle readings, as shewn by a 
 fixed index, may increase as the telescope is moved from the 
 Pole southward, that is, as the polar distance increases. The 
 micrometer-heads are all turned in the direction of the gradua- 
 tion, so that Figure 30 (1), in page 155, represents the position 
 of the micrometer-head of that microscope which is near the 
 south end of the horizontal diameter of the Circle. In the field 
 of view of the microscope, represented by Figure 30 (2), is seen 
 a portion of the gold band on the rim of the Circle, with lines 
 of graduation across at intervals of 5', the graduation proceeding, 
 since the microscope inverts, in the downward direction. One 
 of the lines, that which is nearest the hole of the comb and 
 towards the micrometer-head, is bisected \)y the cross-wires of 
 the microscope. There are screws (not seen in the Figure) by 
 
THE MURAL CIRCLE. 169 
 
 which the axis of the microscope is adjusted so that the inter- 
 section of the cross-wires is brought to the middle of the gold 
 band. Also when the tongue pointing to the intersection of the 
 cross-wires is placed, by turning the micrometer-head, across 
 the centre of the circular hole in the comb, the index q by which 
 the graduation on the rim r is read off, should point very nearly 
 to zero. If this be not the case, the micrometer-head is to be 
 set to the zero reading, and then the comb is to be shifted, by 
 turning a screw provided for that purpose, till the tongue 
 crosses the centre of the circular hole. This adjustment of 
 the comb is to be performed for each of the microscopes, the 
 Circle being in the mean time clamped. 
 
 176. After adjusting the combs, and while the micrometer- 
 indices still point to the zero readings, a line of graduation is 
 to be brought by the tangent-screw to be bisected by the cross- 
 wires of one of the microscopes. This having been done, and 
 the Circle remaining clamped, the cross-wires of the other 
 microscopes are to be made to bisect, at the middle of the gold 
 band, the lines of graduation to which the tongues point, by 
 means of screws provided for adjustments of the microscope-axes 
 in the vertical plane, in addition to those for the horizontal 
 adjustment mentioned in Art. 175. The original mounting of 
 the microscopes is intended to secure that the lines of gradua- 
 tion thus bisected shall be separated by equal arcs, for instance, 
 of 60 in the case of six microscopes. If an inspection of the two 
 graduation bands should shew that the proper line has not in 
 every case been selected, new adjustments of the axes of the 
 discordant microscopes will have to be made, every thing else 
 remaining as before. What is done by these operations is called 
 the setting of the microscopes. 
 
 177. The rim of the micrometer-head being, as stated in 
 Art. 162, divided into 60 equal parts, and the common interval 
 between the graduation lines of the Circle being 5', suppose the 
 cross-wires of any microscope to be moved from the above- 
 mentioned position for bisecting the line opposite the hole of 
 the comb, through five complete revolutions, as indicated by 
 the teeth of the comb and the micrometer-index, and let the 
 
170 
 
 PRACTICAL ASTRONOMY. 
 
 direction of the motion be that of increasing index-readings, 
 and therefore towards the micrometer-head. Then if the next 
 line of graduation be found to be exactly bisected, the interval 
 between the divisions of the micrometer-head will correspond 
 to one second of the arc of the Circle. This condition is ap- 
 proximately satisfied at the first mounting of the microscope ; 
 but as it is liable to gradual change, or sudden disarrangement, 
 the astronomer is provided with means of making the adjust- 
 ment himself, or, at least, of securing that the deviation from 
 its exact fulfilment shall always be small. The way in which 
 this is effected by means of the apparatus described in Art. 162, 
 may be shewn as follows by reference to that description, and 
 to Figure 30 in pages 155 and 170. The first step is to ascer- 
 
 Fig. 30. 
 (1) (2) 
 
 tain whether the image of the graduation in the field of the 
 microscope, and the cross-wires, can be seen at the same time 
 distinctly with the eye-piece /. If not, the distance of the 
 object-glass at h from the graduation has to be altered, after 
 releasing the clamping nut i, by turning the tube g (see Art. 
 162) till the required condition is fulfilled, and then clamping 
 again. Next, the number of the division-intervals of the mi- 
 crometer-head corresponding to the graduation-interval of 5' is 
 
THE MURAL CIRCLE. 171 
 
 to be found by bisecting two consecutive lines in the manner 
 just stated, and noting the difference of the index-readings in 
 the two cases. Supposing that in this way the difference is 
 found to be 5 x 60 + x, and that a; is a considerable number 
 whether positive or negative, the distance of the microscope 
 from the graduation will in that case have to be changed. The 
 quantity x is called the Run of the microscope. According as 
 x is positive or negative the image of the graduation-interval is 
 too large or too small, and the microscope requires to be moved 
 bodily from or towards the circle. Suppose that by means of 
 the tapped nuts d, d' the microscope is moved, in the direction 
 required for correcting the Run, through a given space k, which 
 might be the interval between consecutive threads of the 
 external screw. Then after adjusting the object-glass for seeing 
 the graduation-lines distinctly together with the micrometer- 
 wire, let the interval of 5' be again measured as before, and 
 suppose the number of micrometer-intervals now to be 300 -J- x. 
 Then the additional space through which the microscope re- 
 quires to be moved to correct the Run x has to x the same 
 
 kx 
 
 ratio that Jc has to x x, or is equal to - , . According as 
 
 X ~~ X 
 
 this quantity is positive or negative, the second movement is to 
 be in the same direction as the first, or in the contrary direction. 
 By such means the Run of a microscope can always be approxi- 
 mately corrected. For reasons that will be given in the next two 
 paragraphs, the correction is not required to be made with ex- 
 treme accuracy. After correcting the Runs of all the micro- 
 scopes, the settings of the microscopes should be repeated. 
 
 178. It is now required to define precisely a microscope- 
 reading of the Circle-graduation. The reference position of the 
 cross-wires of the microscope is that for which the micrometer- 
 index is at zero when the intersection of the wires is opposite 
 to the hole of the comb. As shewn in Figure 30 (2), the line 
 selected for bisection is that which is nearest the hole of the 
 comb on the same side as the micrometer-head, the reasons for 
 the selection being that the order of the graduation, as seen in 
 the microscope, is from the micrometer-head, and the purpose 
 
172 PRACTICAL ASTRONOMY. 
 
 of the microscope-reading is to measure the interval from 
 the bisected line to the reference position just defined. The 
 degrees and minutes reckoned from the zero of the Circle- 
 graduation to the bisected line are given at once by the two 
 graduated bands described in Art. 164, and the addition of the 
 microscope-reading, consisting of the minutes inferred from the 
 number of indents of the comb between the hole and the pro- 
 jecting tongue, together with the seconds read from the micro- 
 meter-graduation, makes up the whole of the arc from the zero 
 of the Circle-graduation to the reference pointing of the micro- 
 scope. Let, for example, the arc of the circle from zero to the 
 bisected line be 130. 25', and the microscope-reading (which 
 as represented in the Figure is nearly 2') be exactly 1'. 54",7; 
 then the Circle-reading given by the particular microscope is 
 130. 26'. 54",7. It is to be noticed that the micrometer-gradua- 
 tion is always read off to estimated tenths of a second. 
 
 179. Here it is to be observed that the above microscope- 
 reading is not accurate unless the value of the micrometer- 
 intervals be exactly 1", or the Run of the microscope be zero. 
 If it be found that the arc of 5' is measured by 300 + x micro- 
 meter-intervals, the value of one interval will be ^^ x 1", 
 
 oOO -f- x 
 
 Hence if the recorded microscope-reading be h, the required 
 
 ,. . 300/z, 7 -hx , . , 
 
 correction is h, or ^--r nearly, x being by supposition 
 
 oUU ~r" X oUU 
 
 small compared to 300. If h be expressed approximately in 
 
 hx 
 
 minutes and tenths of a minute, this amount is , which is the 
 
 5 
 
 formula for the correction of the Run of any one of the micro- 
 scopes. In the example given in the preceding article, h = l',9, 
 so that if, for instance, x were found to be + 3",5, the correction 
 
 1 9 
 
 would be - x 3",5 = 1",3 to tenths of a second. The rule 
 
 o 
 
 is the same if h should somewhat exceed 5'; but if, as is some- 
 times the case, a line on the negative side of zero be bisected, 
 the micrometer-reading is set down as if the reading at zero 
 were 5', Then if h be the recorded reading, and if h' be the 
 
THE MURAL CIRCLE. 173 
 
 distance in micrometer-measure of the positive line from 
 zero, we have Ji = x + h, and the correction for Runs = 
 
 = _ (^ + #) - . Consequently the corrected micrometer-reading 
 5 
 
 for the positive line is h + x (h + x) ^ = h + x (1 ^ J nearly, 
 
 x x ^- being omitted, because x does not exceed a small number 
 oUU 
 
 of seconds. Thus the correction for Runs in this case is 
 -f x fl gli an d is called a "Negative Correction." 
 
 The Circle-reading. 
 
 180. We are now prepared to obtain rules for calculating 
 accurately the mean of all the microscope-readings, or, as it is 
 called, a complete Circle-reading. Supposing the number of the 
 microscopes to be six, let the recorded readings be m^ w 2 ,...w e , 
 and the Runs of the respective microscopes be x^ x^...x & . Then 
 since by the setting of the microscopes h has nearly the same 
 
 value for all, the corrected microscope-readings are m A ^x lt 
 
 m 9 v# 2 , &c., and their sum may be written 2.ra -2.ce. 
 
 o ' o 
 
 The Runs of all the microscopes, as being subject to continual 
 changes, probably owing chiefly to the effect of changes of 
 temperature on the radius of the graduation-band, are taken 
 once or twice a week, or even nightly, and the values of 2. x 
 thereby obtained are used for observations made both before 
 and after the several times of taking the Runs. The quantity 
 2 . x is called the Correction for Runs ; a proportional part of 
 which in the ratio to the whole of h to 5, or, in the case of a 
 negative line being bisected, in the ratio of h 5 to 5, is added 
 up with the six microscope-readings, and the sum, divided by 6, 
 gives the complete Circle-reading. 
 
 181. In consequence of the setting of the microscopes, it 
 suffices to record the Circle-reading for the bisected line in the 
 .case of only one of the microscopes, and the one most convenient 
 
174 PRACTICAL ASTRONOMY. 
 
 for this purpose is that at the north end of the Circle's horizon- 
 tal diameter. As a Circle-reading has of itself no astronomical 
 signification, it may be reckoned from the zero of the graduation 
 to an index in an arbitrary position ; and hence, for the sake of 
 convenience in recording, we may take, instead of the Circle- 
 reading for the bisected line, that for a line distant from the 
 bisected line by a constant arc. In order that this may be 
 readily done, the pointer spoken of in Art. 164, as being used 
 for setting for an observation, is so placed that it points to a 
 graduation-line when there is one coincident with the zero 
 position of the selected microscope. Then the Pointer-reading 
 to be recorded is the circle-reading for the line which has the 
 same position relative to the pointer, as the bisected line of 
 graduation has to the zero of the microscope, and is, therefore, 
 the circle-reading for the line which in the order of graduation 
 comes next before the pointer (see Art. 178). In the case of the 
 Cambridge Circle the pointer is placed below the north microscope, 
 or farther on in the order of graduation, and the arc between the 
 zero of the microscope and the pointer is 10. 40'. Accordingly 
 in the instance in Art. 178, the pointer-reading would be 
 130. 25' + 10. 40', or 141. 5', and the concluded Circle-reading 
 141. 6'. 54",7 for the particular microscope. It is to be un- 
 derstood that the graduation proceeds from to 360 round 
 the Circle. 
 
 182. Besides the error of Kuns, the microscope-readings 
 may be affected by two other kinds of error due to the following 
 causes. First, the centre of the graduation, defined to be the 
 centre of a circle coinciding with the middle of the gold band, 
 may not be on the axis of motion, in which case the Circle is 
 said to be eccentric; and again, the axis of motion may not have 
 a fixed position in space in consequence of a non-cylindrical form 
 of the pivots. If the pivots be exactly cylindrical the path of 
 the centre of graduation is a circle; otherwise it is a re-entering 
 curve the deviations of which from the circular form depend wholly 
 on the shape of the pivots. In Figure 35 let OPQR represent 
 the path described by the centre of graduation when the Circle 
 is turned on its axis completely round ; and supposing the centre 
 
THE MURAL CIRCLE. 175 
 
 of the graduation to be at any point of the path, let COG' be 
 drawn through this point in the common direction of the axes of 
 the opposite microscopes A and B\ and at the same time, the 
 
 Fig. 35. 
 
 opposite limbs being at MC and NC r , let the Runs of the two 
 microscopes be taken. Suppose also that by turning the Circle 
 about its axis the position of the centre of graduation is changed 
 from to Q, and that the limbs are brought into the new 
 positions at D and D'. The motion of the Circle by which 
 these changes are produced may be conceived to consist of a 
 motion of rotation about the centre of graduation at 0, and a 
 motion of translation of every point of the Circle equal and 
 parallel to the translation of the centre of graduation from 
 to Q. The former motion produces no change of the position 
 of the periphery of the Circle ; and the change produced by 
 the other is such that if CE and G'E' be drawn parallel and 
 equal to OQ, the points which by the rotation alone would be 
 brought to C and G' at the end of the interval during which 
 the centre of graduation passes along the curve OPQR from 
 to Q, are brought, at the end of the same interval, by the 
 motions of rotation and translation combined, to the positions 
 E and E'. It is evident since by the construction CD = CD', 
 that the Run of A is as much diminished, as that of B is 
 increased, by the change of the distances of the graduation from 
 the microscopes, and their sum is consequently unchanged. Also, 
 the graduation proceeding in the direction indicated by the 
 arrow-heads, since DE D'E' y the pointing of A is as much in 
 advance of the position E as the pointing of B is behind that of 
 E' t so that the sum of the opposite readings remains unaltered 
 by the change of position of the Circle. Since A and B may be 
 
176 PRACTICAL ASTRONOMY. 
 
 any two opposite microscopes, the same inferences apply to the 
 complete Circle-reading. Hence we may conclude, that after 
 obtaining the correction of the error of Runs for a given posi- 
 tion of the Circle, the remaining errors, due to eccentricity of 
 the graduation and irregularity of the forms of the pivots, are 
 corrected in the means of the readings of opposite microscopes 1 . 
 It is advisable, for greater certainty, to determine the correction 
 for error of Runs for more than one position of the Circle at the 
 same time. 
 
 183. The position of reference in the field of view for 
 observations with the Mural Circle is that of a fixed horizontal 
 wire, which in Figure 31 (p. 157) is represented by the straight 
 line passing through the hole of the comb, and the object 
 observed is bisected by this wire by means of the tangent-screw 
 movement, generally when it is crossing, or very near, the 
 middle of the five vertical wires. As, however, the bisections 
 cannot always be made at this part of the field, the fixed wire 
 requires an exact equatorial adjustment, in order that the 
 reference position which it indicates may be the same for 
 bisections made at different distances from the middle wire. 
 This adjustment is effected by means of the apparatus C (see 
 Art. 163), which serves to give to the fixed wire a motion of 
 rotation about the axis of the telescope. By joining this 
 motion with the movement of the Circle by the tangent-screw, 
 a small star, on or near the Equator, may be made to traverse 
 the wire during its passage through the field. If an equatorial 
 star should be judged to be well bisected by the wire throughout 
 the course, the required adjustment may be considered to be 
 
 1 According to this theory the errors of the graduation-readings of a Mural 
 Circle might be corrected by the use of only two opposite microscopes. In 
 reliance on this principle, Pond made trial for a time of a single pair of micro- 
 scopes: but having found, after comparing the results with those obtained by 
 six microscopes, that the latter were the more consistent and trustworthy, he 
 finally adopted the use of six. Considering the changes of form the Circle may 
 be liable to from the mechanical effect of its own weight, and from differences 
 of temperature at different parts, it might be expected that the larger number 
 of microscopes would have the effect of eliminating, at least in part, residual 
 errors due to those causes. 
 
THE MURAL CIRCLE. 177 
 
 made. That, however, is a condition it is not easy to satisfy, 
 requiring generally a succession of trials by different stars, which, 
 for greater certainty in bisecting, should all be of about the 
 ninth magnitude. When the required condition has been by 
 degrees fulfilled, the wire is to be fixed in position by making 
 the opposite screws of C bear fully on the projecting piece e 
 (Fig. 31). As it is important that this adjustment should be 
 exact, it would be well, after completing the operation, to test 
 the result by additional bisections of small equatorial stars. 
 
 184. The use of the fixed wire might be dispensed with by 
 making all bisections with the micrometer-wire (Art. 161), sup- 
 posing the reference position to be indicated by this wire when 
 it crosses the hole of the comb and the micrometer-index is at 
 zero. For convenience the micrometer-reading for this position 
 may be taken to be exactly 10 r . This method requires the 
 micrometer-wire to be equatorially adjusted by the process 
 described in Art. 183 relatively to the fixed wire. This process 
 may in the present case 'be facilitated by bisecting the star soon 
 after its entrance into the field and shortly before its departure, 
 and judging from the difference of the micrometer-readings for 
 the two bisections, as to the amount of angular deviation of the 
 wire that remains to be corrected. Although simplicity is 
 gained by using only the micrometer- wire, it is still to be said 
 that a fixed wire may be advantageously employed in certain 
 observations, such as measures of the diameters of planets, and 
 generally for measuring small differences of N. P. D. of adjacent 
 objects. In any case, however, it will be required to reduce an 
 observation made by bisecting the object with the micrometer- 
 wire in any position, to what it would have been if the bisection 
 had been made in the reference position. This is done when 
 there is only a micrometer-wire by subtracting 10 r from the 
 micrometer-reading for the bisection, and multiply ing the excess 
 (positive or negative) by the value in arc of one micrometer- 
 revolution, as ascertained by processes that will be given in the 
 next paragraph. The result with its sign changed is to be added 
 to the mean of the microscope-readings to obtain the concluded 
 Circle-reading for the observation. In the case of a fixed wire 
 c. 12 
 
178 PEACTICAL ASTRONOMY. 
 
 equatorially adjusted, the micrometer- wire (Fig. 31) is not 
 assumed to be exactly parallel to it, because, as stated in Art. 
 161, the apparent parallelism depends wholly on mechanical 
 construction. Hence, on the principle of testing and correcting 
 mechanical adjustments by optical means, the readings for 
 coincidence of the micrometer-wire with the fixed wire (see Art. 
 51) are taken at the five vertical wires and at the comb, and 
 those for intermediate positions are inferred by interpolation. 
 Then the distance of an object bisected by the micrometer- wire 
 from the fixed wire is measured by the excess of the reading for 
 bisection above the coincidence-reading corresponding to the 
 place of bisection, and this excess, converted into arc, is to be 
 applied negatively to the mean of the microscope-readings. 
 
 185. For finding the value in arc of one revolution of the 
 micrometer, a meridian mark, such as that described in page 40, 
 note 1, might be made use of in the following manner. The 
 micrometer-wire, being set at a certain number of integral 
 revolutions from zero, as 10 r , is brought by clamp and tangent- 
 screw to bisect the mark, and the circle-reading is taken. Then 
 after another setting, as at + 10 r , the wire is again brought to 
 bisect the mark, and another circle-reading is taken. The 
 difference between the circle-readings is the value in arc of the 
 difference between the micrometer-readings, and being divided 
 by the number of the micrometer-revolutions (20, in the sup- 
 posed instance), gives the value in arc of l r . The adopted 
 value should be the mean of several such measures. In making 
 use of this method I found that the measures were vitiated by 
 variation of terrestrial refraction in the interval between the 
 bisections, the apparent altitude of the mark being thereby 
 altered, especially if the operation was performed in the evening. 
 This source of error may in great measure be got rid of by 
 employing the following process. Supposing the micrometer- 
 wire to be set at 4- 10 r , 10 r , and + 10 r in succession, and the 
 mark to be bisected by the wire in each position, let the respec- 
 tive circle-readings be a v a a , 3 . Then if by the increment of 
 refraction in the interval between the first and second bisections 
 the angular elevation of the mark be increased by e, the reading 
 
THE MURAL CIRCLE. 179 
 
 a g , as compared with a lt is too small by e, or, corrected for 
 change of refraction, becomes a 2 -f e. So if the interval between 
 the second and third bisections be the same as that between 
 the first and second, and the variation of refraction during the 
 two intervals be supposed to be uniform, the corrected third 
 reading, or that which would have been obtained if the refrac- 
 tion had not varied, is a 3 + 2e. We have, therefore, from the 
 three Circle -readings for bisection of the mark, when the micro- 
 meter-readings and the corrections for variation of refraction 
 are taken into account, the following equations : 
 
 a t + 10 r = a z + e - 10" = a 3 + 2e + 10 r . 
 Hence 20 r = a 2 a l + , and 20 r = a 2 a 3 e. 
 
 '' 
 
 If the successive settings of the micrometer be 10 r , + 10 r , 
 10 r , the value of 20 r is 1 3 -a 2 . Supposing the mark to 
 
 be bisected any number of times n exceeding 3, and the micro- 
 meter settings to be alternately + 10 r and 10 r beginning with 
 either, we might obtain, in the manner above shewn, a measure 
 of 20 r from the Circle-readings a v a< 2 , a z ; another from a 2 , a a , a 4 ; 
 and so on. Each of the n 2 values thus calculated would be 
 in great measure free from the effects of variation of refraction. 
 I have found that the several values thus obtained are consist- 
 ent with each other, and that their sum divided by n 2 gives 
 a very probable mean result 1 . 
 
 1 This method of alternate measures for extracting the true measure of a 
 quantity from observations affected with small continuous changes, admits of 
 various applications. We have had an instance in the determination of the 
 azimuth-error of a transit-instrument by observations of the clock-times of 
 transit of Polaris alternately above and below Pole. (See Art. 89 (4).) The 
 method may be generalized as follows. Let a 15 a 2 , a 3 ...a n be quantities recorded 
 in order of time, as given by the alternate observations, and let d^, d z , d 8 .--^n-i 
 be the differences (without respect to sign) between a l and a a , a 2 and G 3 , &c., 
 that is, the several measures uncorrected for the above-mentioned changes. 
 Then if Aj, = d 1 + d z , A 2 = d z + d 3) . . . A n _ 2 = d n _ z + d n _ v the corrected measure is 
 
 2Tn32 ) ( A i + < i >+- + A -^ 
 
 122 
 
180 PRACTICAL ASTRONOMY. 
 
 186. On account of the above-mentioned effect of the. varia- 
 tion of terrestrial refraction, and the general unsteadiness of the 
 image of a distant mark, the use of a Collimator for deter- 
 mining the value of the micrometer-revolution, whether it be 
 temporarily or permanently fixed, is much to be preferred. 
 The collimator may be at a short distance from the Circle- 
 telescope, and even in the same room, and at its geometrical 
 focus there should be two connected parallel wires, like those 
 which in Figure 13 (p. 44) are represented as being in the 
 field of view bb' of the collimator B. In fact, this collimator is 
 well adapted for the present purpose, the parallel wires being 
 first brought into position across the middle of the field by 
 turning the Collimator's micrometer-head. These wires have 
 to be adjusted to geometrical focus, and placed parallel to the 
 micrometer-wire of the Circle-telescope, by the processes indicated 
 in Art. 54. Then the micrometer-wire, after being set either at 
 10 r or + 10 r , is to be brought by clamp and tangent-screw to 
 the middle of the fixed space between the images of the two 
 parallel wires (a condition which by the judgment of the eye 
 can be satisfied with great precision), and the Circle-reading is 
 to be taken. The same operations having been performed with 
 the micrometer-wire set at + 1CT, or 10 r , the difference of the 
 two readings is a measure of 20 r , whence the value in arc of V 
 may be at once inferred. The adopted value should be the 
 mean of several such determinations ; and it is advisable, in 
 this method also, to read off the Circle for observations made in 
 alternate positions of the wire, as probably tending to diminish 
 the effect of incidental errors. 
 
 187. It has been shewn (Art. 182) that corrections for 
 the error of Runs, obtained in an arbitrary position of the Circle, 
 are applicable to any other position, notwithstanding eccentricity 
 of the graduation and irregularity of forms of the pivots, if the 
 graduation-band be always circular and its radius remain 
 unchanged. But if these conditions be not fulfilled, the sum of 
 the Runs might be different for every different position of the 
 Circle, or be different for the same position at different times. 
 Such changes might be produced by the effects of change of tern- 
 
THE MURAL CIRCLE. 181 
 
 perature both on the Circle and the microscopes, or by the defor- 
 mation of the Circle due to flexure caused by its weight, or by 
 the two causes combined. As far as regards the bodily movements 
 of the Circle and the microscopes by the effects of temperature, 
 it is evident that if the heights of the supports be arranged so 
 that there is no motion of the microscopes relative to the Circle, 
 the amount of Runs will not be changed. (See Art. 158, p. 152). 
 Also supposing the change of height of the supports by tempe- 
 rature to be such as to make the Circle move relatively to the 
 microscopes, it appears from the argument in Art. 182, that no 
 change of the amount of Runs will thereby be produced, if only 
 the motions of opposite microscopes be equal and parallel. But 
 so far as by the effect of temperature the displacements of 
 opposite microscopes are not the same in amount and direction, 
 or the radius of the graduation-band is changed, or its curvature 
 is caused to vary from point to point, the amount of Runs 
 undergoes change, because the simultaneous changes of the 
 Runs of two opposite microscopes do not in these cases neutralize 
 each other. The errors with which the Circle-readings will in 
 consequence be affected, can only be obviated by frequently 
 taking the Runs, and in various positions of the Circle, with 
 special reference to changes of temperature. The effect of the 
 deformation of the Circle by its weight, and of the flexure of the 
 telescope-tube thence resulting, will come under consideration 
 in Division III. 
 
 The Zenith Point. 
 
 188. Before proceeding to treat of the mode of using the 
 Mural Circle for observing celestial objects, it is required to 
 find the value of an arc which may be called the index error of 
 a Circle-reading. It has been stated (Art. 181) that a Circle- 
 reading obtained, according to the foregoing rules, for any 
 arbitrary position of the telescope, has no signification of itself ; 
 but a reading obtained (by means which will be presently 
 indicated) for the particular position in which the telescope 
 points to the zenith of the place of observation, gives the means 
 of inferring from other Circle-readings Zenith Distances. It is 
 
182 PRACTICAL ASTRONOMY. 
 
 conceivable that by shifting the telescope on the Circle through 
 an arc exactly equal to the Circle-reading for zenith-direction, 
 that reading might be reduced to zero, and the index error thus 
 be corrected. But there is no need to make this mechanical 
 correction, because the index error is virtually corrected, as will 
 afterwards appear, by the process of calculation applied to each 
 observation. The uncorrected index error, as having reference 
 to the zenith- direction, is called the zenith point. 
 
 189. There are various ways of obtaining the Circle-reading 
 for the zenith-direction. The use of floating collimators for this 
 purpose, proposed by Kater, is simple in principle, and gives 
 trustworthy results, but does not admit of the degree of accuracy 
 attainable by employing the collimating eye-piece in a manner 
 which will presently be indicated. Floating collimators are of 
 two kinds, vertical and horizontal, the principle of floating being 
 applied alike in both. The collimator, consisting simply of an 
 achromatic object-glass, with cross- wires at the geometrical focus, 
 is firmly attached to a heavy iron plate which floats on mercury 
 contained in a shallow iron vessel. In the case of the vertical 
 collimator, the axis of which is nearly vertical, the object-glass is 
 directed downwards through an aperture at the central part of 
 the bottom of a vessel of circular form, the shapes given to the 
 interior containing surface of the vessel, and to the iron plate, 
 being such as to allow of collimating with the telescope through 
 the aperture. A contrivance is added whereby the vessel and 
 its contents, together with the collimator, may be readily turned 
 about a vertical axis through an arc which by a check is limited 
 to 180. For use with a Mural Circle this instrument requires 
 to be supported by an appropriate frame just above the highest 
 part of the Circle, in order to perform the collimation with the 
 telescope directed nearly to the zenith. The collimation is 
 effected when, by moving the Circle and adjusting the position 
 of the cross-wires, their intersection, as seen in the field of the 
 telescope, is brought to be coincident with a defined point near 
 the centre of the field. The Circle-reading is then taken, and 
 after turning the collimator through 180, the collimation is 
 performed as before, the same two points being brought into 
 
THE MURAL CIRCLE. 
 
 183 
 
 coincidence, and the Circle is again read off. Since by the 
 principle of floating the axis of the collimator in the two 
 positions is equally inclined to the vertical in opposite azimuths, 
 the same will be the case with respect to the axis of pointing 
 .of the telescope, and the mean of the two Circle-readings will 
 consequently give the reading for zenith-direction, or the zenith 
 point. In Figure 36, 0, o, c are respectively the positions of 
 
 Fig. 36. 
 
 the optical centres of the telescope's and collimator's object- 
 glasses, and the intersection of the cross-wires, before reversing 
 the floating apparatus, and 0', o, c the positions of the same 
 points after. The three points may in each case be assumed to 
 be on the same straight line representing the common direction 
 of the axes of the telescope and collimator, and the straight 
 line NZ bisecting the angle which the two directions make with 
 each other gives the direction of the zenith. 
 
 190. In the case of the horizontal floating collimator, the 
 iron vessel containing the mercury is of rectangular form, and 
 the float is a rectangular plate of iron, to which the vertical 
 supports of the two ends of the collimator are attached. When 
 floating the collimator has its axis very nearly horizontal, and 
 by means of pins fastened to opposite edges of the plates and 
 
184 
 
 PRACTICAL ASTRONOMY. 
 
 fitting into fixed slits, horizontal displacements of the apparatus 
 are sufficiently prevented. After the collimator has been suitably 
 placed on a level with the centre of the Circle to the North or 
 South, the colliraation with the telescope is effected just as in 
 the case of the vertical collimator, and the Circle is read off. 
 The collimator is then transferred to the opposite side of the 
 Circle and similarly placed, and after a second collimation with 
 the telescope, the Circle is again read off. For the same reason 
 as in the case of the vertical collimator the axis of the telescope 
 in the two positions is equally inclined in opposite azimuths to 
 the vertical, and the mean of the Circle-readings is therefore the 
 zenith point. The way in which the geometrical conditions of 
 
 Fig. 37. 
 
 the operation are illustrated by means of Figure 37 may be stated 
 in the very same terms as those applied above to the vertical 
 collimator by reference to Figure 36. 
 
 (The method of obtaining the zenith point by direct and 
 reflection observations of the same star belongs to the next 
 Division.) 
 
 191. It only remains to shew how to determine the zenith 
 point by means of the collimating eye-piece described in Art. 81, 
 which, as used at the Cambridge Observatory, could be adapted 
 both to the telescope of the Transit-instrument and to that of 
 the Mural Circle. There are several ways of employing the 
 collimating eye-piece for this purpose. Care should first be 
 taken in each instance to perform the adjustment of the distance 
 of the wire-frame from the object-glass by which a wire and its 
 image are made to appear equally well defined (see end of Art. 
 161). If the telescope has only a micrometer-wire, for deter- 
 
THE MURAL CIRCLE. 185 
 
 mining the zenith point it is only required to obtain the micro- 
 meter-reading for coincidence of the wire with its reflected 
 image, the Circle being previously set so that the coincidence 
 may take place near the middle of the field, or the coincidence 
 reading not differ much from 10 r . The coincidence of a wire 
 with its image may be judged of with considerable certainty by 
 noticing the definition when they are being brought into 
 conjunction, which becomes sensibly sharper as the coincidence 
 is more exact. But in general it is best to place the image 
 either in contact with the wire, or at small distances from it 
 judged by the eye to be equal, an equal number of times on one 
 side and on the other, and to take the mean of the micrometer- 
 readings in the different positions for the coincidence reading. 
 The difference between this value and the reference reading of 
 10 r , being converted into arc, and applied to a Circle-reading 
 taken before or after the coincidence is observed, gives the 
 Circle-reading for the pointing of the telescope in the direction 
 indicated by that coincidence, that is, the zenith point. If the 
 telescope has a fixed wire for reference, the wire and its image 
 might be brought into coincidence by movement of the Circle 
 by clamp and tangent-screw, if such apparatus should be 
 conveniently within reach, and the Circle-reading then taken 
 would be at once the zenith point. But it would be a more 
 certain method to obtain, for a given Circle-reading not differing 
 much from the zenith point, the reading for coincidence of the 
 micrometer- wire with its image in the way just stated, and then 
 after finding the reading for coincidence of the micrometer-wire 
 with the fixed wire, and converting the difference of the two 
 readings into arc, to infer the zenith point by applying the 
 difference to the above mentioned Circle-reading. For finding 
 the reading for coincidence of the micrometer-wire with its 
 image, I finally adopted the following method, considering it 
 more accurate than those before mentioned; but it is only 
 applicable in case there is a fixed wire. A Circle-reading differing 
 little from the zenith point having been taken, the micrometer- 
 wire was placed so that either its image, or that of the fixed 
 wire, was mid-way between the two wires, a position which could 
 
186 PRACTICAL ASTRONOMY. 
 
 be determined with nicety by the judgment of the eye. Then, 
 the micrometer-reading for coincidence with the fixed wire being 
 taken, if a be the reading for that wire whose image is mid-way 
 between the two, and b the reading for the other, the reading 
 r for coincidence of the micrometer-wire with its image is 
 
 evidently given in all cases by the equation r = a -\- -^(b a). 
 
 As the zenith point is a reference quantity, for the sake of 
 accuracy it should be frequently observed, regard being had at 
 the same time to changes of temperature, and on each occasion 
 the Runs should be taken. 
 
 Plate II. represents the Troughton Mural Circle, which, 
 in conjunction with a similar one by Jones, was used at the 
 Greenwich Observatory till both were superseded by the erection 
 of the present Transit Circle. An historic interest pertains to 
 Troughton's instrument, it being the first of the kind. The 
 Cambridge Mural Circle is very like this both as to construction 
 and mounting, but differs in being eight feet, instead of six 
 feet, in diameter, and in some minor details. The six micro- 
 scopes (for the description of which see Art. 162) are repre- 
 sented as mounted on brass supports satisfying the condition 
 spoken of in Articles 158 and 187. The designations of the 
 microscopes, in the order from North over the highest part of 
 the Circle to South, are A, C, E, B, D, jF'and the readings are 
 recorded in this order in two rows A, G, E and B, D, F. the 
 latter under the other, so that the mean of the readings of each 
 set of opposite microscopes can be readily inferred, and any 
 considerable change of value be detected. The object-end of 
 the telescope is at a, and the eye-end at e, and the positions of 
 the two Ramsden's Ghost Apparatus (described in Articles 
 168 170) are at /and g. The friction wheels nn are suspended 
 for the counterpoise action by the rods mm (see Art. 165). 
 For the purposes of clamping and slow motion, a circular flat 
 ring, somewhat larger than the graduated rim of the Circle, is 
 fastened to the wall, and on this ring a clamp-and-screw c for 
 slow motion slides, and may be clamped to it at any point, 
 a clamp b, which bites the Circle and gives it motion, being at 
 
LIBRARY 
 
 TVER SIT'' 
 
THE MUEAL CIRCLE. 187 
 
 the other end of the moving screw. The necessity for this 
 additional ring is superseded in the case of the Cambridge 
 Circle by the construction of the clamp and tangent-screw, as 
 explained in Art. 160, by reference to Figure 29. The letters 
 P, Q, R mark the positions at which the three apparatus used 
 with that Circle are attached to the wall, the position of P 
 being about 30 below the microscope A. The small squares 
 near the microscopes are sockets for inserting the support of 
 the lamp by which the graduation read off by the microscopes 
 is illumined. The lamp-supports of the Cambridge Circle are 
 connected by hinges with attachments to the wall, and can be 
 let down, or put up, as the purpose they are intended for re- 
 quires. 
 
 III. METHODS OF OBSERVING WITH THE MURAL CIRCLE, AND 
 CALCULATION OF APPARENT ZENITH DISTANCES. 
 
 192. It will be assumed, as in the case of taking observa- 
 tions with the transit-instrument, that in every instance the 
 right ascension and polar distance of the object to be observed, 
 and the error of a sidereal clock, suitably placed in the observ- 
 ing room, are at least approximately known. (See Art. 108.) 
 Let us, first, suppose that the index at that end of the level of 
 the setting-circle which is northward when the bubble is in 
 mid-position, is made to point to the zero of the graduation 
 when the telescope is directed to the zenith by any of the 
 means described in Arts. 189 191, the graduation proceeding 
 both ways from zero to 180. Then another index attached to 
 the level, and at a distance below the first equal to the colati- 
 tude X, will point to the graduation-reading for polar distance 
 of zenith, and, therefore, indicate the actual pointing of the 
 telescope. The same index will serve for setting the telescope 
 to any given polar distance X 4- 2, because if the setting-angle 
 be increased by z t the telescope, in order to bring the bubble 
 to mid-position, must be turned to point farther from the Pole 
 by the same angle. This, in fact, is the usual method of setting 
 for direct observations made with a Transit-instrument. But 
 
188 PRACTICAL ASTRONOMY. 
 
 with the Mural Circle, besides direct observations, many are 
 taken by reflection at the surface of mercury, and it is, there- 
 fore, required to find the position of an index which may be 
 used for setting in such observations. 
 
 193. If the telescope be moved northward from the zenith 
 direction through an angle equal to the colatitude X, it will be 
 made to point to the Pole, and in order that the bubble may 
 in that case be in mid-position, the level must be turned in the 
 opposite direction through X, by which operation the index for 
 setting in polar distance (Art. 192) will be made to point to 
 the zero of the graduation. If now the telescope be turned 
 farther in the same direction through an angle equal to twice 
 the latitude, it will point below the north horizon by an angle 
 equal to the latitude, and, therefore, in the direction proper for 
 observing by reflection an object at the Pole. Then in order 
 to bring the bubble into mid-position the index and level must 
 be moved farther in the same direction as before through twice 
 the latitude. Hence, L being the latitude, if a second index be 
 attached to the level so as to be farther on in the direction of 
 the graduation than the first by 2L, when that index points to 
 zero, the bubble being in mid-position, the telescope will point 
 in the direction proper for observing by reflection an object at 
 the Pole. The same index will serve for setting for the reflec- 
 tion observation of a star of any polar distance ; only it is to be 
 noticed that the setting is to be performed as if the polar 
 distance were negative. For if the telescope be moved to a 
 depression below the north horizon greater than L by 8, that 
 is, if it be set for the reflection observation of a star whose 
 polar distance is S, the new index will have to be moved from 
 the zero through 8 in the negative direction to put the bubble 
 in mid-position. In accordance with the above explanations 
 the Cambridge Mural Circle has a setting-circle furnished with 
 two indices, movable with the level, and separated by an 
 interval of 104 26' (= 2Z,), and one is marked for use in direct 
 observations, and the other for use in reflection observations. 
 
 194. But in the use of that instrument, a direct and a re- 
 flection observation of the same star, one quickly after the other, 
 
THE MURAL CIRCLE. 189 
 
 are taken in the same meridian transit (by a method which 
 will presently be indicated), and there is, consequently, not 
 sufficient time for setting between the observations by means 
 of the setting-circle. On this account the following process is 
 adopted. (See Art. 164.) Let Zloe the zenith point (inclusive 
 of pointer reading), obtained as stated in Arts. 189 191, and 
 in Figure 38, let PZa (= 8) be the star's polar distance, and 
 
 PZ (= X) the colatitude of the Observatory. Ov is the line of 
 pointing of the telescope for the direct observation, and Oa-' 
 that for the reflection observation, the angle ZOcr being equal 
 to the angle NOcr' by the law of reflection. The pointer read- 
 ing for the direct observation is greater than that for the zenith 
 by the angle Z0a, or B X, and is, therefore, equal to Z+ S \ 
 which, if A be put for the constant angle Z X, becomes A +8. 
 The pointer reading for the reflection observation is less than 
 that for nadir point by the angle NOo' , or 8 X, and is, there- 
 fore, equal to 180 + Z (8 X), or B 8, the constant B being 
 put for 180 + Z + X. In general the pointer reading for the 
 direct observation is obtained by adding the approximate polar 
 distance to the constant A just before proceeding to make the 
 observation. But for bodies of the Solar system, the motions 
 of which are given by ephemerides, and for stars which are to 
 be observed directly a considerable number of times, it is ad- 
 visable to tabulate for use the values of A -f ; and in all 
 cases of stars selected for frequent observation directly and by 
 reflection at the same transit, the values both of A -f 8 and 
 
190 PRACTICAL ASTRONOMY. 
 
 B 8 should be calculated beforehand, and tabulated accord- 
 ing to the approximate right ascensions. 
 
 195. According to what is stated in a Note to Art. 34 
 (p. 28), the use of a polar distance index may be dispensed with, 
 and the zenith distance index be used in its place, if the 
 setting-circle be turned relatively to the level with bubble in 
 mid-position through an angle equal to the colatitude. For 
 instance, let the telescope be pointed to the zenith, and the 
 bubble be in mid-position, so that the zenith distance index 
 points to zero and the polar distance index to the colatitude. 
 To make the former index indicate polar distance when the 
 bubble is in mid-position, it is only required to turn the zero 
 of graduation through an angle equal to the colatitude from 
 the northward horizontal direction towards the vertical direc- 
 tion. The mechanical construction allows of this being readily 
 done. By this adjustment the radius drawn to the zero from 
 the centre of the setting-circle makes an angle equal to the 
 latitude with a radius drawn parallel to the axis of the tele- 
 scope towards the object-end. After this alteration the setting 
 is performed exclusively by the zenith distance index. In the 
 case of the Cambridge Transit-instrument, which had two 
 setting-circles, the change of the place of zero was made, in 
 the above manner in each, when it was westward, after which, 
 as before, either circle could be used for setting when on the 
 west side of the telescope tube. 
 
 196. When, however, a meridian Instrument has two set- 
 ting-circles, and, as in the instances of the Greenwich and 
 Cambridge Transit-Circles, is not reversed, the second one may 
 be made use of in one or the other of these two ways : either 
 by the above-stated arrangements the setting for polar distance 
 might be performed in exactly the same manner with both, in 
 which case they could be conveniently used for setting for the 
 direct observations of two objects coming in quick succession 
 one after the other ; or the second circle might be adapted for 
 setting for reflection observations. The way in which the latter 
 purpose is effected by the eastern setting-circle of the Cam- 
 bridge Transit-Circle is such as follows. Suppose the telescope, 
 
THE MURAL CIRCLE. 191 
 
 as before, to point at first to the zenith, the bubble being in 
 mid-position, and of the two indices of the level that which is 
 northward to point to the zero of the graduation. Then let the 
 telescope be turned from the zenith northward through 180 X, 
 and the index be turned in the opposite direction till it points 
 to 180 - X of the graduation. The bubble will thus be brought 
 to mid-position, and the telescope will be set for the reflection 
 observation of a star at the Pole. After this there are two ways of 
 proceeding. The circle may be shifted, every thing else remaining 
 the same (see Art. 194), either so as to bring the zero of the 
 graduation to the north index, or so as to bring it to the south 
 index. In the former case the north index would be used for 
 the setting, which, however, must be performed as for a negative 
 polar distance, or for a star sub polo, because if that index be 
 moved upwards from zero through an arc 8 equal to the polar 
 distance, the depression of the telescope below the north horizon, 
 in order to bring the bubble to mid-position, has to be increased 
 by 8, and the setting is, consequently, that required for the 
 reflection observation of a star whose polar distance is S. In 
 the other case, the south index would have to be used for the 
 setting, and, as in the ordinary setting for a direct observation, 
 the movement of the index from zero will be downwards, so that 
 the setting for a reflection observation by the south index of the 
 east circle is performed in exactly the same manner as the setting 
 for a direct observation by the north index of the west circle. 
 
 197. The rules for setting the telescope having been laid 
 down, we may proceed to consider the different processes 
 employed in taking Circle observations. For the direct obser- 
 vation of a star, the telescope is first to be set, according to the 
 star's polar distance, either by the setting-circle (Art. 192), or by 
 the pointer directed to the graduation-reading A + 8 (Art. 194), 
 and the Circle is then to be detained by the clamp of that 
 tangent-screw which will be most conveniently reached when 
 the telescope is pointed to the star. Knowing approximately 
 the right ascension of the star and the error of the clock, the 
 observer knows when the entrance of the star into the field is 
 to be looked for, and the setting has secured that the direction 
 
192 PRACTICAL ASTRONOMY. 
 
 of the path shall be nearly across the middle of the field. 
 Then, if no circumstances prevent, the star is bisected, just as it 
 crosses the middle one of the five vertical wires, by the fixed 
 wire, or, if there is no fixed wire, by the micrometer-wire in the 
 reference position, the bisection being effected by moving the 
 Circle with the tangent-screw. In order to satisfy the con- 
 dition mentioned in Art. 160, of always performing the bisection 
 by turning the screw in the direction indicated by the rough- 
 ness of the milled head, if the star should not enter into the 
 half of the field suitable for this operation, it has previously to 
 be placed in that half by means of the tangent-screw. The 
 Circle-reading is then to be taken (Arts. 180 and 181), and 
 this completes the instrumental process. 
 
 198. If it should happen that the bisection is not made 
 when the star is on or very near the middle wire, the place of 
 bisection is recorded by stating its distance from that wire in 
 integer wire-intervals and fraction of an interval, and prefixing 
 a minus or plus sign according as the bisection was made 
 before or after passing the mid-wire. This is done for the 
 purpose of calculating a correction for the curvature of the path 
 of the star in the field, to be applied to the Circle-reading to 
 reduce it to what it would have been if the star had been 
 bisected when on the meridian. This mode of indicating the 
 place of bisection is sufficiently accurate if the plane of collima- 
 tion of the telescope has been well adjusted mechanically to 
 the plane of the meridian by the methods explained in Arts. 
 168 174, and if the curvature of the star's path be small. But 
 for observations of stars near the Pole, as especially Polaris, 
 8 UrssB Minoris, and 51 (Hev.) Cephei, and generally for 
 observing with precision a star, the curvature of whose path is 
 considerable, the following process should be adopted. The 
 clock-time of making the bisection should be set down, and 
 then by knowing approximately the error of the clock and the 
 star's right ascension, the hour angle from the meridian can be 
 calculated. From this datum (which may be supposed to be 
 accurate to within two or three seconds), and from the polar 
 distance of the star, the correction for curvature of path can 
 
THE MURAL CIRCLE. 193 
 
 be derived with accuracy by formulae investigated in the next 
 paragraph. If the R. A. of the stars be not known by contem- 
 poraoeous transit-observations, they may be immediately in- 
 ferred, in the instances of stars contained in the Catalogue of 
 the Nautical Almanac, from apparent R A. computed in that 
 work, but in other cases they will have to be approximately 
 deduced from mean R. A. given in standard catalogues. 
 
 199. The above-mentioned formulas for calculating cor- 
 rections for curvature of path are obtained as follows. In 
 Figure 39, P is the pole, the arc PNS is in the plane of the 
 
 meridian, a is the place of the star when its image is bisected, 
 and the bisecting wire, being straight and horizontal, is pro- 
 jected into a great circle of which <rN perpendicular to PS is a 
 part. The dotted arc o-S is part of the star's diurnal path, so 
 that PS is equal to Per the polar distance. Hence, since the 
 projection of the wire cuts the meridian at ^instead of $, the 
 error to be corrected is NS. Let NS = e, Pa = , <rN D, and 
 /. crPN (in time) = t, this hour-angle being obtained in the 
 manner stated in Art. 198. Then since PN=S e, the right- 
 angled triangle PNcr gives 
 
 cos (8 e) = cos 8 sec Z>, 
 and tan (8 - e) = tan B cos lot. 
 
 The first of these equations is applicable to observations of 
 stars not near the Pole, D being the noted interval of the place 
 of bisection from the mid-wire. Supposing, as is usually very 
 nearly the case, the wire-intervals to be all equal, and the 
 common equatorial interval in time to be T*, then if m = the 
 distance of the star from mid-wire, recorded as stated in Art. 
 c. 13 
 
194 PRACTICAL ASTRONOMY. 
 
 198, we shall have D (in arc) = 15mr. As this is a small arc, 
 
 D 2 
 
 sec D = 1 H nearly, R being the number of seconds in an 
 
 arc equal to radius, and since e is very small, 
 
 p 
 
 cos (8 e) = cos 8 + -^ sin 8 nearly. 
 
 Putting, for the sake of convenience in the sequel, the com- 
 plement of the declination A for 8, we obtain, with sufficient 
 approximation, 
 
 g ^ ggl tan A, or e = ^g^ tan A = [6,73672] mV 2 tan A. 
 
 In the instance of the Telescope of the Cambridge Circle 
 T = 16*,6, and the formula employed is 
 
 e = 0",1503xm 2 tanA. 
 
 200. The second of the above equations is applicable to 
 the case of those observations of stars near the Pole which are 
 accompanied by determinations of the hour-angle t. This 
 equation gives the exact result 
 
 sin 28 sin 2 -^- 
 tan e = - . 
 
 P 
 Hence, putting -=. for tan e, neglecting the second term in 
 
 the denominator, which for circumpolar stars will always be 
 very small compared to unity, and substituting 90 A for 8, 
 there results the sufficiently approximate formula 
 
 e = E sin 2A sin 2 ^ = [5,31443] sin 2A sin 2 ^ . 
 
 L 
 
 Let us now suppose the declination A to be increased by 
 the small arc n, and, in consequence, e to be changed to e. 
 Then, for a given value of t, we get by differentiation 
 
 e = e + 2n cos 2 A sin 2 ~ = (R sin 2 A + 2n cos 2A) sin 2 -%- . 
 
THE MURAL CIRCLE. 195 
 
 For examples, let A =88. 40' + ft" for Polaris, and A = 
 86. 36' + n" for S Ursa3 Minoris. Then the formula gives 
 
 1 *" 4- 
 
 Cor. for Polaris {[3,98212] - [0,30056] x n] sin 2 ^ . 
 
 1 x/ 
 
 Cor. for 5 Ursse Minoris... {[4,38780] - [0,29796] x 71} sin 2 . 
 
 2i 
 
 These formulae are calculated for the year 1879, and might, 
 for hour angles not exceeding 6 m , be used for two years before 
 and after that date. 
 
 201. The foregoing investigation gives formulae for calcu- 
 lating the amount of the correction without respect to its sign. 
 The proper sign is affixed according to rules derived as follows. 
 The telescope, when pointed to an object between the pole and 
 the equator at a little distance from the meridian, is turned, by 
 reason of curvature of path, more towards the pole than when 
 the object is on the meridian ; the Circle-reading is conse- 
 quently too small, and the sign of the correction is plus. If the 
 object is either below the equator, or below the pole, the 
 telescope for the same reason is turned farther from the pole 
 than for bisection on the meridian, so that the Circle-reading 
 is too great, and the sign of the correction is minus. 
 
 202. We have, next, to consider in what manner Circle- 
 observations of the zenith distances of stars are made by means 
 of reflection at the surface of mercury. As such an observation 
 is almost always accompanied (for a reason that will be given 
 afterwards) by the direct observation, I shall state at once the 
 method of taking what is called the double observation. At the 
 Greenwich Observatory under Pond's direction, two Circles 
 exactly alike, named from their makers "Troughton" and 
 "Jones," were handled by two observers for this purpose, one 
 observing directly and the other by reflection, and the difference 
 of the index errors of the two instruments had to be very 
 exactly determined. The present Astronomer Royal introduced, 
 first, at Cambridge in 1833, and adopted at Greenwich in 1839, 
 the practice of taking both observations with a single Circle at 
 the same meridian Transit by the following process. The 
 
 132 
 
196 PRACTICAL ASTRONOMY. 
 
 Circle is first set and clamped for the reflection observation, 
 according to a prepared pointer reading (Art. 193), and the 
 setting, if carefully performed, secures that the image of the 
 star shall cross the field not far from the centre. Also the 
 mercury-trough, which is usually of wood, and of rectangular 
 form with the longer side north and south, has to be placed so 
 that the light from the star falls near the middle part of the 
 surface of mercury. This may be done in day-time by placing 
 the trough where the light of the sky through the eye-piece is 
 seen reflected at that part, while at night-time light from the 
 illuminating lamp is sent back from the eye-end in sufficient 
 quantity to be used for the same purpose. The bisections of 
 the graduation-lines by the cross-wires of the microscopes are 
 next made, but, as the observation may fail, are not usually 
 read off; after which the star, as it crosses the field, is bisected 
 by the micrometer-wire one or two intervals before passing the 
 mid-wire, in order that the reflection observation may be made 
 when the star is at about the same distance from the mid-wire 
 on the opposite side. That this may be successfully done, the 
 Circle is undamped immediately after the reflection-observa- 
 tion, is then quickly set and clamped for the direct observation 
 according to a prepared pointer reading, and lastly by means of 
 the tangent-screw the star is bisected, if there is a fixed wire, 
 either by the fixed wire or the micrometer-wire, otherwise of 
 necessity by the micrometer-wire. The intervals between the 
 places of the two bisections and the mid -wire, being fresh in 
 memory, should next be recorded. The microscope-readings 
 for the reflection observation are now written down, if omitted 
 before, new bisections for the direct observation are made and 
 recorded, and the micrometer is read off. This completes the 
 operation. In the case of a Transit-Circle which has two setting- 
 circles and is not reversed, the settings for the double observa- 
 tion would be performed by the two circles in the manner 
 indicated in Art. 195, the telescope being pointed for the re- 
 flection observation by the east circle after preparing the west 
 circle for the subsequent direct observation. It remains to 
 remark respecting observations by reflection, that the trough 
 
THE MURAL CIRCLE. 197 
 
 ought to be placed on a solid basis of brickwork or masonry, 
 and to be screened from wind, in order that the surface of 
 mercury may be as free as possible from tremor ; also that for 
 obtaining satisfactory reflections the surface has frequently to be 
 cleared of dross and dust. This is found to be, effectually done 
 by passing the mercury through a small hole in the side or 
 bottom of a containing trough or vessel into the trough used for 
 the observations. 
 
 203. The correction for curvature of path is plainly the 
 same in amount for the reflection, as for the direct, observation, 
 at the same distance from the mid-wire, but has always the 
 opposite sign, the reason for which will be seen by remarking 
 that the error of position of the telescope due to this cause is 
 in opposite directions in the two observations relatively to 
 the order of a graduation proceeding from to 360. (See 
 Art. 201.) 
 
 204. The interval between the place of bisection and the 
 mid-wire, which, except for small polar distances, may be taken 
 to be the distance from the meridian, has to be noted in every 
 instance of the bisection of a moving body out of the meridian, 
 for calculating, as well as the correction for curvature of path, 
 an additional correction for change of polar distance. As this 
 change may in all cases be assumed to be proportional to the 
 time-interval between the bisection and meridian -passage, let 
 m be the space-interval between the place of bisection and the 
 mid-wire, as recorded in wire-intervals, and T the common 
 equatorial interval between the wires ; then if A be the decli- 
 nation of the body, and /' its horary change of declination, 
 derived either from the Nautical Almanac or from actual obser- 
 
 .,, , m rsec A T , T - , 
 
 vation, the correction in amount will be x 1 . If the 
 
 ouUU 
 
 polar distance be increasing, or the value of T in the Nautical 
 Almanac be negative, the Circle-reading is too small if the 
 bisection be made before meridian passage, and too great if 
 made after y so that in the former case the sign of the correction 
 is plus, and in the other, minus. If the polar distance be de- 
 creasing, or /' in the Nautical Almanac be positive, the sign is 
 
198 PKACTICAL ASTRONOMY. 
 
 minus before meridian passage and plus after. Consequently 
 if m be the interval recorded with sign attached, as stated in 
 Art. 198, and the sign of /' be taken as in the Nautical Alma- 
 nac, the above expression gives the correction in all cases with 
 its proper sign 1 . 
 
 205. The Sun, Moon, and Planets, on account of the forms 
 of their disks, require special modes of observation, the several 
 descriptions of which are here subjoined. For observing the 
 Sun, either the North, or the South Limb, is first to be set for 
 according to a pointer reading derived, to the accuracy of about 
 1', from the Declination of centre and semi-diameter given in the 
 Nautical Almanac ; next, the Circle being (not forcibly) clamped, 
 the microscope-bisections are made, but need not be read off; 
 then the Limb is tangentially bisected by the micrometer-wire a 
 little before the meridian passage of centre, immediately after 
 which the Circle is undamped and turned by hand till the 
 image of the other Limb is placed near the middle of the field, 
 and this Limb, after again clamping, is bisected tangentially, 
 either by the fixed wire or the micrometer-wire, by means of 
 the usual tangent-screw movement (Art. 160). The Circle- 
 reading for the former observation, if not already recorded, is 
 now written down, fresh microscope-bisections are made and 
 read off for the second observation, and lastly the micrometer- 
 reading is recorded. Generally the second observation, like 
 the first, may be made when the Sun's centre is at a small 
 distance from the meridian, and it might, therefore, not be 
 worth while to state the distances. It would, however, be more 
 exact always to bisect a Limb tangentially just when, according 
 
 1 In Volume xvn. of the Cambridge Observations, for the years 1846 1848, 
 pp. Ixvi. Ixviii., are given, together with others, two Tables (iv. and v.), one of 
 which is adapted for calculating by interpolation the correction for curvature of 
 path for all declinations from 5 to 78.40', and for every quarter of a wire- 
 interval from to 4f ; and the other, which is computed to four places of 
 decimals, for calculating by interpolation the change of decimation, for one 
 interval, for all horary variations from V to 10", and multiples or submultiples 
 of the same by 10, and for all declinations from to 40, the equatorial interval 
 being 16 S ,6. Such tables shorten to an important amount the time occupied in 
 reducing a continuous series of Circle observations. 
 
THE MURAL CIRCLE. 199 
 
 to the judgment of the eye, it is cut symmetrically by one of 
 the vertical wires, and to note the wire for the purpose of cal- 
 culating corrections for reducing the observation to the meridian. 
 The operations above detailed, together with the calculation of 
 corrections for distances of the bisections from the meridian, 
 and the reduction in arc, according to the rules given in Art. 
 184, of the micrometer-reading to the reference position, or to 
 the fixed wire, furnish two Circle-readings, one applicable to the 
 North Limb, and the other to the South Limb. The mean 
 between these is the Circle-reading for the Sun's centre, and 
 their difference is a measure of the Sun's diameter, both deter- 
 minations involving, to an amount which will afterwards be 
 taken into consideration, effects of atmospheric refraction. 
 
 206. From the above description of the mode of taking an 
 observation of the Sun with the Mural Circle, it will be seen 
 that on account of the Sun's large diameter it might be practi- 
 cable for one observer, without assistance, to observe transits of 
 the first and second Limbs, and bisect the North and South 
 Limbs between the transits, at the same meridian passage if the 
 Transit-instrument and Circle be in contiguous rooms. This, 
 in fact, was occasionally done at the Cambridge Observatory. 
 The two meridional observations may also be made simul- 
 taneously with a Transit-circle, but only with the aid of an 
 assistant for making and recording the first set of microscope- 
 bisections and making (without recording) the second set, in 
 the interval between the transit-observations of the first and 
 second Limbs. In the use of the Greenwich Transit-circle for 
 the same purpose, the observer is aided by two assistants, who 
 make and read off bisections with the microscope-micrometers, 
 the observer himself reading off the telescope-micrometer. By 
 these means, when, besides, " the transits are made by galvanic 
 touch (Art. 131), it is found perfectly easy to take those of the 
 first and second Limbs over nine wires, still leaving time enough 
 for the observations of both Limbs in N. P. D." (See Greenwich 
 Observations for 1874, p. ix.) It may here be remarked that 
 according to Greenwich arrangements there are two clamps, 
 acting one on the North side, and the other on the South side, 
 
200 PRACTICAL ASTRONOMY. 
 
 of a clamping circle, but they have no slow motion, all bisections 
 of a limb or star being made with the micrometer-wire by 
 turning its screw. By this process errors that might arise from 
 deformation of the Circle are avoided. 
 
 207. In observations of the Moon, it is the practice, on 
 account of the generally rapid change of polar distance, to 
 apply the micrometer-wire to the Limb at each of the vertical 
 wires, or at other positions, and to record the micrometer-reading 
 and position immediately after each bisection. The several 
 observations have then to be corrected for micrometer-reading, 
 and for distance from the meridian according to rules already 
 given (Arts. 184, 199 and 204), and the mean of the several 
 results is the concluded Circle-reading. It is, however, to be 
 remarked that although in the expression given in Art. 204 for 
 the correction on account of change of declination, the time- 
 interval between the bisection and meridian passage is with 
 sufficient accuracy mr for the Sun and Planets, in the case of 
 the Moon this quantity has to be multiplied by two factors 
 sensibly affecting the result, one on account of the retardation 
 of the diurnal motion by the Moon's motion in her orbit, and 
 the other on account of retardation due to parallax. The two 
 factors, which have already been obtained in Arts. 123 and 124, 
 are respectively 
 
 3600 , sin(^-6-ff) 
 
 3600-1* sin(Z-e) ' 
 
 and consequently the complete expression for correction to the 
 meridian in observations of the Moon is 
 
 T 3600 sin^-e- 
 
 X 
 
 3600 
 
 /' as well as I being taken from the horary variations given in 
 the Nautical Almanac under the head of Moon-culminating 
 stars. The Local Zenith Distance Z, contained in this expres- 
 sion and in the formula obtained in Art. 124 for calculating p, 
 may be readily deduced, to the accuracy of 1', from the differ- 
 ence between the Circle-readings for Zenith Point and the 
 bisection of the Limb, corrected by a prepared Table of approxi- 
 
THE MURAL CIRCLE. 201 
 
 mate refractions with zenith distance for argument. The 
 above-mentioned formula, after putting p and P for their sines, 
 and unity for p, becomes p = P sin (z e), which is sufficiently 
 approximate 1 . Also by expanding, and neglecting terms in- 
 volving e 2 , ep, &c., it will be found that 
 
 sin (Zep) sm(Z-p) ^ ,, 
 
 ^TV \ ' ir 2 Hence finally 
 
 sm (Z e) sin Z 
 
 T , rar sec A sin of Vs Geoc. Zen. Dist. 
 Correction = J x -- T x 
 
 3600 - 1 sin of ))'s Local Zen. Dist. ' 
 
 which formula is used at Greenwich and Cambridge, having 
 been originally proposed by the Astronomer Royal when at 
 Cambridge. 
 
 208. In the use of the Transit-circle, a transit of the 
 Moon's first Limb may be observed, and different bisections of 
 the full North or South Limb be made in the manner described 
 in Art. 207, by a single observer at the same meridian passage. 
 So also if the full North or South Limb be carefully set for, 
 and the microscopes be read beforehand, the different micro- 
 meter-bisections of the Limb may be made and recorded, and a 
 transit of the second Limb be taken, by one observer. When it 
 happens either that the first and second Limbs, or that the 
 North and South Limbs, are both sufficiently illumined for 
 observation, in the former case transits of the two Limbs and a 
 Circle-reading for one Limb, and in the latter a transit of one 
 Limb and Circle-readings for the two Limbs, may be obtained 
 with the aid of one assistant, as will be seen from the account 
 given in Art. 206 of the method of observing the Sun with a 
 Transit-circle. 
 
 209. The Circle- observation of a Planet which has a con- 
 siderable disk is made by bisecting the North or South Limb 
 with the fixed wire by tangent-screw movement, generally as it 
 passes the mid-wire, and bisecting at the same time the other 
 
 1 This is also true with respect to the calculation of parallax mentioned in 
 Art. 124 (p. 115). It is incorrectly said there that p and P should not be 
 substituted for their sines, this rule applying only when an exact value of the 
 Moon's parallax is required. 
 
202 PRACTICAL ASTRONOMY. 
 
 Limb with the micrometer-wire. The Circle-readings for the 
 two bisections, obtained according to rules already indicated, 
 give by their mean the Circle- read ing for bisection of the 
 Planet's centre, and by their difference a measure of its 
 diameter. The same results may be arrived at, if there should 
 be no fixed wire, by bisecting alternately an equal number of 
 times the North and South Limbs by the micrometer-wire, and 
 recording the several places of bisection and micrometer- 
 readings. From these data the Circle-reading for each bisection 
 has to be reduced in the usual way to meridian passage, and 
 then, after inferring the mean of the results for each Limb, the 
 concluded Circle-reading for centre is the mean between the 
 two means, and their difference measures the diameter. The 
 Circle-observations of Jupiter, the globe of Saturn, Mars, and 
 Venus are conducted as above stated ; those of the other planets, 
 inclusive generally of Mercury, are made by simple bisections 
 of their estimated centres. When the observation is made by 
 bisections of opposite Limbs, it is often required to correct the 
 inferred Circle-reading for centre, and the measure of the 
 diameter, for form of the illumined disk, both when the disk is 
 horned and when it is gibbous. We have, therefore, in the 
 next place to enquire how such correction is calculated. 
 
 210. The investigation contained in Art. 113, and applied 
 in obtaining a formula for correcting for defect of illumination 
 in transit-observations of a gibbous disk, will serve for the 
 present purpose, whether the disk be horned or gibbous. It 
 will, first, be supposed that the disk is horned, which can be the 
 case only in the instances of Yenus, Mercury, and the Moon. 
 The observation of Venus is not unfrequently made by bisec- 
 tions of a Limb and the opposite cusp ; but this process is 
 rarely employed in observing Mercury (see Art. 209) and the 
 Moon. The following investigation of the correction required 
 in the case of a horned disk applies equally to the three bodies. 
 In Figure 26 (1), the arcs PS ( = A) and PV ( = d) are re- 
 spectively the polar distances of the Sun and the observed 
 body, and the angle SPV (=P) is the difference of their Right 
 Ascensions, at the given time of observation. These three 
 
THE MURAL CIRCLE. 
 
 203 
 
 quantities may be supposed to be derived from the Nautical 
 Almanac. The angle 6 between the arc of the meridian through 
 the centre and the line joining the cusps (Fig. 26 (2)) is equal 
 
 (5) 
 
 to 90 Z PVS. Hence from a known formula of Spherical 
 Trigonometry, which, according to the usual notation, is 
 
 cot a sin b = cot A sin C + cos b cos (7, 
 we derive, mutatis mutandis, 
 
 tan cot A sin S cosec P cos 8 cot P, 
 
 from which equation 6 is to be calculated. Then if 2D be the 
 full diameter of the body, and D be the difference of zenith 
 distance measured by the observation (inclusive in the case of 
 the Moon of a small correction for refraction), we shall have 
 
 2) = DV + Vn (by Fig.) = D + D cos = 2D cos 2 1 . Hence 
 
 D ft 
 
 J) = sec 2 , which is the semi-diameter as deduced from the 
 
 observation, and the required amount of the correction of D is 2Z> 
 n 
 
 D, or D tan 2 ^ . Also the local zenith distance of centre is 
 
 M 
 
 obtained from that of the full Limb (calculated as will be sub- 
 
20-t PRACTICAL ASTRONOMY. 
 
 D ft 
 sequently shewn), by applying the correction + sec 2 -= , ac- 
 
 ~ 
 
 cording as the Limb is the upper or lower one. 
 
 211. In Art. 113 it has been shewn that if R and r be 
 respectively the tabular distances of the Earth and the observed 
 body from the Sun, and e be the eccentricity of the elliptic 
 boundary of illumination in the case of a gibbous disk, the 
 
 7} 
 
 value of e is sin$F. But by the spherical triangle SPY, 
 sin$F sinSPV OTr sin A sin P 
 
 na a > 
 
 sin PS smPVS ' cos 6 
 
 sin A sin P 
 
 Hence e = 
 
 r cos 6 
 
 Now suppose the angle & to be derived from the equation 
 sin & = e sin 0, which equation, by substituting the above value 
 of e, and putting for tan 6 the expression obtained in Art. 210, 
 
 r> 
 
 becomes sin & = (cos A sin 8 cos 8 sin A cos P), giving the 
 
 value of & in known quantities. Hence the value of the per- 
 pendicular Vn on the tangent en (Fig. 26 (3)), which, by Conic 
 Sections, is equal to D cos & ' , can be calculated. Then, as & 
 simply takes the place of 0, the inferences relating to a gibbous 
 disk are precisely analogous to those obtained at the end of 
 Art. 210 relative to the horned disk. 
 
 212. When, as is most frequently the case, the Moon is gib- 
 bous, the calculation of & may be simplified in a manner like 
 
 that indicated in Art. 115 with reference to a transit observation 
 
 p 
 of opposite Limbs. As there shewn, = 1, and e = sin SV 
 
 nearly. Hence, if S'M be the arc joining M the Moon's 
 centre and S' the point opposite the Sun's centre, so that SV ' = 
 180- S'M, it follows that e = sin (ISO - S'M) = sin S'M, and 
 sin & = sin S'M sin 6. Since, therefore, in this kind of observa- 
 tion S'M is always a small arc, we have nearly & S'M sin 6 
 MN in Fig. 26 (5). Thus to obtain & with sufficient approxi- 
 mation, it is only required to find the difference of the polar 
 
THE MURAL CIRCLE. 205 
 
 distances of the Moon and the point opposite the Sun at the 
 time of the Moon's transit. The upper or lower Limb is defective 
 according as the polar distance of the Moon's centre is greater 
 or less than that of the point opposite the Sun's centre. 
 
 213. It is evident from Figure 26, (1) (2) and (3), that the 
 lower or upper Limb is defective according as the angle PVS is 
 less or greater than 90, or according as 6 is positive or negative, 
 and that this rule is applicable both to a horned and a gibbous 
 disk. Hence from the expression for tan 6 in Art. 210 we may 
 infer that the lower or upper limb is defective according as 
 cot A tan 8 is greater or less than cos P. 
 
 214. For observing with the Cambridge Circle asteroids 
 and small stars, when they were too faint to admit of illumina- 
 ting the field sufficiently for bisections with a fine wire, a broad 
 rectangular metal bar was attached (in September 1853) to the 
 east side of the frame carrying the micrometer-wire, with an 
 edge parallel to this wire and distant from it nearly 10 r in the 
 direction/rom the micrometer-head, and a vertical edge nearly co- 
 incident with the middle vertical wire. A faint object could then 
 be bisected by the edge of the bar with very little, and some- 
 times without any, illumination of the field by the lamp. Under 
 these circumstances, suppose the micrometer to be read for a 
 bisection by the bar just as if the micrometer-wire were in the 
 place of the bar-edge, and, M being such reading, let B and W 
 be respectively the readings for coincidence of the bar and wire 
 with the fixed wire at an interval from mid-wire equal to that 
 of the object when bisected. Of these readings W is usually 
 known already (see Art. 184), and B, if not known, should be 
 taken immediately after the observation. Now the micrometer- 
 reading for the actual position of the micrometer- wire when 
 the bar-bisection was made, is plainly M + 10*", and, by inference, 
 the reading for bisection of the object, if made by the micro- 
 meter-wire, would have been M + I0 r (B W). Hence the 
 correction to be added to the recorded reading M t for reduction 
 to the wire-bisection, is the algebraic excess of 10 r above the 
 measured interval B W between the wire and bar-edge. After 
 applying this correction, the reduction to the fixed wire for 
 
206 PRACTICAL ASTRONOMY. 
 
 micrometer-reading is made in the usual way. It may be 
 observed that the use of the fixed wire for the purpose above 
 mentioned might be superseded by measures of the interval 
 between the bar-edge and micrometer-wire obtained from 
 time to time, at the vertical wires, by bisecting with each a 
 meridian mark, or point of a collimator. The interval at other 
 positions might then be obtained by interpolation. 
 
 Finding that the bisection of a faint object by the .edge of 
 a bar is rendered uncertain by the diffraction of the light, I 
 adopted the method of placing the edge, by micrometer-move- 
 ment, as exactly as possible in a line with the path of the 
 object just before its arrival at the mid-wire, or the vertical 
 edge. By noting simultaneously the clock-time of the transit at 
 the edge, an approximate Right Ascension of the object might 
 also be obtained, which, in taking Circle-observations of faint 
 objects, as the asteroids, is often required for the purpose of 
 identifying them. 
 
 Calculation of Local Zenith and Polar Distances. 
 
 215. The difference between the Circle-reading for the 
 direct observation of any object and that for Zenith Point, gives 
 the apparent Zenith Distance of the object northward or 
 southward. As inferred from a reflection observation, the same 
 zenith distance is the difference between the Circle-reading for 
 the observation and that for the Nadir Point. The arc so 
 measured is called apparent zenith distance, because its value 
 involves the amount by which the zenith distance is diminished 
 by atmospheric refraction. The calculation of this effect of 
 refraction is usually performed according to Bessel's Refraction 
 Tables, which are contained in pages 538 542 of the Tabulce 
 Regiomontance, and are founded on an investigation resting on 
 Laplace's solution of the Problem, given in pages lix. Ixiii. of 
 the same work. The data required in using these Tables are, 
 the height of the Barometer, the temperature of the external air, 
 the temperature of the mercury of the Barometer, and the 
 above-defined apparent zenith distance as general argument. 
 
THE MURAL CIRCLE. 207 
 
 The readings of the Barometer and Thermometers are recorded, 
 if not at each observation, frequently enough to take account of 
 all changes that might sensibly affect the calculation of the 
 refractions. A zenith point given immediately by the collima- 
 ting eye-piece (Art. 191) maybe used for inferring the apparent 
 zenith distances, as it will only differ slightly from a finally 
 adopted value derived (as will afterwards be shewn) from direct 
 and reflection-observations of stars. Bessel's Tables have been 
 adopted both at Cambridge and Greenwich, altered in form and 
 expanded, as they are given in an Appendix to the Volume of 
 Greenwich Observations for 1836, and more recently at Green- 
 wich, as given in an Appendix to the Volume for 1853, with a 
 small correction of Bessel's mean refraction. In these compu- 
 tations the external thermometer only is used, unless the Zenith 
 Distance exceeds 85. 
 
 216. After adding the amount of refraction to the ap- 
 parent zenith distance obtained as above stated, the result is 
 the zenith distance of the object measured from the astro- 
 nomical zenith of the place of observation, or from the local 
 direction of the action of gravity. This is the angle which is 
 designated by Zm Arts. 124 and 207, and in the latter Article 
 is called 'local zenith distance.' In order to deduce from this 
 angle local polar distance, or the angle which the direction of 
 the object, as seen from the place of observation, makes with a 
 parallel to the earth's axis, it is required to find the zenith 
 distance of the Pole. This is done by taking Circle-obser- 
 vations of the same star above and below Pole, and correcting 
 the observations for refraction, the mean between the results 
 for the upper and lower culminations being the zenith distance 
 of the Pole. But before using zenith distances of stars for this 
 purpose, account has to be taken of an instrumental irregularity 
 which is named the Discordance of Zenith Points. It is found, 
 in fact, that the mean between the Circle-readings for the direct 
 and reflection observations of a star corrected for refraction, 
 which mean should be the reading for the horizontal direction 
 northward or southward, is somewhat different as determined 
 by different stars, and, consequently, that the zenith point, 
 
208 PRACTICAL ASTRONOMY. 
 
 which is the reading for the horizontal direction 90, varies 
 with the zenith distance. It becomes necessary, therefore, to 
 ascertain by what process the calculation of polar distances 
 may be performed so as to be free from effects of this irregu- 
 larity, that is, to obtain corrections for discordance of zenith 
 points. The following method, described in pages xxxi. and 
 xxxii. of Vol. xx. of the Cambridge Observations (for 1855 
 1860), is for the most part the same as that which was adopted 
 by Professor Airy for the observations of 1833, the year in 
 which the Circle was first brought into use. 
 
 217. The discordance is such that the Circle-reading for 
 zenith is in general less by a star observed to the south of 
 zenith than by a star observed to the north of zenith, the 
 telescope, whether directed to the heavens or the trough of 
 mercury, requiring, when the object is to the south of zenith, 
 to be turned for bisecting it a little farther in the direction of 
 the graduation than if the inequality did not exist, and when 
 the object is to the north of zenith a little in the contrary 
 direction. Whatever may be the cause of the discordance, the 
 law of the error it produces may be presumed to be approxi- 
 mately represented by the differences between an ascertained 
 zenith point for a given zenith distance and the zenith points 
 corresponding to other zenith distances, if these be sufficiently 
 consecutive and numerous. It will be assumed that the colli- 
 mating eye-piece gives correctly the zenith point corresponding 
 to the zenith direction, and that the double observation gives 
 zenith points that require corrections. Let, therefore, M be a 
 zenith point given by the eye-piece, and Z l a zenith point 
 obtained nearly contemporaneously by a double observation. 
 Since, if there were no discordance Z^ would be equal to M, it 
 follows from the foregoing principles that the Circle-reading for 
 the direct observation and that for the reflection-observation 
 both require to be corrected by M Z v it being assumed that 
 they are alike affected by the inequality both in amount and as 
 respects the direction of the graduation. In order, therefore, to 
 obtain the true zenith distance, or that which, except for the 
 discordance, would be given by adopting the zenith point M, 
 
THE MURAL CIRCLE. 209 
 
 the quantity M Z has to be added to the Circle-reading for 
 the direct observation, and subtracted from that for the reflection 
 observation. Hence the algebraic excess of the latter Circle- 
 reading above the other is twice M Z. This quantity is also 
 equal to the algebraic excess of the polar distance derived 
 from the reflection observation above that derived from the 
 direct observation, before taking account of the discordance of 
 zenith points, polar distances below Pole being considered 
 negative. In short, the values of M Z are most conveniently 
 obtained by halving the excesses thus deduced from polar 
 distances provisionally calculated with the values of Jf. 
 
 218. This being understood, suppose the values of MZ 
 derived from all the double observations made in a year to be 
 collected, the mean, in each instance of several being given by 
 the same star, to be taken, and the separate results, whether a 
 single value or the mean of several, to be divided into groups 
 such that the stars in each group shall not greatly differ in 
 zenith distance. Then let each value in a group be multiplied 
 by the number of observations by which it was determined, and 
 the corresponding zenith distance by the same number. After 
 dividing the sum of each series of products by the whole number 
 of observations in the group, the resulting value of MZ may 
 be considered to belong to the resulting zenith distance. Con- 
 ceive these zenith distances to be set off on a straight line of 
 abscissae, and ordinates proportional, according to an arbitrary 
 scale, to the values of MZ corresponding to the several 
 abscissae, to be drawn at right angles to the line. The extre- 
 mities of the ordinates will thus mark a certain number of 
 positions, the probable error of each of which may be supposed 
 to be smaller the greater the number of the observations by 
 which it was determined. This having been done, the next 
 step is to trace by hand among the points laid down a con- 
 tinuous curved line, making it at the same time approach nearer 
 to any position the less the probable error. Now let intervals of 
 5 be marked off on the line of abscissae, commencing at the 
 zero of zenith distance, and proceeding both in the positive and 
 negative directions, and let ordinates to the curve be raised at 
 c. 14 
 
210 PBACTICAL ASTRONOxMY. 
 
 the points of division. Then these ordinates, their scale-measure 
 being known, will give the values ofMZ for every fifth 
 degree of zenith distance, from which the value for any other 
 zenith distance may be found by interpolation. It appears 
 from the foregoing reasoning, that the required correction is 
 M Zwiih its proper sign, or with the sign changed according 
 as the zenith distance is inferred from a direct or a reflection 
 observation. A Table of the corrections for discordance of 
 zenith points, both for direct and reflection observations, for 
 every fifth degree of zenith distance, or polar distance, should 
 be formed to facilitate the interpolations. 
 
 219. We may now proceed to consider how the colatitude 
 of the Observatory may be exactly determined by observations 
 of circumpolar stars above and below Pole. The mean between 
 the zenith distances of the same star at the upper and lower 
 culminations, after correcting both for refraction and dis- 
 cordance of zenith points, is the exact zenith distance of the 
 Pole, or the Colatitude. As this is an essential element in 
 calculating polar distances, the adopted value should be the 
 mean result of a large number of observations ; and on account 
 of the uncertainties of refraction, stars should be selected that 
 are not far from the Pole. Polaris is especially adapted for 
 this purpose, both because of its nearness to the Pole, and 
 because it is visible in day-time in telescopes usually attached 
 to a Mural Circle, or a Transit-circle. In calculating the mean 
 result, less weights should be given to individual results in 
 proportion as the stars rendering them are more distant from 
 the Pole. I proceed now to exemplify the above method by 
 stating in some detail the processes by which the colatitude of 
 the Cambridge Observatory at present adopted was obtained. 
 
 220. The first accurate determination was made by Pro- 
 fessor Airy in 1833 from observations, taken in that year, of the 
 polar distances of ten circumpolar stars, extending from the 
 polar distance of Polaris (=1,35') to that of a Cassiopeiae 
 (= 34 . 23'). The data for the calculations are given in p. xxxv 
 of Volume VI. (for 1833) of the Cambridge Observations. The 
 whole number of observations employed was 917, consisting of 
 
THE MURAL CIRCLE. 211 
 
 observations above and below Pole in unequal portions, and 
 also of unequal portions of direct and reflection observations. 
 The weight attached to the result given by each star was 
 proportional to the product of the number of observations from 
 which it was deduced by an estimated factor varying with the 
 polar distance. The value of the colatitude finally obtained 
 was 37 . 47' . 8",43, inclusive of a correction + 0",15 required hi 
 consequence of its being subsequently found that the barometer 
 reading was O in -,l too small. By similar means I calculated 
 (taking into account the barometer-correction) the colatitude 
 from observations made in 1836, 1837, and 1838, consisting of 
 very nearly equal numbers of direct and reflection observations, 
 but of observations above and below Pole in unequal numbers. 
 The data and results of the calculations are contained in pages 
 liii iviii of Volume XL (for 1838). The weight given to the 
 value deduced from the observations of any star was calculated 
 in the following manner. If the number of observations of the 
 star above and below pole be respectively a and 6, the weight, 
 so far as it depends only on the number of the observations, was 
 
 taken to be proportional to 7, the reciprocal of- + r . It was 
 
 also considered that an observation below pole of a star of more 
 than 40 polar distance would be of no value for determining 
 the exact colatitude ; and as the correction for refraction is less 
 uncertain the less the zenith distance, it was assumed that the 
 
 factor by which 7 should be multiplied to take account of 
 
 the increment of the weight due to greater certainty in the 
 amount of correction for refraction, might be increased by 2 for 
 every 1 decrease of polar distance, beginning with zero at 40 
 of P. D. Accordingly, A being the star's P. D., the adopted 
 
 weight is equal to 2 (40 A) ^ . Suppose now the cor- 
 rected zenith distance of the star to be z 1 above pole and z 2 
 below pole, and X + x to be the true colatitude, X being put for 
 37 . 47' . 8",43. Then the polar distance is equal to X + x z t 
 and also to z z X x ; whence it follows that (\ zj + (X 2 2 ) 
 
 142 
 
212 PRACTICAL ASTRONOMY. 
 
 = 2ic, and that x is obtained by halving, and changing the 
 sign of, the sum of the calculated polar distances above and 
 below pole, the latter being negative. Every such determi- 
 nation was multiplied by a weight computed according to the 
 foregoing formula, and for each of the three years the sum of 
 the products was divided by the sum of the weights. In this 
 manner the value of x was found to be + 0",01 by 354 obser- 
 vations in 1836, 0",18 by 446 observations in 1837, and - 0",12 
 by 392 in 1838. By giving to these results weights proportional 
 to the sums of the weights of the respective years, the final 
 value of x was found to be 0",11, and the corrected colatitude 
 37. 47'. 8",32. This result depends on 1190 observations, and 
 that obtained in 1833 on 917. If the weights of the two results 
 be taken to be proportional to these numbers, the colatitude 
 concluded from their combination is 37. 47'. 8",37, which is the 
 value I have adopted for all the observations made in the years 
 1838 1860, and contained in Volumes xi. xx. 
 
 To shorten the calculation of Circle observations the colati- 
 tude printed in the Calculation Books is 37. 47'. 8",00, and the 
 correction + 0",37 for error of assumed colatitude is incorporated 
 in the Table of corrections for discordance of zenith points 
 mentioned in Art. 218. 
 
 221. I propose to conclude this Division III. by considera- 
 tions respecting the effect of the flexure of the telescope of the 
 Cambridge Circle, and the extent to which it might have given 
 rise to the discordance of zenith points. For conducting the 
 enquiry as to the amount of flexure, I employed the two col- 
 limators mentioned in Art. 172, the construction of which, and 
 the mode of using them for finding collimation error, are 
 described in Arts. 53 55. For the present purpose they were 
 mounted in such manner that the Circle-telescope could be 
 made to collimate with either, both horizontally and in directions 
 inclined at any angles to the horizontal direction. This was 
 effected by firmly attaching the collimators to a wooden plank 
 11,5 feet long, 12 inches wide, and one inch and a half thick, 
 with 6 feet of the length at the middle part of double thick- 
 ness, then fastening symmetrically the plank, thus loaded and 
 
THE MURAL CIRCLE. 213 
 
 strengthened, to the end of a stout axle 52 inches long, having 
 two cylindrical pivots, and finally placing this apparatus on a 
 braced wooden stage, so as to be capable of being moved about 
 an axis nearly in a line with the axis of motion of the Circle. 
 The collimators were fixed to the vertical face of the plank near 
 its ends, and pointed in opposite ways in the direction of its 
 length, and their axes could, by means of screws provided for 
 this purpose, be adjusted so as to revolve very nearly in the 
 plane of the motion of the telescope-axis. Thus for any angular 
 elevation the Circle-telescope could collimate with both, one 
 after the other, and their collimation with each other could be 
 effected through the aperture made at the middle part of the 
 telescope-tube (Art. 172). For steadying the collimators' 
 support while a set of collimations was being taken, the ends 
 of the two arms were clamped to the pier by means of brass rods 
 and appropiate clamping gear. The wooden stage supporting 
 the revolving part of the apparatus rested at first on the floor 
 of the Circle-room. I made trial of these arrangements for 
 measuring the effects of flexure on various occasions from 
 November 16, 1855, to February 13, 1856, and obtained values 
 for every tenth degree of zenith distance South from 20 to 90. 
 The results of this preliminary investigation shewed that the 
 floor was not a sufficiently firm support of the apparatus, and 
 that the brass clamping rods had an injurious effect, owing to 
 variations of temperature. To get rid of these sources of error, 
 the brass rods were replaced by wooden ones, and the stage was 
 placed on a mass of brick-work surmounted by a thick stone 
 slab. After these changes the measures of flexure were much 
 more consistent with each other, and seemed to be sufficiently 
 trustworthy. The instrumental operations for determining the 
 flexure, and the calculation of its amount, were conducted in 
 the following manner. 
 
 222. Ifc will be assumed that the plane of collimation of the 
 Circle-telescope has been placed very nearly in the plane of the 
 meridian, and that its wire-frame has been adjusted to geometrical 
 focus by the method described in Art. 83. Under these circumstances 
 the wires of the collimators are placed exactly at the geometrical 
 
214 PRACTICAL ASTRONOMY. 
 
 foci by adjusting the wire-frames, one after the other, so that the 
 images of the wires may be seen distinctly at the geometrical focus of 
 the Circle-telescope; for when this condition is satisfied the images 
 are formed by parallel rays issuing from the collimators' object-glasses. 
 Supposing these adjustments to be made in positions of the telescope 
 which are approximately opposite, or differ, according to the gradua- 
 tion of the Circle, by nearly 180, the collimators will by this means 
 be made to point towards each other, so that the images of the wires 
 of either will be seen distinctly at the geometrical focus of the other. 
 If, for instance, the centre of the field of each be marked by the in- 
 tersection of a vertical and a horizontal wire (see Fig. 13 in pp. 44 and 
 215, and Arts. 54 and 55), the two lines joining the points of intersec- 
 tion and the respective optical centres of the object-glasses will be very 
 nearly parallel, and the image of the point of intersection of the wires 
 of one collimatorwill be formed near the point of intersection of the wires 
 of the other. (I provided screw-adjustments for correcting, if there 
 was need, defect of parallelism of these optical axes, and also for 
 having the means of placing both in a line with the optical axis of 
 the telescope, which latter condition, although not optically necessary, 
 it was considered desirable to be able to fulfil approximately.) At 
 the same time that these adjustments are made, the horizontal wires 
 of the collimators might be placed, according to judgment of the eye, 
 parallel to the image of the horizontal wire of the telescope, by being 
 slightly rotated by hand and then clamped. If, however, all bisections 
 be made near the centres of the collimators' fields, small deviations of 
 the wires from horizontally will have no sensible effect. After the 
 foregoing arrangements the operation for measuring the flexure pro- 
 ceeds as follows. 
 
 223. The collimators were first made to collimate with each other 
 through the aperture in the telescope-tube. In order to make this 
 adjustment with precision the double micrometer- wire of B (Fig. 13) 
 was moved to a position such that the image of the horizontal wire of 
 A was seen, near the middle of the field, exactly between the two 
 wires, and the micrometer-reading of B was recorded. This was 
 done six or more times, and after calculating the mean of all the 
 readings the micrometer was set to this mean. The telescope having 
 been turned to collimate with B and the Circle clamped, the telescope- 
 micrometer-wire was placed six or more times between the images of 
 the two wires of B, the several micrometer-readings were recorded, 
 
THE MURAL CIRCLE. 215 
 
 and the mean being computed was taken to be the concluded 
 micrometer-reading (x^). The telescope was then put in collimation 
 with A by placing its micrometer-wire in coincidence with the image of 
 the horizontal wire of A, the coincidence reading (x 2 ) being ascer- 
 tained by the method of alternate contacts (Art. 191). The Circle- 
 reading having been taken in the usual manner for each position of the 
 
 Fig. 13. 
 
 C 
 
 B 
 
 telescope, suppose the reading to be C l for the first position, the 
 telescope pointing northward from Zenith, and G z for the second 
 position, the telescope pointing southward. Then if a be either 
 the micrometer-reading for coincidence with the fixed wire, or the 
 reference micrometer-reading, and m the value in arc of one micro- 
 meter revolution, the concluded Circle-reading for the first bisection 
 is C l m (x l - a\ and that for the other C a -m (x a - a). Now if 
 there should be no effect of flexure, the excess of the second reading 
 above the first would, by reason of the collimating process, be exactly 
 180. Also it is to be considered that the effect of flexure might be 
 such as to cause the optical centre of the object-glass and the reference 
 point in the field of the telescope to be equally displaced in parallel 
 directions, in which case the flexure would produce no angular 
 displacement of the optical axis, and the difference of the Circle- 
 readings would still be 180. But if the arc (G 2 mx. 2 ) (C l mx l ) 
 through which the telescope-axis is turned differs from 180, this will 
 shew that sensible effect is produced by the bending of the telescope- 
 tube, and it may readily be seen that according as that arc is less or 
 greater than 180, the drooping of the object-end of the tube is greater 
 or less than that of the eye-end. As the difference of the droopings 
 may be assumed to be the same in the two positions of the telescope, 
 
216 PRACTICAL ASTRONOMY. 
 
 the error of the Circle-reading due to flexure is in amount equal to 
 half the difference between 180 and the arc C 2 - C^- m(x 2 -x l ). 
 Also since, for a given pointing of the optical axis southward, the 
 greater drooping of the object-end would diminish the Circle-reading, 
 the correction of the error is a positive or negative quantity, according 
 as the object-end droops more or less than the eye-end, that is, by 
 what is argued above, according as 180 is greater or less than 
 C^ C 1 m(x z x i ). From mechanical considerations we may con- 
 clude that the correction is equal in amount with opposite sign 
 for a pointing equally distant from the zenith northward. Hence, 
 
 generally, flexure-correction = { 1 80 (C 2 C 1 ) + m(x 2 x t )} ac- 
 
 cording as the telescope points southward or northward from the 
 zenith. The following examples will serve to illustrate the process of 
 calculation. 
 
 224. On September 10, 1856, at 3J h , I took measures as follows 
 for ascertaining the flexure-correction of the Cambridge Mural Circle 
 for the horizontal pointing of the telescope southward. 
 
 Object-glass. Circle -reading. Micr, reading. 
 
 N. ...... 246 40 38,95 (,) ...... 10,451 (xj. 
 
 S ....... 426 40 39,67 (<7J ...... 10,486 (a,). 
 
 N ....... 246 40 38,81 (tfj ...... 10,563 (x 3 ). 
 
 The value of m, the micrometer revolution, being 20", 8 50, 
 ife-a; 1 ) = 2<)",85xO r ,035 = +0",73j and 0,-^= 180 + 0", 7 2. 
 
 Hence + {180 9 - (C % -CJ+m (as, - x,)} = (- 0",72 + 0",73) 
 
 = + 0",005 = flexure-correction by the first and second measures. 
 Also + i {180 - (<7 2 - C 3 ) + m (r,-r 3 )} = \ {180 - (180 + 0",86) 
 
 + 20"85 x - O r ,77} = ~ (- 0",86 - 1",63) = - 1",245, which is the flexure- 
 
 2t 
 
 correction by the second and third measures. The mean between the 
 two results is 0",62, which is the concluded correction as given by 
 these measures. It will afterwards appear, notwithstanding the 
 discrepancy between the separate results, that this mean is entitled to 
 considerable weight. The sign of the correction thus obtained proves 
 that the eye-end droops most. See Art. 223. 
 
THE MURAL CIRCLE. 217 
 
 225. As the above-mentioned discrepancy appeared to be due to 
 gradual change of the reading of the J5-micrometer for coincidence 
 with the wire of A, the measures were repeated on September 12, 
 and the coincidence reading was taken three times. The telescope 
 was pointed horizontally and the B collimator was northward. After 
 making trial of different methods of conducting the collimations it 
 appeared to be most satisfactory to put one of the wires of B in 
 collimation with the wire of A, by means either of alternate contacts 
 or of actual coincidences, and then to place the images of these two 
 wires, one after the other, mid- way between the telescope's fixed and 
 micrometer wires separated by an arbitrary small interval. Suppose 
 that for such collimation northward the Circle-reading and micrometer- 
 reading are C r and x^ and for the southward collimation C 2 and x. 2 , 
 and that the reference micrometer-reading is a. Then the reduced 
 
 Circle-reading for the northward collimation is C l - -= (x l - a), and 
 
 *W? 
 
 that for the other is C 2 -^ (x 2 a). Hence by the same argument 
 
 2t 
 
 as that in Art. 223, the flexure-correction is ^{ISO-(C Z -C.) 
 
 A 
 
 + (x 2 x )} according as the telescope is pointed southward or 
 2 
 
 northward, -= simply taking the place of m in the former expression. 
 
 A 
 
 The following data were obtained by such measures on September 12. 
 
 Pointing of Reading of Circle-reading. Reading of 
 
 telescope. B-micrometer. Circle-micrometer. 
 
 N 10,012 246 40 3,43 (0,) 10,559 (a?,). 
 
 S 10,018 426 40 4,18 (C 2 ) 10,582 (as,). 
 
 N 10,033 246 40 3,72 (C 3 ) 10,703 (x 3 ). 
 
 As the readings 10 r ',018 and 10 r ',033 were taken before and after 
 the south pointing of the telescope, the coincidence reading at the 
 time of pointing is assumed to be the mean between these, or 10 r ',025. 
 Compared with this the reading 10^,012 is too small by O r ',013, 
 and consequently, as the micrometer-head was downwards, the wire 
 of B was too high, and the Circle-reading too great, by this quantity. 
 It was found by taking the difference of the Circle-readings for 
 collimations of the fixed wire of the telescope with the double- wire of 
 B at positions separated by 24*", that l r ' = 58",07. Hence the 
 required correction of C l =- 0,013 x 58",07 =- 0",76, the corrected 
 
218 PRACTICAL ASTRONOMY. 
 
 seconds of C l = 2",67, and <7 2 - C 1 = 180 + 4", 18 - 2",67 = 180 + 1",51. 
 Also (aj a - ajj) = ^ x 20",85 x 0,023 = + 0",24. Hence the flexure- 
 
 correction = + o ( - 1",51 + 0' / ,24) = - 0",63. Similarly, as compared 
 
 with 10 r ',025 the reading 10 r ',033 is too great by (K,008, and the 
 correction of G a = + 0,008 x 58",07 = + 0",46. Hence the corrected 
 seconds of C a are 3",72 + 0",46 = 4", 18, and the flexure-correction 
 
 '-$^ 
 
 = 0",63. Hence the mean result from the two sets of measures is 
 -0",63, agreeing nearly with that of Sept. 10. If the calculation 
 had been made without reference to the change of reading of the 
 ^-micrometer, the result would have been - 0",26 by the first and 
 second measures, and - 0",86 by the second and third, the mean being 
 -0",56. This result shews that the effect of that change is nearly 
 eliminated by adopting the principle of alternate measures. 
 
 226. October 23, 1856, 3 b 5 h , the following measures were 
 taken for determining the flexure-correction when the telescope is 
 horizontal and looks northward. 
 
 Pointing of Reading of Circle-reading. Reading of 
 
 telescope. B -micrometer. Circle-micrometer. 
 
 S ....... 9*916 (a?/) ...... 66 43 57,"lO (C,) ...... 9^659 fo). 
 
 . N ....... 9,928 (<) ...... 246 43 55,57 (<7J ...... 9,585 (*,}, 
 
 S ....... 9,935 (x*) ...... 66 43 57,05 (C 3 ) ...... 9,634 (scj. 
 
 N ....... 9,938 (<) ...... 246 43 56,52 (<? 4 ) ...... 9,669 (a$ 
 
 Calculating, first, without regard to the change of the micrometer- 
 reading of B, the flexure-correction by the first and second measures 
 
 -^,- 0.) + (, -.) = - (- 1",53 + 0",77) = + 0",38; 
 by the second and third measures = - - -j 180 - (G 3 - C 2 ) + ~ (x 3 - x a ) i 
 
 = o (- 1/ '? 4:8 + 0",51) = + 0",48 ; and by the third and fourth measures 
 a 
 
 ,, ,. 
 
 The mean of the first and second results is + 0",43, and that of the 
 second and third is + 0",47, and the concluded mean is .+ 0",45. 
 
 For taking account of the change of the micrometer-reading of B, 
 
^>> 
 
 > 
 
 THE MURAL CIRCLE. 219 
 
 94 
 
 the times of taking the Circle readings and those of both micrometers 
 were noted. Hence it was found by interpolating that the reading 
 of ^-micrometer was 9 r ',920 corresponding to (7,, and 9 r ',936 cor- 
 responding to (7 3 . Hence x 2 ' is too great by 9 r '928 - 9 r ',920, which 
 in arc = 58",07 x 0,008 = + 0",46. Consequently, since the ^-micro- 
 meter was northward, the wire was too low by 0",46 and the cor- 
 rected seconds of <7 2 = o5",57 + 0",46 = 56",03. Thus by the formula 
 
 - ||l 80- (C l ~ <? 2 ) + ? fa - x 2 )\ , the flexure-correction = - ^ (- 1",07 
 
 ** V ) 
 
 + 0",77) - + 0",15. So in comparing the second and third measures, 
 xi is too small by 9 r ',936 - 9^,928, or in arc 0,008 x 58",07 ( = 0",46), 
 and the corrected seconds of (7 2 = 55 // ,57 -0",46 = 55", 11. Hence 
 
 flexure-correction = - ^|l80 - (G 3 - C t ) + y fa - x 2 )\ = - ^ ( ~ 1", 
 
 + 0",51) = + 0",71, and the mean of the two results = -= (0",15 + 0",71) 
 
 = + 0",43. Again, in comparing the third and fourth measures sr 4 ' is too 
 greatly 9 r ',938 - 9 r ',936, or in arc 0,002 x 58",07(= 0",12). Hence cor- 
 rected seconds of (7 4 = 56",52 + 0",12 = 56",64, and flexure-correction 
 
 = _ Ml80 - (a, - ^ 4 ) + f fa - aO } = - \ (~ " 5 41 ~ "; 36 ) = + O" 39 - 
 Hence the mean of the two results by the second and third measures 
 and by the third and fourth = ^ (0",71 +0",39) = + 0",55. Con- 
 sequently concluded mean = - (0",43 + 0",55) - + 0",49, which differs 
 
 little from + 0",45 obtained by the other calculation. 
 
 227. I propose now to give two examples of the calculation of 
 the flexure-correction for oblique positions of the telescope, one for a 
 southward, and the other for a northward, zenith distance. In the 
 first example it was required to find the flexure-correction when the 
 telescope is pointed 70 from the zenith southward ; for calculating 
 which, on September 13, 1856, at 3 h , the following measures were 
 taken : 
 
 Pointing of Reading of Circle-reading. Reading of 
 
 telescope, ^-micrometer. Circle-micrometer. 
 
 N 10^88 fa') 226 39 49^93 (C T ) 10,684 fa). 
 
 S 10486 fa') 46 39 50,22 (C7.) ...... 10,648 fa). 
 
 N 10,211 fa') 226 39 49,50 (<7J 10,624 fa). 
 
220 PRACTICAL ASTRONOMY. 
 
 The readings of ^-micrometer being left out of consideration, we 
 have (7,-^= 180+ 0",29, fa-a^i x 20",85 x - 0,036 - - 0",37, 
 
 and flexure-correction = ( - " 29 ~ "> 37 ) = - "> 33 . Also <7 2 - C7 8 
 
 777 
 
 = 180 + 0",72, y fa - 3 ) = ^ x 20",85 x 0,024 = + 0",25, and flexure- 
 correction = + - ( - 0",72 + 0",25) = - 0",23. Concluded value = 
 ~ ( - 0",33 - 0",23) = - 0",28. Taking account, now, of the readings 
 
 A 
 
 aij', o; 2 ', OJ 3 ', since the Circle reading for the south pointing of the 
 telescope was obtained between x 2 ' and x a ' t the corresponding reading 
 of the ^-micrometer will be assumed to be the mean of these, or 
 10 r ',198. The value of a/ being less than this by O r ',010, and the 
 collimator B being northward and its micrometer-head downwards, 
 the wire was too high by 0,010 x 58",07, or 0",58, and the Circle 
 reading C l requires the correction 0",58. Similarly, C 3 requires 
 the correction +0,013 x 58",07 or + 0",75. These corrections being 
 included, it will be found by calculating as before, that the flexure- 
 correction is 0",33 0",29, or 0",62 by the first and second 
 measures, and -0",23 + 0",37, or +0",14 by the second and third. 
 
 The mean of the two results is ^ (+ 0",14 - 0",62), or - 0",24, agreeing 
 
 well with the value obtained above. I am not sure that the former 
 method of calculation is not the best. 
 
 In the second example, the following measures were taken on 
 October 21, 1856, at 22 h 23J h , for calculating the flexure-correction 
 for the pointing of the telescope 70 from the zenith northward. 
 
 Pointing of Reading of Circle-reading. Reading of 
 
 telescope. .B-micrometer. Circle-micrometer. 
 
 S 9J65 ) 86 42 55*43 (C,) 9^659 fa). 
 
 N 9,760 (<) 266 42 54,95 (C 2 ) 9,701 fa). 
 
 S 9,760 (<) 86 42 55,90 (<7 3 ) 9,724 fa). 
 
 N 9,758 ) 266 42 55,35 ((7J 9,677 fa). 
 
 Calculating without reference to the readings of the J5-micrometer, 
 the following values of the flexure-correction are obtained from the 
 first and second measures, the second and third, and the third and 
 fourth, respectively : 
 
THE MURAL CIRCLE. 
 
 221 
 
 -l{l80-(C 1 -(7 2 ) + |(a ;i -a ;2 )}^-i(-0",48-0",44) = + 0", 
 - I {l80 - (C 3 - C 2 ) + 1 (x 3 - * f )} = - i (- 0",95 + 0",24) = + 0", 
 
 46 
 
 36 
 
 _ * J 180-((7 3 - Q + ^ (aJ a -a? 4 )U - ~ (-0 /r ,55+ 0",49) = + 0",03. 
 
 The mean of the first and second results is + 0",410, and the 
 mean of the second and third + 0",195, and the concluded mean 
 = + 0",30, which is the required flexure-correction. By calculating 
 according to the other method I obtained + 0",31, differing very little 
 from the previous result, because, in fact, the jS-micrometer readings 
 differ little from each other. I have not thought it worth while to 
 give the details of the second calculation, which was conducted exactly 
 as in like cases already adduced. 
 
 228. In the interval from September 10 to October 24, 1856, I 
 took thirty -three different sets of measures to determine the flexure- 
 correction for zenith distances separated in some instances by 5, but 
 generally by 10, and extending from 90 south to 90 north. The 
 subjoined Table contains the results for every tenth degree of zenith 
 distance, as deduced from these measures. It is to be understood 
 that in all the calculations micrometer-readings of the collimator B 
 were taken into account. The corrections for discordance of zenith 
 points (Arts. 217 and 218) adopted in the years 1837 and 1856 are 
 inserted in the Table for the purpose of drawing inferences from a 
 comparison of the law and amount of the two classes of corrections. 
 
 Z. D. 
 
 South. 
 
 Flexure- 
 corr n . 
 
 Discordance-corr". 
 
 Z. D. 
 
 North. 
 
 Flexiire- 
 corr n . 
 
 Discordance-corr n . 
 
 1837 
 
 1856 
 
 1837 
 
 1856 
 
 
 
 ii 
 
 // 
 
 // 
 
 
 
 n 
 
 // 
 
 it 
 
 90 
 
 -0,50 
 
 
 
 
 
 -0,08 
 
 0,00 
 
 + 0,02 
 
 80 
 
 -0,58 
 
 
 
 10 
 
 + 0,13 
 
 -0,46 
 
 -0,57 
 
 70 
 
 -0,33 
 
 + 0,44 
 
 + 1,25 
 
 20 
 
 + 0,27 
 
 -<y>6 
 
 -0,83 
 
 60 
 
 + 0,15 
 
 + 0,30 
 
 + 0,88 
 
 30 
 
 + 0,29 
 
 -0,51 
 
 -0,51 
 
 50 
 
 + 0,24 
 
 -0,08 
 
 + 0,53 
 
 40 
 
 + 0,11 
 
 -0,32 
 
 + 0,02 
 
 40 
 
 + 0,32 
 
 + 0,23 
 
 + 0,51 
 
 50 
 
 + 0,16 
 
 + 0,15 
 
 + 0,35 
 
 30 
 
 + 0,18 
 
 + 0,55 
 
 + 0,84 
 
 60 
 
 + 0,43 
 
 -0,14 
 
 
 20 
 
 + 0,34 
 
 + 0,55 
 
 + 1,16 
 
 70 
 
 + 0,31 
 
 -0,34 
 
 
 10 
 
 -0,14 
 
 + 0,37 
 
 + 0,71 
 
 80 
 
 + 0,38 
 
 
 
 
 
 -0,08 
 
 0,00 
 
 + 0,02 
 
 90 
 
 + 0,49 
 
 
 
222 PRACTICAL ASTRONOMY. 
 
 The measures from which the four values of the flexure-correction 
 for the zenith distances 20 50 S. were calculated, were taken with 
 all the shutters of the room closed, and lamps were suspended at the 
 ends of the arms supporting the collimators for the purpose of 
 throwing light on the reflectors of diagonal eye-pieces whereby the 
 wires were made to appear dark on bright fields for collimation with 
 the telescope. It was found, however, that the heat from the lamps 
 had a perceptible effect on the reading of B for collimation with A , 
 producing apparently anomalous values of the correction in the above- 
 mentioned instances; on which account the use of the lamps was 
 discontinued. The other measures were taken with the shutters 
 open, the illumination of the collimators' fields being effected by light 
 from the sky. That they might be the better comparable with each 
 other all the measures were taken by myself, often, however, under 
 considerable difficulties arising from effects of wind and changes of 
 temperature. The results for the north zenith distances, as depending 
 on measures taken after I had had experience in overcoming the 
 difficulties incident to the experiment, are more trustworthy than 
 those for the south zenith distances, and, I believe, may be considered 
 to be a pretty close approximation to the law and amount of the 
 flexure. 
 
 The corrections for discordance of zenith points in 1837 were 
 interpolated from the values for the direct observations given in the 
 Table in page Ixi of the Volume of Cambridge Observations for that 
 year. The adopted corrections of 1856 were formed by first taking 
 the means of the values obtained for 1852, 1853 and 1854, the forms 
 of the discordance-curves for the three years being very nearly the 
 same, and then correcting these means by - 0",32. This correction 
 was applied because the discordance-correction for Z.D. deduced from 
 the three years' observations was 4- // ,32, whereas by the results of 
 numerous observations made in 1856 near the zenith, as given in 
 page Ivi of Yol. xx., the correction was found to be zero. Discord- 
 ance-corrections for zenith distances greater than 70, as depending 
 on few observations, were considered too uncertain for the present 
 purpose, and in the years 1852 1856 none were obtained for north 
 zenith distances greater, than 42. I proceed now to state the 
 inferences which appear to be deducible from the foregoing dis- 
 cussion. 
 
 229. It may, in the first place, be concluded that the 
 
THE MURAL CIRCLE. 223 
 
 discordance of zenith points cannot be accounted for by flexure 
 of the telescope-tube, inasmuch as the flexure-correction agrees 
 neither in law nor amount with the discordance-correction. 
 Hence, since from mechanical considerations the flexure must 
 under any circumstances produce its effect, there must be some 
 cause which, by operating according to a different law and in 
 greater degree, veils this effect. From the consideration I have 
 given to the question I incline to the opinion that the sudden 
 and somewhat violent movement of the instrument from the 
 position for the reflection observation to that for the direct 
 observation gives rise to a temporary deformation of the Circle, 
 of the nature of flexure, from which it does not recover in the 
 short interval between the observations, especially if it be 
 clamped for the direct observation. This view receives support 
 from the following facts. (1) The change is most rapid at small 
 zenith distances, where the difference between the pointings 
 of the telescope for the reflection and direct observations is 
 greatest. (2) The discordance is greater for the Cambridge 
 8-feet Mural Circle than for the Greenwich 6-feet Transit-circle, 
 and follows different laws in the two instruments. (This will 
 be seen by comparing the values for 1856 in the above Table 
 with corresponding values in page Ivii of the Volume of 
 Greenwich Observations for that year.) These facts seem to 
 indicate that the discordance is due as to law and amount to 
 the construction of the instrument, or to mode of using it, or to 
 these conditions combined. (3) As far as regards the Cambridge 
 Circle, the discordance increased in amount from 1837 to 1856, 
 the law remaining nearly the same ; which might be the result 
 of an increasing liability of the Circle to deformation consequent 
 upon prolonged use. (4) The discordance-curves from which 
 the corrections in the above Table for the years 1837 and 1856 
 were deduced, as well as all the curves that I formed from the 
 observations of other years, exhibited a minimum value at about 
 50 Z. D. south, and a maximum value at the same Z. D. north. 
 Possibly at these positions the effect of the flexure due to the 
 action of gravity, and the temporary effect due to the movement 
 impressed on the instrument, might most nearly counterbalance 
 
224 PRACTICAL ASTRONOMY. 
 
 each other. From the aggregate of the values of the flexure- 
 correction in the above Table, with the exception of those for 
 20 60 south, all of which appear to have been affected by 
 disturbing causes (see Art. 228), it may be inferred that the 
 law and amount of the flexure are very nearly the same on the 
 opposite sides of zenith (as from mechanical considerations might 
 be expected), and that they may be supposed to be approxi- 
 mately expressed by the formula, horizontal flexure x sin Z. D. 
 This formula is, in fact, assumed in the reduction of the 
 Greenwich observations, the zenith points obtained after the 
 application of the flexure-correction being supposed to depend 
 on some unknown cause different from flexure. It is plain that 
 according to the supposed law of flexure, the correction is the 
 same in amount, and in direction as regards the order of the 
 graduation, for a reflection as for a direct observation, and 
 consequently that its effect on the zenith point is taken into 
 account in the usual treatment of the double observation, when, 
 as in Cambridge practice, it has not been previously eliminated. 
 If it had been eliminated from the results of the observations 
 of 1856, the residual discordance would have been much greater 
 for the Cambridge instrument than any hitherto obtained at 
 Greenwich, proving, I think, that the 8-feet Mural Circle was too 
 large, and that a decided advantage has been gained by substi- 
 tuting for it a 3-feet Transit-circle. If the foregoing theory be 
 accepted, it will follow that in the case of a double observation, 
 the Z. D. deduced from the reflection observation, corrected 
 only for flexure, is likely to be more accurate than that 
 similarly deduced from the direct observation, and that the 
 uncertainty of the direct observation increases with the size 
 of the Circle 1 . 
 
 1 As far as I am aware the experiment of which I have here given an 
 account is the only one in which a determination of the flexure-corrections for 
 oblique positions of the Telescope has been attempted to he made. I intended 
 originally to offer a communication of all the details for publication in the 
 Memoirs of the Royal Astronomical Society, instead of inserting them in a 
 Volume of the Cambridge Observations, but press of other occupation prevented 
 my carrying out this intention. The details and explanations, as far as they 
 are given in the present work, may suffice to make known the character of 
 
THE MURAL CIRCLE: 225 
 
 230. No account lias been taken in the preceding explana- 
 tions of any effect on the Circle-reading that may be due to 
 errors of graduation, which at Greenwich are carefully deter- 
 mined, and allowed for in the reduction of the observations. 
 Respecting this source of error as affecting observations with the 
 Cambridge Circle, I am only able to say, that since the correc- 
 tions of the assumed colatitude obtained in the years 1836, 1837, 
 and 1838 by observations made with the telescope in three differ- 
 ent positions on the Circle were + 0",01, - 0",18, and - 0",12 
 respectively (see Art. 220), the close agreement of these results 
 may be regarded as evidence that the Circle-readings were very 
 little, if at all sensibly, affected by errors of graduation (see Art. 
 158), considering that a difference of latitude equal to 0",1 cor- 
 responds to only 10 feet on the earth's surface. 
 
 IV. CALCULATION OF THE MEAN NORTH POLAR DISTANCES 
 OF STARS, AND OF THE GEOCENTRIC NORTH POLAR 
 DISTANCES OF THE SUN, MOON, AND PLANETS. 
 
 231. After correcting the observed zenith distance of any 
 object for refraction, and adding an assumed value of the colati- 
 tude (Art. 220), the result is its apparent North Polar Distance 
 subject to farther correction for the error of the assumed colati- 
 tude and the discordance of zenith points. As these two correc- 
 tions are always of very small amount, instead of taking account 
 of them for each observation, it is preferable, when the object is 
 a Star, to determine the star's mean polar distance for the be- 
 ginning of the year by applying to the uncorrected apparent 
 polar distance the value of 4- (&' 8) calculated by the formulae 
 given in Arts. 146 and 147, and afterwards to obtain the mean 
 
 the experiment, and so much of its results as may possess scientific value. 
 Nothing relating to them has been previously published, except a brief abstract 
 of results in a Letter inserted in Vol. v., No. 100, page 28, of The Astronomical 
 Journal, and addressed to Dr Gould the Editor. In the same letter I took 
 occasion to state that the experiment gave the means of determining the law 
 of flexure of a loaded beam for different angular elevations, and that this law 
 was shewn to be approximately expressed by the formula A sin Z, but more 
 exactly by the formula A sin Z + B sin 2 Z t Z being the angular distance from 
 the vertical. 
 
 c, 15 
 
226 PRACTICAL ASTRONOMY. 
 
 of all such determinations made in the course of the year. It 
 will then suffice to apply to this mean a correction for error of 
 assumed colatitude and discordance of zenith points interpolated 
 from a Table including both kinds of correction, with polar 
 distance for argument, the discordance-correction having been 
 deduced from all the double observations of the year. (See 
 Arts. 217 and 218.) The values of mean polar distance thus 
 derived from the observations of different years can subsequently 
 be referred to January 1 of a given year, by applying annual 
 variations calculated by the formula, n cos a, in Art. 145, and 
 thus a concluded mean N. P. D. be obtained depending on a 
 large number of observations. This process, combined with 
 that for calculating concluded mean K. A, described in Art. 154, 
 gives the means of forming a Catalogue of the mean places of 
 stars for a given epoch. 
 
 232. In observations of the Sun, Moon, and Planets, the 
 corrections for discordance of zenith points have to be applied to 
 the polar 'distances severally, and may either be deduced, in 
 conjunction with the correction for error of assumed colatitude, 
 from the above-mentioned Table, or be added to that correction 
 after being taken by measurement with compasses and scale 
 from the discordance-curve (Art. 218). By these means the 
 local polar distance of a bisected north or south limb of any of 
 these bodies is accurately determined, and to obtain the geocen- 
 tric polar distance, it is only required to subtract therefrom the 
 parallax, which is the angle subtended at the observed limb by 
 the radius from the earth's centre to the place of observation. 
 Let p be this angle, P the equatorial horizontal parallax of the 
 body's centre, P 4- a that of the limb, Z the zenith distance by 
 the observation, p the local terrestrial radius, and e the local 
 angle of the vertex ; then (Art. 124), 
 
 sinp =p sin (P-f a) sin (Ze). 
 
 In using this formula for the Moon, the value of P, as obtained 
 by interpolation to second differences from the Nautical Almanac, 
 may at present be adopted without correction ; but it would not 
 be sufficiently exact to substitute p and P for their sines, on 
 
THE MURAL CIRCLE. 227 
 
 which account it is convenient to construct a Table from which, 
 within the limits of the values of P, the arc may be readily 
 inferred from the sine, and the sine from the arc. (Such a 
 Table is given in the Greenwich Observations of 1874, p. cxxix.) 
 The small quantity a, required to be added to Pto obtain exactly 
 the parallax of the Limb, may be derived by interpolation from 
 the subjoined values, which have been extracted from a, general 
 Table contained in the Cambridge Observations, Vol. IV. for 
 1831, p. 147, having been constructed by Professor Airy to be 
 used in the calculation of Occultations by the Moon for obtaining 
 the parallax of any point of her periphery : 
 
 Zenith Distance... 30 40 50 60 70 80 
 
 Cor n .forN.L... -0",03 -0",05 -0",06 -0",OS -0",09 -0",09 
 Cor n .forS.L... +0",10 + 0",12 +0",13 +0",15 +0",16 + 0",16 
 
 For all the bodies except the Moon the formula p = P sin (Z-r e) 
 is sufficiently approximate. The Equatorial Horizontal Paral- 
 laxes may either be adopted from the Nautical Almanac, or 
 computed from that of the Sun and the Log. of true distance 
 from the Earth given in Ephemerides. (For the Cambridge 
 Observatory p = [9,9990916] and e = 11'. 12".} 
 
 233. Parallax having been taken into account in the manner 
 above stated, the geocentric polar distance of the centre of the 
 body, being the mean between the polar distances of the limbs, 
 may be at once inferred when the north and south limbs have 
 been both observed, and corrections have been applied for defect 
 of illumination. The difference between these polar distances 
 is a measure of the geocentric diameter. If only one limb be 
 bisected, the polar distance of centre is obtained by adding the 
 value of the geocentric semidiameter to the polar distance of 
 the north limb, and subtracting it from that of the south limb. 
 This value, if not adopted from the Nautical Almanac, might be 
 computed either from a sufficient number of Circle measures of 
 diameter, such as that just mentioned, or from the mean result 
 of measures taken expressly for the purpose with an Equatorial 
 instrument, it being supposed that, the Log. distance of the body 
 from the Earth, which is required for such computation, is 
 
 152 
 
228 PRACTICAL ASTRONOMY. 
 
 supplied by the theory of gravity. For instance, from 68 
 measures of the Moon's diameter obtained in the years 183G 
 1842, partly by differences of transits of limbs taken with the 
 transit-instrument and five-feet Equatorial, and partly by 
 Circle measures of diameter, I found that the semidiameter 
 adopted at that time in the Naut. Aim. required the correction 
 + 2",21. (Cambridge Observations, Vol. xiv. for 1842, p. xxxviii ) 
 It will be proper to mention here that the corrections of 
 Mr Baldrey's transits of limbs spoken of in Art. 135, were de- 
 duced by taking as standards my own transits of limbs, which, 
 in the case of the Sun, gave values of diameters agreeing very 
 nearly with those of the Naut. Aim., while the latter, according 
 to the results of Mr Glaisher's Circle-measures in the 5 years 
 1836 1840, required a correction not larger than 0",16. (See 
 Vol. XIII. pp. xxii and xxiii.) 
 
 234. In the reduction of meridional observations of the 
 Moon I have taken account of the above correction, + 2",21, of 
 semidiameter; but for reduction to centre, in the cases of obser- 
 vations of a single Limb of the Sun, I adopted without change 
 the semidiameters, and sidereal intervals occupied by transits of 
 semidiameters, given in the Nautical Almanac. At the same 
 time yearly comparisons of numerous diameters derived from 
 observation with tabular values, have furnished data for ascer- 
 taining what amount of correction the assumed value of the 
 Sun's semidiameter at the earth's mean distance may require. 
 The tabular semidiameters (as also the parallaxes) in the Naut. 
 Aim., are at present, in consequence of corrections applied from 
 time to time for reasons drawn both from theory and observa- 
 tion, more likely to be trustworthy than at the period referred 
 to in Art. 233. In the Greenwich calculations of the Apparent 
 R. A., and Geocentric N. P. D., of centre from observations 
 with the Transit Circle made in 1874, the adopted parallaxes 
 (those of the Moon corrected as stated in Art. 232) and the 
 semidiameters are all interpolated from the Naut. Aim., with 
 the following exceptions pertaining to cases in which only one 
 Limb was observed. For these the tabular semidiameter of the 
 Sun is corrected by 0",53, to agree with the value derived 
 
THE MURAL CIRCLE. 22.9 
 
 from the observations of several previous years ; those of Mars, 
 Jupiter, and Saturn are corrected from the results of neighbour- 
 ing observations ; and to the semidiameter of Venus a special 
 correction is applied, the amount of which, as given in the 
 Greenwich Observations of 1874, page Ixxii, is + 0",392 + 0,027 
 x tabular semidiameter. This expression was deduced by 
 Mr Stone from 370 vertical measures of diameter taken with 
 the Greenwich Transit Circle from 1851 to 1862. (See 
 Monthly Notices of the R. Ast. Society, Vol. xxv., No. 3, p. 57.) 
 The reasons for applying such a correction may be gathered 
 from the following investigation, which, however, is founded on 
 observations of Venus made with the Cambridge Mural Circle. 
 
 235. It being understood that each vertical measure of the 
 Planet's diameter is compared with the diameter computed in the 
 Naut. Aim., let e be the excess, in any particular instance, of the 
 tabular above the observed value. Then if the latter should be 
 affected by a constant error such that the observed always exceeds 
 the true value by the constant JE, the excess of the tabular above the 
 true diameter will be e + E. Again, suppose the tabular diameter to 
 be A, and the true diameter, by reason of error in the value assumed 
 for the unit of distance, to be A (1 -e). Then cA, being the excess 
 of the tabular diameter above the true, is equal to e + E ; or, 
 e eA + E = 0. Hence if values of e and A, extracted from a Table 
 of the above-mentioned comparisons, be substituted in this equation, 
 as many different equations will be formed as there are observations 
 for comparison, and if the number be large, by dividing them into two 
 groups, values of e and E may be deduced, the probable errors of 
 which may be presumed to be small. For example, I formed 78 such 
 equations from the Circle measures of diameter in 1838, and as the 
 Planet passed through conjunction in the first half of the year and 
 through opposition in the other, the equations pertaining to the two 
 halves were put in separate groups, the mean result from which gave, 
 = -0,0267 and .# = + 0",94. From 74 equations derived from 
 measures in 1839, and similarly treated, it was found that - 0,0375 
 and J = + 0",40. Hence the mean results of the 152 measures are, 
 = -0,0321, and JZ= + Q",G7. As the tabular diameter for unit of 
 distance was 8",25, the true diameter according to this investiga- 
 tion is 8",25(1+ 0,032) = 8",51. Also the half of E, 0",34, is the 
 
230 PRACTICAL ASTRONOMY. 
 
 apparent enlargement of the semi diameter. This effect I supposed 
 to be due to irradiation 1 . (See Cambridge Observations, Vol. xi. for 
 1838, p. li, and Vol. xn. for 1839, p. xx.) 
 
 236. It only remains to make mention of a correction of 
 the observed N. P. D. which may be required on account of 
 error of position of the plane of collimation. It has already 
 been stated in Art. 174 that transits of known stars are usually 
 taken from time to time with the Circle-telescope in order to 
 determine the deviation (in time) of its pointing from the meri- 
 dian for any polar distance. A correction for reduction to the 
 meridian is in strictness required on account of such deviation ; 
 but after adjusting the Circle in the manner described in Arts. 
 167 172, the amount can be sensible only in observations of 
 the Moon, and in case the change of her N. P. D. is rapid. (In- 
 stances of data from observations of stars for calculating this 
 correction are given in the Cambridge Observations, Vol. xvm. 
 p. xlix, and Vol. xix. p. xxx.) 
 
 237. Having now gone through the treatment of Circle 
 observations under the four Divisions, I propose to add exam- 
 ples of the complete reduction of observations taken with the 
 Cambridge Mural Circle, both when the object is a star, and 
 when it is a moving body. The subjoined instances will serve 
 to exemplify the application of most of the various calculations 
 that have been under consideration. 
 
 1856, April 23, 10 h , 8 Ursse Majoris was observed by reflection (7?) 
 at the mid-wire, and directly (D) at three intervals from the mid-wire. 
 The micrometer-reading both for R and D was 8 r ,176, and the 
 reference reading 9 r ,000. One micrometer revolution 20",869. 
 Barometer, 29 in> ,880 ; External Thermometer, 44, 3 ; Approximate 
 
 1 The investigation by Mr Stone cited in Art. 234 is similar to the above. 
 From 219 observations made with the Greenwich Mural Circle in 1839 1850, 
 and 370 made with the Transit Circle in 18511862, he obtains for the semi- 
 diameter at the unit of distance, 8", 47. He also finds that the apparent 
 enlargement of semidiameter is 0",729 by tbe former instrument, and 0",392 by 
 the other, and remarks that as these quantities are nearly in the inverse ratio 
 of the diameters (4 inches and 8 inches) of the object-glasses, they might be 
 attributable to irradiation. 
 
THE MURAL CIRCLE. 
 
 231 
 
 K. A. of >fc, 12 h . 8 m ; Approximate Decl., + 57. 50'. For the direct 
 observation " Negative Correction for Runs" was noted. 
 
 Pointer Reading 
 
 8 Ursse M^j. (R) 
 
 ' It 
 
 ... 162 15 
 
 8 Ursaj Maj. (D) 
 
 ' It 
 
 331 
 
 Microscope A 
 
 3 63,1 
 
 4 52,6 
 
 B. 
 
 63 6 
 
 48,8 
 
 ti C. 
 
 641 
 
 50,8 
 
 D... 
 
 59 2 
 
 43,4 
 
 E.... 
 
 60 6 
 
 49,3 
 
 F 
 
 63 7 
 
 50,0 
 
 Correction for Run of + 0",9 
 
 -4-0 7 
 
 (N. C. ) 00 
 
 
 
 
 Mean of Microscope Readings 
 
 ... 162 19 2 50 
 
 331 4 49,15 
 
 Micrometer correction 
 
 ... +17,20 
 
 + 17,20 
 
 Reduction to Meridian 
 
 000 
 
 +2,14 
 
 
 
 
 Concluded Circle Heading ... 
 
 ... 162 19 19,70 
 
 331 5 8,49 
 
 Adopted Zenith Point 
 
 ... 336 42 12,73 
 
 336 42 12,73 
 
 
 
 
 Apparent Zen. Dist. South ... 
 
 - 5 37 6 97 
 
 -5 37 4,24 
 
 Refraction-correction 
 
 -5 80 
 
 -5,80 
 
 Assumed Colatitude 
 
 ... 37 47 8,00 
 
 37 47 8,00 
 
 
 
 
 Apparent North Polar Distance 
 
 ... 32 9 55,23 
 
 32 9 57,96 
 
 Reduction to Mean N. P. D. 
 
 4-2 31 
 
 + 2 31 
 
 
 
 
 Mean N. P. D., 1856,0 
 
 ... 32 9 57,54 
 
 32 10 0,27 
 
 Cor n . for discordance of Z. P 
 
 -f-035 
 
 -0,35 
 
 Cor tt . of assumed Colatitude 
 
 + 037 
 
 + 0,37 
 
 
 
 
 Concluded Mean N. P. D., 1856,0 .. 
 
 ... 32 9 58,26 
 
 32 10 0,29 
 
 The micrometer correction - (9,000 - 8,176) x 20",869. The cor- 
 
 4 
 rection for Runs applied in the reflection observation = -= x + 0", 9; 
 
 for the other observation the correction is 0",0 because "Negative 
 correction" was recorded (see Art. 179). The reduction to meridian 
 for three intervals in the direct observation = 0",1503 x 3 2 xtan57.50'; 
 in the other the reduction is zero because the mid-wire was nearly in 
 the meridian (see Art. 198). The observer noted that the reflected 
 image of the star was " much diffused," and that the reflection 
 observation was "worth little." I find, in fact, by comparison with 
 other observations of the star, both at Greenwich and Cambridge, that 
 this observation is discordant. 
 
 1856, October 2, both Limbs of the Sun were observed, N. L. at 
 two intervals before, and S. L. at two intervals after, mid- wire. One 
 
232 
 
 PRACTICAL ASTRONOMY. 
 
 equatorial interval =16 8 ,6. The micrometer reading = 16 r , 48 2 for 
 each Limb; reference reading = 9 r ,000. One micrometer revolution 
 = 20",869. Approximate declination of the Sun's centre, - 3. 45'. 
 Horary variation of declination =-58", 12. Barometer, 29 iIX ,766 ; 
 External Thermometer, 61, 3. 
 
 Pointer Beading 
 
 Sun N. L. 
 o i 
 32 25 
 33,4 
 37,0 
 33,6 
 27,4 
 30,0 
 33,2 
 40,2 
 
 Sun S. L. 
 
 f If 
 
 32 55 
 2 31,3 
 37,0 
 30,4 
 24,4 
 27,3 
 29,8 
 + 1,1 
 
 Microscope A 
 
 B. 
 
 0. 
 
 D. .. 
 
 E. 
 
 P. 
 
 Correction for Run of + 2",3 . . . 
 
 Mean of Microscope readings ... 
 Micrometer correction 
 
 32 25 32,47 
 -2 36,14 
 -0,04 
 + 0,54 
 
 32 57 30,22 
 -2 36,14 
 -0,04 
 -0,54 
 
 Cor", for curvature of path 
 
 Cor n . for change of declination... 
 Concluded Circle Reading 
 
 32 22 56,83 
 336 42 15,68 
 
 32 54 53,50 
 336 42 15,68 
 
 Adopted Zenith Point .... 
 
 Apparent Zen. Dist. south 
 
 + 55 40 41,15 
 + 1 22,88 
 37 47 8,00 
 
 + 56 12 37,82 
 + 1 24,54 
 37 47 8,00 
 
 Refraction- correction 
 
 Assumed Colatitude .. 
 
 Apparent N. P. D. of N. L 
 S.L 
 
 Apparent N. P. D. of centre 
 
 93 29 12,03 
 94 1 10,36 
 
 App*. Diam r . 31 58,33 
 Parallax cor n . - 0,04 
 
 93 45 11,20 
 -7,08 
 + 0,68 
 + 0,37 
 
 Geoc. Diam'. by) 31 6g 29 
 observation j 
 Tabular 32 1,76 
 
 Cor n . for discordance of Z. P. ... 
 Cor n . of assumed Colatitude 
 
 Concluded Geocentric N. P. D. ... 
 Seconds of Tabular N. P. D 
 
 Excess of Tabular N. P. D 
 
 93 45 5,17 
 4,53 
 
 Excess of Ta- } ,0/17 
 bularDiam'. j 
 
 -0,64 
 
 The corrections for Runs are + 2",3 x -^- and + 2",3 x -1- . The 
 
 micrometer correction =(9,000- 16,482) x 20",869. Correction for 
 curvature of path =-0",1503 x 2 2 xtan3.45' (Arts. 199 and 201), 
 
 ,. f 2 x 16,6 x sec. 3. 45' _,, 10 
 
 correction for change of decl. = vac\c\ x ^" >** anc ^ 1S 
 
 obUU 
 
THE TRANSIT CIRCLE. 233 
 
 plus for N. L. and minus for S. L. (Art. 204). For these, as for all 
 other observations, the refractions were calculated by Bessel's Tables 
 (see Art. 215). The Parallax, taking 8",5776 to be the Sun's Equat. 
 Hor. Parallax, is [0,9325] x sin (Z- e) (see Art. 232), which gives 
 7",06 for N. L., 7",10 for S. L., 7",08 for centre, and -0",04 for the 
 parallactic correction of measured diameter. The corrections for 
 discordance of zenith points applied to the observations both of April 
 23 and October 2, were derived from a Table calculated for the year 
 1856, and contained in page Ivi. of Vol. xx. of the Cambridge 
 Observations. 
 
 THE TRANSIT CIRCLE. 
 
 238. Although in treating of the Transit Instrument and 
 Mural Circle separately all explanations have been given that 
 are essential for understanding the taking and reducing of 
 meridian observations, it will be proper to add some account of 
 the construction and use of a Transit Circle, this instrument 
 being adaptable by itself to every kind of meridional observa- 
 tion, and at the present time very generally employed in large 
 Observatories. In speaking of the latest improvements that 
 have been made in the instrumental means of taking meri- 
 dian observations, I have already had occasion to advert to 
 various particulars relating to the observational apparatus of 
 the Greenwich and Cambridge Transit Circles 1 : my object at 
 present is to give a description in detail of the latter instru- 
 ment, making use for that purpose of Plate III. (taken from a 
 Photographic Picture), with accompanying explanations and 
 remarks. [The Greenwich Transit Circle is described in full 
 detail in Appendix I. of the Greenwich Observations of 1852.] 
 
 239. The Cambridge Transit Circle was constructed and 
 mounted by Mr Simms, and in December 1870 was completely 
 
 1 Notices respecting the parts and uses of a Transit Circle, as distinguished 
 from those of the Transit Instrument and Mural Circle, are contained in the 
 following Articles: 6, 7, 29, 34 (note p. 28), 37, 38, 43 (note p. 36), 47, 48 
 (note p. 40), 3356, 58, 64, 86, 102 (note p. 98), 112, 116, 128, 137 (note p. 148), 
 164 (note p. 159), 173, 196, 202, 206, 208. 
 
23 4i PRACTICAL ASTRONOMY. 
 
 ready for use. The following account of it is extracted from 
 the Report of the Council of the Royal Astronomical Society, 
 February 10, 1871, under the head of The Cambridge Observa- 
 tory. (Monthly Notices, Vol. xxxi. p. 106.) " Professor Adams 
 believes that this instrument, for which the Observatory is 
 indebted to the munificence of Miss Sheepshanks x , will prove 
 to be one of the finest of its class. The object-glass of the 
 telescope is an excellent one, by Cooke, of 8 inches aperture 
 [and about 9 feet focal length]. There are two divided circles 
 of 3 feet in diameter, one of them being fixed relatively to the 
 axis of the telescope, and the other moveable and capable of 
 being clamped to the axis in any position. Each of the divided 
 circles is read off by means of four micrometer microscopes, 
 and additional microscopes, if required, may be readily applied. 
 There are two collimating telescopes each of 6 inches aperture, 
 which can be directed upon each other through an opening in 
 the central cube. A powerful apparatus is likewise provided 
 by which the instrument may be readily and safely reversed 2 . 
 There are no screw adjustments for azimuth and level, but the 
 axis of the instrument is brought into its definitive position by 
 scraping the Y supports 8 ." 
 
 The following additional particulars are given in the Monthly 
 Notices, Vol. xxxn. p. 151. " Small buildings have been erected 
 over the collimators, which are now, therefore, virtually in the 
 same room with the Transit Circle, and the consequent steadi- 
 ness in the images of the collimator wires is very remark- 
 able." 
 
 "Mr Graham [First Assistant at the Cambridge Observa- 
 tory] has adopted a method of determining the intervals of the 
 right-ascension wires which has been found to be very accurate 
 
 1 The donation in 1863 of 2000 by Miss Sheepshanks, to be employed for 
 the benefit of the Cambridge Observatory, with special preference for its 
 application in the purchase of one or more large instruments, and the legacy 
 in 1858 of 10,000, left by her brother, the Rev. R. Sheepshanks, for the pro- 
 motion of the science of Astronomy in the University, are recorded in the yearly 
 issues of the Cambridge Calendar under the head of The Observatory. 
 
 2 See Note 2 to Art. 48, p. 40. 
 
 3 See at the end of Art. 47. 
 
f Ij I B R A R \ 
 
 UNIVERSITY OF jj 
 
 CALIF [A 
 
THE TRANSIT CIRCLE. 235 
 
 and convenient. The eye-piece is turned through 90, and each 
 of the right-ascension wires, now become horizontal, is in turn 
 directed to the horizontal wires of either collimator, and the 
 corresponding readings of both circles are taken. The values 
 of the revolutions of the micrometer-screws have been found in 
 a similar way with great accuracy." 
 
 240. In Plate III., is the object-end of the telescope, 
 and a, 6, c, d, e,f are the positions of mechanical arrangements 
 at the eye-end adapted to the following purposes : at a are two 
 graduated micrometer-heads with fixed indices, one for reading 
 off the integral micrometer revolutions, and the other the parts 
 of a revolution, whereby the position of the wire-frame in the 
 vertical direction is recorded. The latter micrometer-head is 
 turned by hand, and by tooth-and-wheel action turns the other 
 in the required ratio. Apparatus of exactly the same kind is 
 provided for moving the wire-frame in the horizontal direction, 
 and both are covered by the angular plate 6, which is attached 
 for preventing accidental disturbances of the micrometer-heads, 
 and may be readily detached as occasion requires. In taking 
 an observation the eye-piece is shifted horizontally by turning 
 the milled head c, and by turning that at d it may be shifted 
 vertically and independently of the horizontal movement, so 
 that by the two movements the position of the eye-piece, within 
 certain limits, may be changed ad libitum. Turning the milled- 
 head at e regulates the illumination of the field, or of the wires. 
 (See Arts. 37 and 38.) At / there are two sets of antagonistic 
 screws, acting in transverse directions, one for turning the eye- 
 end bodily about the axis of the telescope, and the other for 
 changing its distance from the object-glass. (See Arts. 36 and 
 83.) The former action is explained in Art. 163, and in p. 157 
 is represented by the part of Figure 31 indicated by the letter 
 G. At g, g are the setting-circles, respecting which and the 
 mode of using them, in the case of non-reversion of the instru- 
 ment, see what is said in Art. 196. 
 
 241. Relatively to the wires used in taking transit obser- 
 vations it is requisite to state, that as there is no vertical 
 micrometer-wire, the abbreviated method of observing transits 
 
236 PRACTICAL ASTRONOMY. 
 
 of stars near the Pole, described in Art. 112, cannot be con- 
 veniently made use of; on which account six wires, in addition 
 to the usual number of seven, have been inserted and arranged 
 as follows. Supposing the whole system of wires to be A, B, C, 
 a, A 7, D, 8, e, E, F, G, the intervals between A, B y C, D y E, 
 F, G are quam proxime equal and of usual magnitude, while 
 those between (7, a, /3, 7 and those between S, e, E are all 
 equal but much smaller, and considerably smaller than the 
 remaining intervals between 7, D y 8. Polaris and other stars 
 in proximity to the Pole are observed at the wires C, a, /5, 7, 
 D, 8, e, f, E, and the larger intervals from 7 to D and from D 
 to 8 allow of observing them both by reflection and directly 
 at the same meridian transit. 
 
 242. After what has been said in Arts. 53 56 respecting 
 the construction and use of collimators, it will suffice to men- 
 tion the following particulars respecting those of the Cambridge 
 Transit Circle. The focal length of each is very nearly 6J- feet. 
 In Plate III., / is the opening in the central cube through 
 which the collimators are pointed towards each other (Art. 239). 
 At the geometrical focus of the south collimator, there are two 
 parallel wires, separated by a small interval, and crossing the 
 middle of the field vertically, and also two other wires intersect- 
 ing near the middle of the field, and inclined by equal small 
 angles to a horizontal direction passing through their point of 
 intersection. These two sets of wires are attached to the same 
 frame, which is moveable in the horizontal direction by turning 
 a graduated micrometer-head. At the geometrical focus of the 
 north collimator there are two like sets, but the parallel wires 
 are horizontal and the crossing wires are inclined by equal 
 small angles to the vertical direction. For determining colli- 
 mation error, first, the parallel wires of the south collimator are 
 moved by turning the micrometer-head into a position such 
 that the image of the point of intersection of the cross wires of 
 the north collimator is exactly mid-way between them, and at 
 the same time near the place of their intersection by the image 
 of the north collimator's parallel wires. Then, after pointing 
 the telescope to the south collimator, the part of the wire D 
 
THE TRANSIT CIRCLE. 237 
 
 near the centre of the field is placed by the telescope-micro- 
 meter mid-way between the images of the collimator's parallel 
 wires, and upon turning the telescope to the north collimator, 
 the same part of D is made by the micrometer-movement to 
 bisect the image of the cross wires of that collimator. These 
 operations having been performed, and the micrometer-readings 
 recorded (six or more times for each pointing of the telescope), 
 from the difference of the means of the readings the error of 
 collimation may be calculated, as shewn in Art. 55, and if of 
 excessive amount might be mechanically corrected. The pa- 
 rallel horizontal wires of the north collimator, and the vertical 
 cross wires of the south collimator, might be employed in an 
 analogous manner for determining by the process explained in 
 Arts. 222 and 223 the amount of horizontal flexure. 
 
 243. The East and West Piers, which are those of the old 
 Transit-instrument reduced in height, are found to give suffi- 
 ciently steady support to the much heavier Transit Circle. They 
 are surmounted by strong iron pillars K, with which the Ps in 
 which the pivots rest have rigid connection, so as not to be 
 susceptible of any screw adjustment (see Art. 47). The arrange- 
 ments relative to the graduations and microscopes are the same 
 on the East and West sides of the telescope, being, in fact, 
 mutually convertible by a reversion of the instrument. (The 
 reference letters are generally the same for both sides.) G, G 
 are the graduated circles, turning with the instrument about 
 the horizontal axis, that on the West side having a permanent 
 position relative to the axis, and that on the East side capable 
 of being shifted rotationally into any position relative to the 
 graduation of the other. To the wheels F with axle and spokes 
 are attached four micrometer-microscopes A, B, C, D, pointing 
 to the graduations on the circles G. The weight of F is 
 counterpoised by two weights W, one on each side of the axis, 
 the fulcrum being the pillar K. H is an additional microscope 
 also directed to the graduation, but only used as a pointer, or 
 for more exact setting, its reading for the setting being pre- 
 viously calculated to the nearest minute. For instance, to set 
 for a reflection observation so as to secure that the star shall 
 
23cS PRACTICAL ASTRONOMY. 
 
 cross very near the middle of the field, the setting would first 
 be performed in the usual manner by the setting circle, and 
 then, if on looking into microscope H it should be found to be 
 not sufficiently exact, it might by the tangent-screw movement 
 be readily changed ; after which the bisections of the gradua- 
 tions by the four microscopes would be proceeded with. (See 
 note to Art. 164 in p. 159.) The rods p, p acting upwards by 
 weights and leverage, as already described at the end of Art. 29, 
 sustain, by intervention of the friction wheels ra, n, the greater 
 part of the weight of the instrument, and thus lessen the wear 
 of the pivots by friction. Relative to the apparatus for the 
 Circle-readings it only remains to mention, that the parts 
 marked x, x furnish means of adjusting the axes of the wheels 
 F so that they shall be nearly in a line with the axis of motion 
 of the instrument. 
 
 244. The apparatus for slow-motion about the horizontal 
 axis is attached to the West Pier. At i there is a small screw 
 the end of which abuts on a collar of the axis, and by turning 
 the milled-head at h this end is made to act on the collar, and 
 thereby brings down the parts o, o, o, the uppermost of which 
 is a half-ring fitting into an angular groove, the pressure on the 
 faces of which clamps the instrument. After this the slow 
 movement is effected by turning the handle^", which by screw- 
 and-wheel action with Hooke's joint, gives horizontal motion to 
 the lower end of the apparatus o, o, o. On unclamping by 
 turning h, the two lower parts o, o are upheld by the upper 
 half-ring by means of intermediate screws with capstan heads. 
 The handle j is usually supported by a moveable upright stand 
 (not represented in the Figure), which is furnished with a row 
 of hooks for placing the handle in a position convenient to the 
 observer. Similar apparatus at h', with handle and Hooke's 
 joint, was added, at Mr Graham's suggestion, for being occa- 
 sionally used by an assistant, especially for clamping the instru- 
 ment in different positions in taking zone observations of stars, 
 the changes of setting being quickly made by the observer by 
 simply pulling a string tied to the Circle and passing over a 
 fixed pulley. 
 
THE TRANSIT CIRCLE. 239 
 
 245. At L a large lens, placed on a stage, not only throws 
 light from a paraffin lamp on a lens near the end of the trans- 
 verse axis for the illumination of the field and wires 1 (Art. 240), 
 but also, by means of transmission of light through glass prisms 
 suitably formed and placed, illumines the graduations towards 
 which the five microscopes are directed. S, R, r are parts of a 
 rail and balustrade for the observer to ascend by steps t, t to 
 take his position for reading off the microscopes ; p is part of 
 the bracing of the stage for supporting the lamp ; q, q are iron 
 wires for propping the flap U, on which, as there is occasion, 
 the observer places his memorandum book and hand-lamp. 
 The arrangements for illumining the field of view and the 
 graduations, and reading off the microscopes, are precisely the 
 same on the eastern, as on the western side, the interior plate 
 which reflects light towards the field being adaptable, by turn- 
 ing the screw e, to the incidence of the light from either the 
 western or eastern lamp. At X is the clock-face, which is 
 illumined by a large lamp moveable by a hinged support, and 
 can be looked at in most of the positions occupied by the 
 observer. 
 
 246. With respect to the means employed for obtaining 
 corrections for forms of the pivots of the Transit Circle, little 
 need be said in addition to the statements made in the Note 
 to Art. 102 in page 98. The dot is marked in the centre of a 
 small metallic disk, which is affixed to a piece of plate-glass 
 covering the end of the opening for the passage of the rays that 
 illumine the field of view. After removing the lens at the end 
 of the axis (Art. 245), a powerful microscope-micrometer is 
 placed so that an image, or ghost, of the dot (suitably illumined) 
 is formed where its horizontal and vertical coordinates can be 
 measured by the micrometer-wires. The arrangements are 
 precisely the same at the two ends of the transverse axis, two 
 microscope-micrometers being made use of. For adjusting the 
 
 1 Mr Graham informs me that he has found the visibility of very faint 
 stars to be sensibly improved by interposing between the lens at L and that at 
 the extremity of the axis a plate of red glass, which by means of an attached 
 string can be readily put in place or removed. 
 
240 PRACTICAL ASTRONOMY. 
 
 micrometer-wires horizontally and vertically, the spirit-level 
 apparatus mentioned in clause (3) of the Note in page 98 is had 
 recourse to, but not till its indication has been corrected or 
 verified by a small plumb-line used in the manner described in 
 Art. 95 with reference to the Transit-instrument. 
 
 247. After the explanations that have now been given of 
 the processes of taking and reducing meridian observations, it 
 only remains, for the complete determination of celestial 
 positions, to deduce from observations of the Sun the exact 
 position, at any epoch, of the origin of right ascension, assumed 
 to be the point of intersection of the Equator and the Ecliptic, 
 or First Point of Aries, and also to find exactly, for the same 
 epoch, the inclination of the Ecliptic to the Equator, or The 
 Obliquity of the Ecliptic. In the determination of these two 
 Elements by Flamsteed's Method (given in Arts. 137 and 138), 
 only observations of the Sun taken near the times of the 
 Equinoxes are employed ; by which means provisional values 
 may be obtained sufficiently accurate for adoption by the theo- 
 retical astronomer for the purpose of expressing his theoretical 
 places in right ascension and polar distance for immediate 
 comparison with the results of observation. It has already 
 been intimated (Art. 141) that it is usual for the practical 
 astronomer to employ such comparisons for obtaining mean 
 errors of the Sun's Tabular right ascension and polar distance 
 from all the meridian observations of the Sun in the course of a 
 year, and thence to infer the error of the assumed Obliquity, 
 and the mean error of the assumed right ascensions of the 
 fundamental stars. These corrected values may still admit of 
 being farther corrected by a continuation of the series of ob- 
 servations. But to effect these corrections so as to obtain 
 the most accurate values of the Solar Elements that obser- 
 vation is capable of giving, it is necessary to have recourse to 
 the Solar Tables themselves ; which is done on principles that 
 are quite logical, as will be seen from the following expla- 
 nations, which are applicable to the method employed in Vols. 
 x xx. of the Cambridge Observations. (See Vol. xx. pp. xxxvi. 
 and 96.) 
 
THE TRANSIT CIRCLE. 241 
 
 248. The true longitude X and true Polar Distance A of 
 the Sun's centre, and the true obliquity of the Ecliptic 7, are 
 related to each other at any epoch by the equation 
 
 cos A = sin X sin 7; 
 
 and the Tabular Longitude X + SX, the Tabular Polar Distance 
 A-f 8A, and the Obliquity I +81 assumed in the Nautical 
 Almanac, for the same epoch, by the equation 
 
 cos (A + 8A) = sin (X + SX) sin (7+ SI). 
 
 Hence, assuming that the Tabular errors SA, SX, 7 are already 
 so small that powers of these quantities higher than the first 
 may be neglected, we have by expanding, 
 
 gA cosXsm/^ siu^cos/ 
 
 sm A sm A 
 
 The reasoning now proceeds on an assumption which is 
 justified by the existing state of the Solar Theory, namely, that 
 the changes of the values of X and 7 in the course of a year are 
 in agreement with the indications of theory, and consequently 
 that the values themselves as given in the Nautical Almanac 
 are affected, if by any, by constant errors, which it is proposed 
 to find. 
 
 249. The actual errors of the Solar Tables in Polar Distance 
 cannot be immediately derived from comparisons of the cal- 
 culated with the observed values, because, although mere errors 
 of observation may be supposed to be eliminated in the mean 
 result of a large number of observations, there may still remain 
 uncorrected instrumental errors- and errors of reduction. Re- 
 presenting, therefore, by a. any excess of the tabular above the 
 observed polar distance, and by p the unknown excess of the 
 observed above the true polar distance, we shall have 
 
 Excess of tabular above true polar distance (= 8A) = a + p. 
 As we are ignorant of the causes to which p may be owing, 
 we can only undertake to deduce from observation the mean 
 of its values within the limits of the Tropics. (See what is 
 said in Art. 229.) By putting m for sin 7 x 8X, n for cos 7 x 67, 
 and a.-}-p for 8A, the equation (A) becomes 
 
 a + m cosec A cos X + n cosec A sin X + p = 0. 
 c. 16 
 
242 PRACTICAL ASTRONOMY. 
 
 Instead of forming a separate equation from this formula for 
 each value of a, let the whole number of observations be 
 divided into twelve groups, the mean of the values of a in 
 each group be calculated, and considered to correspond to the 
 day nearest the numerical mean of the days of observation in 
 the group, and let X and A be taken for the mean noon of the 
 mean day from the Nautical Almanac. In this manner twelve 
 different equations will be formed, which may be divided into 
 four groups of three, corresponding roughly with four quarters 
 of the year. Then three equations for determining the three 
 unknown quantities m, n, p may, as respects probable errors, be 
 advantageously obtained as follows : 
 
 First Quarter -f Second + Third + Fourth = 0, 
 First Quarter + Second - Third - Fourth = 0, 
 First Quarter - Second - Third + Fourth = 0. 
 250. For the purpose of exemplifying the results to which 
 the foregoing reasoning conducts, those obtained from the 
 Cambridge Meridian Observations of the Sun in the year 
 1855 are here subjoined. The total number of the Circle 
 observations was 125, inclusive of 24 observations of single 
 Limbs. These were divided into four quarter-groups contain- 
 ing severally 28J, 27J, 28^ and 28J- observations, one of a 
 single Limb being taken to be half an observation. Each of 
 these groups was subdivided into three in such manner that 
 the twelve groups all contained nearly the same number of 
 observations. In calculating the value of a for any group, 
 half-weight was given to an observation of a single Limb. The 
 rest of the reasoning was conducted by the process explained 
 in Arts. 247 and 248, and the following results were obtained 
 by the solution of the three final equations : 
 
 m = - 1",239, n = - O f/ ,233, jp = + 0",329. 
 
 Hence SX (= m cosec I) = - 1" 5 239 cosec 23. 27',5 = - 3",112, 
 which is the mean excess for the year of the Tabular, above 
 the true, Longitude of the Sun ; 
 
 87 (= n sec J) = - 0,"233 sec 23. 27',5 = - 0" 5 254, 
 which is the excess of the Obliquity assumed in the Tables above 
 
THE TRANSIT CIRCLE. 243 
 
 the true Obliquity ; and p the mean excess within the Tropics 
 of the N. P. D. determined by the reduction of the Circle 
 observations of 1855 above the true N. P. D. (See Art. 229.) 
 
 If 8R represent the mean excess for the year of the Tabular 
 R.A. above the true R.A. in arc, we may assume that SR = 8\. 
 Hence in the present instance 8R (in time) = 8 ,207. The 
 comparisons of all the observations of the year with the Tabular 
 values of the Nautical Almanac gave s , 187 for the mean 
 excess (A) of the Tabular above the observed R.A. by 128 
 comparisons, and 0",364 for the mean excess (D) of the 
 Tabular above the observed N.P.D., by 113 comparisons. Hence, 
 putting q for the mean excess of the assumed R A. of the funda- 
 mental stars above the true, we have the following results : 
 
 SA = (Tab. N.P.D. - Ob. N.P.D.) + (Ob. N.P.D. - True N.P.D.) 
 
 = D + p = - 0",364 + 0",329 = - 0",035, 
 q = (Tab. R.A. - True R.A.) - (Tab. RA. - Ob. R.A) 
 = 8R-A=- O s ,207 4- O s ,187 = - O s ,020. 
 
 Hence the observed R.A. were too small by the mean quan- 
 tity O s ,020 ; or the assumed position of the first point of Aries 
 was in advance of the true by this quantity. 
 
 251. By 127 observations of the Sun in R. A., and 
 117 observations in N.P.D. in the year 1856, it was found 
 that m = -0",254, n = -0",199, _p = + 0",451, Z> = -0",447, 
 SA = + 0",004, 81 = - 0",217, SR = - 8 ,043, A=- O s ,155, and 
 # = + O s ,112. Hence the assumed R.A. of the fundamental 
 stars were too great by O s ,112. (See Camb. Obs. Vol. XX. 
 pp. 204 and 205.) It may be noticed that these results, with 
 the exception of the value of m and the consequent values 
 of SB and q } agree closely with those given by the observations 
 of 1855. On account of the discrepancy between the values 
 of 8R (= m cosec 7), the corrections of the assumed R.A. of the 
 fundamental stars are different for the two years, being + O s ,020 
 for 1855 and O s ,112 for 1856, although the mean excess of the 
 assumed RA. above those of the Naut. Aim. was almost exactly 
 the same for both. (See Introduction of Vol. XX., p. xlix.) 
 
 162 
 
244 PRACTICAL ASTRONOMY. 
 
 Hence as BE is the excess of the tabular above the true RA. of 
 the Sun, it might be inferred that the RA. of the Tables, as 
 calculated for the years 1855 and 1856, require the respective 
 corrections O s ,207 and O s ,043. Such an inference, however, 
 cannot be safely drawn, in the existing state of practical and 
 physical astronomy, unless the amount and variation of the 
 error be determined by the mean results of the observations of 
 a large number of years. The foregoing explanations will, how- 
 ever, have indicated by what kind of process small residual 
 errors of the Tables are ascertained. 
 
 252. The values of the corrections p and q having been 
 determined by observations of the Sun, those of BB (= A + q) 
 and SA (= D +p) are readily deducible, inasmuch as A and D 
 are respectively the excesses of the tabular above the observed 
 R. A. and N. P. D. usually computed by the observer. From 
 the values of the errors R and SA thus obtained, mean errors 
 may be derived from arbitrary groups of the individual errors, 
 and may severally be supposed to correspond in epoch to the 
 mean of the times of the observations furnishing the group of 
 errors. These rules for calculating mean tabular errors are 
 applicable alike to meridian observations of the Sun, Moon, and 
 the older planets, in all cases in which a continuous series of 
 such observations have been taken. For the sake of the 
 theoretical astronomer, the mean errors in R A. and N. P. D. 
 are converted into errors in Longitude and Ecliptic Polar 
 Distance by known formulae 1 , the application of which is much 
 facilitated by making use of the factors contained in Appendix 
 II. of the Volume of Greenwich Observations for 1836. In the 
 same volume (p. Ixii) are given formulae for calculating errors of 
 Heliocentric Longitude and Ecliptic Polar Distance from the 
 errors of Geocentric Longitude and Ecliptic Polar Distance 
 obtained in the manner just indicated. The results of such 
 calculation in the cases of the minor planets Ceres, Pallas, 
 Juno, and Vesta, and all the larger planets, are given in the 
 Volume just named, and in all succeeding Volumes, of the 
 Greenwich Observations. 
 
 1 See the Rev. R. Main's Practical and Spherical Astronomy, Chap. in. Art. 
 7 (P. 74). 
 
THE TRANSIT CIRCLE. 245 
 
 253. The practical astronomer is required to give, together 
 with his observations, the times at which they were taken. 
 With respect to fixed stars, when their places are observed 
 with a Transit-circle, it generally suffices to name the year and 
 day of the observation ; but for the moving bodies the precise 
 time of taking each observation has to be calculated. The 
 science of Time, as consisting of measurements of time-intervals, 
 and determinations of epochs, depends essentially on data fur- 
 nished by Practical Astronomy. This will appear from the 
 following discussion, the purpose of which is to shew how to 
 calculate the epochs of observations. 
 
 254. Bessel, by a comparison of his own observations of 
 the Sun in 18201825, and those of Bradley in 1753 and 
 1754, with Carlini's Solar Tables, obtained the Sun's mean 
 longitude at a given epoch ; from which the following funda- 
 mental quantity 1 , used in the computations of the Nautical 
 Almanac for the years 1836 1863, was derived : 
 
 At the Greenwich Mean Noon of January 1, of the year 
 1800 + 1, the Sun's Mean Longitude (Jf ) is 
 
 283. 53'. 32",71 + . 27",605844 + * 2 . 0",0001221805-/- 14'. 47",083, 
 
 where 14'. 47",083 is the change of Mean Longitude in 6 h of 
 mean time,/ denotes, for the nineteenth century, the number of 
 years from the year immediately preceding 1800 + 1 which is 
 divisible by 4 without remainder. It is to be observed that 
 this value of the Mean Longitude includes the effect of aber- 
 ration. 
 
 A sidereal day is the interval between two consecutive 
 transits of the first point of Aries across the meridian of any 
 place. A mean solar day is the interval between two con- 
 secutive transits of a fictitious Sun supposed to move in the 
 Equator with the Sun's mean motion in Longitude, or Eight 
 Ascension. 
 
 It has been agreed by astronomers to call the sidereal time 
 
 i See the Astronomische Nachrichten, No. 133 ; Bessel's Tabula Regiomon- 
 tana, p. xxiv ; and the Nautical Almanac for 1863, p. vi. 
 
246 PRACTICAL ASTRONOMY. 
 
 at any place, the arc intercepted between the actual first point 
 of Aries and the point of the Equator which is on the meridian 
 of the place, converted into time at the rate of 15 to an hour. 
 But the first point of Aries, since its position is determined by 
 the intersection of the plane of the' earth's Equator with the 
 plane of the Ecliptic, moves relatively to stars on the plane 
 of the Ecliptic, and, on account of nutation, by an irregular 
 motion. Hence according to the adopted reckoning astro- 
 nomical sidereal time is not strictly uniform. No sensible 
 error, however, arises from this circumstance, because the 
 irregular motion takes place relatively to a mean position 
 which has a uniform motion, and the fluctuations of position 
 (called the Equation of the Equinoxes) are very slow, and 
 much slower than those to which the rate of the best con- 
 structed time-piece is liable from incidental causes. The par- 
 ticular advantage of this conventional reckoning is that the 
 sidereal time at which a celestial object passes the meridian 
 becomes identical with its apparent Eight Ascension. The 
 calculated apparent R. A. of the fundamental stars are ac- 
 cordingly referred to the true Equinox. Hence, since the 
 error of the clock is the difference between its indication and 
 the calculated Apparent Right Ascension of a fundamental star, 
 it follows that the time-piece is virtually regulated to point to 
 O h whenever the first point of Aries (affected by aberration as 
 a star) is apparently on the meridian. 
 
 255. From the definition above given of Sidereal Time it 
 follows that the Sidereal Time at Mean Noon = Sun's Mean 
 R. A. + Nutation in R. A., the addition of the Nutation giving 
 to the sidereal time the fluctuating value it is required to have 
 by being referred to the true Equinox. By means of this 
 equation the relation between the sidereal time and the mean 
 solar time of the same epoch is ascertained by the following 
 process. From the calculations of Bessel already referred to, 
 it was found that the mean motion of the Sun in 3G5 mean 
 solar days was less than 360 by 22",617656 ; whence it fol- 
 lows that the sidereal year, or complete revolution of the 
 Sun with respect to fixed space, is 365 d ,6 h ,9 ra . 10 S ,7496, or 
 
THE TRANSIT CIRCLE. 247 
 
 365,256374417 mean solar days. Taking, according to the same 
 authority, the mean amount of the precession of the equinoxes 
 in the t years succeeding 1800 to be 
 
 50",22350 + 0",0001221805 2 , 
 
 the mean length of the tropical year between 1800 + and 
 1800 + t+ 1 is 
 
 365 d . 5 h . 48 m . 47 S ,8091 - O s ,00595, 
 or 365 d ,242220013 - O d ,00000006686. 
 
 Dividing 360 by the length of the tropical year, the mean 
 motion of the Sun in Longitude in a mean solar day will be 
 found to be 59' . 8",3302, and consequently the mean daily 
 motion in K. A. expressed in time, 3 m . 56 S ,555348. Hence by 
 the foregoing equation we have at the Greenwich mean noon 
 of any day (n) of the year 1800 + t, 
 
 M 
 
 Sidereal time = -- + 3 m . 56 S ,555348^ + Nutation in R A. 
 
 JLO 
 
 It thus appears that from one mean noon to the next 
 following the sidereal time increases by the mean quantity 
 3 m .56 s ,555348, and consequently that 24 h of mean time are 
 equivalent to 24 h . 3 m . 56 3 ,555348 of sidereal time. The above 
 formulae for M and for sidereal time are adopted in the 
 Nautical Almanacs for the 29 years 1835 1863. 
 
 256. In the Nautical Almanacs for 1864 and following 
 years the tabular places of the Sun are taken from Leverrier's 
 Tables contained in Vol. IV. of the Annals of the Imperial Obser- 
 vatory of Paris. According to results deduced by Leverrier 
 from 8911 observations of K A. of the Sun made in the years 
 1750 1850, the mean sidereal motion of the Sun in a Julian 
 year of 365J days is 1295977",38234, or 22",61766 less than 
 360. Also he obtains 50",23572* + 0",00011289r for the 
 general precession in the interval t reckoned in Julian years 
 of 365^ days, and commencing at the Paris mean noon of 
 January 1, 1850. Hence he finds (by a process equivalent to 
 that indicated in Art. 255) that the mean day is equal to the 
 sidereal day multiplied by 1,002737909, or 24 h . 3 ra . 56 8 ,555338, 
 
248 PRACTICAL ASTRONOMY. 
 
 nearly agreeing with Bessel's value above. If therefore M' be 
 the Sun's Mean Longitude for the meridian of Greenwich for 
 the mean noon of January 1, 1850 + , as inferred from 
 Leverrier's Tables of the Sun, allowance being made for the 
 difference between the meridians of Paris and Greenwich, we 
 shall have 
 
 Sidereal time = ^ + 3 m . 56 s . 555338 x n + Nutation in R. A. 
 
 JLo 
 
 257. After shewing how the foregoing two sets of formula? 
 were obtained, I propose now to give an example of the applica- 
 tion of each of them for calculating the sidereal time of mean 
 noon. Supposing that it is required to calculate the sidereal 
 time of the Greenwich mean noon of January 24, 1863, the fol- 
 lowing is the process according to Bessel's formulae contained in 
 Arts. 254 and 255. In this example t = 63, and therefore /= 3. 
 Hence it will be found that 
 
 27",605844 x t = 28'. 59",168, 0^0001221805 x f = 0",485, 
 and - 14'. 47",083 x/= - 44'. 21",249. 
 
 Consequently 
 
 M = 280. 5a'. 32",71 + 28'. 59",168 + 0",4S5 - 44'. 21",249 
 = 280. 38', 11",11 = (in. time) 18 h . 42 m . 32 S ,741. 
 
 Also since mean noon was earlier than apparent noon, the day 
 before is to be taken, so that n = 23. Hence 
 
 3 m . 56 S ,555348 x n = l h . 30 m . 40 8 ,773 ; 
 
 and since the nutation-correction, as given in the Naut. Aim. 
 (p. 242), was + 1 8 ,09, it follows that the sidereal time 
 
 = 18 h . 42 m . 32 8 ,74 + l h . 30 m . 40,77 + l s ,09 = 20 h . 13 m . 14 8 ,60, 
 
 which agrees with the value in page 3 of the Naut. Aim. for 
 1863. 
 
 To exemplify the application of Leverrier's formula, let 
 it be required to find the sidereal time at mean Greenwich 
 noon of 1879 January 20. The Sun's Mean Longitude (M'), as 
 
THE TRANSIT CIRCLE. 249 
 
 given in p. 102 of Vol. IV. of the Annals of the Paris Observa- 
 tory, is 
 
 280. 46'. 43",51 + 1296027",6784 + 0",00011073* 2 , 
 
 the time t being reckoned in Julian years of 365J days, and 
 commencing at the Paris- mean noon of January 1, 1850. 
 Hence t = 29 (= 4 x 7 + 1), and /= 1. These values give 
 
 1296027",6784 x 29 = 10440. 13'. 22",736, 
 and 0",00011073 x 29 2 = 0",093. 
 
 Hence, rejecting 29 x 360, we have 
 
 M f = 280. 46'. 43",51 + 13'. 22",736 + 0",093 - 1 x 14'. 47",083 
 = 280.4a'.19",194 = (in time) 18 h .43 m . l s ,279. 
 
 Since the mean noon, at the epoch, is earlier than apparent 
 noon, the day before is to be taken. Hence n = 19 and 
 
 236 S ,555338 x n= l h . 84 m . 54 S ,551. 
 
 As Paris is 9 m . 20 S ,63 East of Greenwich, the correction of M ' 
 for difference of meridians is +23",033 (= +l s ,535). Also 
 correction for nutation in R. A., derived from 15",16, the equa- 
 tion of the equinoxes in Longitude is O s ,927. Consequently 
 the sidereal time 
 
 = 18 h . 43 m . 1 8 ,279 + l h . 14 m . 54 S ,551 + 1 8 ,535 + O s ,927 
 = 19 h . 57 m . 58 8 ,292, 
 
 which agrees with the time given in page 3 of the Naut. Aim. 
 for 1879. In calculating sidereal times for the formation of 
 Ephemerides, much assistance would be derived from the 
 Tables arranged by Leverrier for this purpose in the before 
 cited Volume. The object of the foregoing discussion has been 
 to indicate processes, and furnish data, whereby the sidereal 
 time of any epoch might be calculated independently of such 
 aid. 
 
 258. For calculating epochs in mean time, it is required to 
 know the mean time of the transit of the first point of Aries; 
 which may be inferred as follows from the sidereal time of 
 
250 PRACTICAL ASTRONOMY. 
 
 mean noon. Let or be this sidereal time ; that is, the horary 
 angle, at mean noon of any day, of the first point of Aries 
 westward. This point is, accordingly, on the meridian at an 
 interval before noon equal to cr reckoned in sidereal time, and 
 therefore equal to /z,cr reckoned in mean time, yu, being the ratio 
 of the Sun's mean diurnal motion to that of a star. Hence, as 
 mean time is reckoned from mean noon, the required mean 
 time is 24 h per. Selecting, for example, the sidereal time at 
 mean noon of January 20, 1879, namely, 19 h . 57 m . 5S S ,292, and 
 either taking 1,002737909 for the ratio of the mean day to the 
 sidereal day, as given in Art. 25"6, or employing the Table of 
 equivalents in pages 490 and 491 of the Naut. Aim. for 1879, it 
 will be found that pa- = 19 h . 54 m . 42 S ,030. Hence 24 h - p<r, or the 
 mean time required, is 4 h . 5 m . 17 S ,970. As, however, this mean 
 time is less than the sidereal time at mean noon of January 20, 
 it must be the mean time of transit of the first point of Aries 
 on January 19. It agrees, in fact, with the value given in the 
 Naut. Aim. for this date. 
 
 259. By knowing the sidereal time of mean noon, and the 
 mean time of transit of the first point of Aries, which may be 
 called sidereal noon, mean time and sidereal time are mutually 
 convertible by calculating according to the following rules: 
 Sidereal time required = that at the preceding mean noon + the 
 equivalent in sidereal time of the given mean time. Mean time 
 required = that at the preceding sidereal noon + the equivalent 
 in mean time of the given sidereal time. 
 
 To convert, for example, 2 h . 22 m . 25 8 ,62 Greenwich Mean 
 Time, January 20, 1879, into sidereal time, the process is as 
 follows. The sidereal time at the mean noon of January 20, 
 1879, (calculated as is shewn in Art. 257) = 19 h . 57 m . 58 8 ,292, 
 and the sidereal equivalent of the mean time 
 
 2 h . 22 m . 25 S ,62 = 2 h . 22 m . 49 S ,018, 
 
 as calculated by the Tables in pages 488 and 489 of the Naut. 
 Aim. for 1879. Hence the sidereal time required 
 
 = 19 h . 57 m . 58 8 ,292 + 2 h . 22 ra . 49 S ,018 = 22 h . 20 m . 47 S ,310. 
 To obtain, conversely, the mean time of January 20, 1879> 
 
THE TRANSIT CIRCLE. 251 
 
 corresponding to the sidereal time 22 h . 20 m . 47 8 ,310 on that day, 
 the mean time of sidereal noon on January 19 is to be taken, 
 in accordance with a rule which, to assist the memory, has been 
 expressed in the terms, " If the sidereal time be more, take the 
 day before" (see Art. 257); the result of the computation in 
 Art. 257, as also the contents of pages 3 and 4 of the Naut. 
 Aim., shewing that for January 19 the sidereal time at mean 
 noon exceeds the mean time of sidereal noon. Now the latter, 
 either as obtained independently by the calculation in Art. 258, 
 or taken immediately from the Naut. Aim., is 4 h . 5 m . 17 8 ,970. 
 Also the equivalent of 22 h . 20 m . 47 S ,310 in mean time is 
 
 22 h . I7 m . 7 8 ,654 
 
 by the Tables in pages 490 and 491. Hence the mean time 
 required = 4 h . 5 m . 17 8 ,970 + 22M7 m . 7 8 ,654-24 h = 2 h . 22 m . 252,624, 
 being for January 20 the excess above 24 h . 
 
 260. For astronomical purposes it is convenient to express 
 the epochs of all events in mean times of a fixed meridian, and 
 the English astronomer selects, of course, the meridian of 
 Greenwich. If the longitude of any observatory eastward from 
 the meridian of Greenwich be L (in time), a given star, or the 
 first point of Aries, passes the meridian of that Observatory 
 before it passes the meridian of Greenwich by a sidereal 
 interval equal to L. Hence if S be the sidereal time of any 
 event, as determined by transit observations at the Observatory, 
 that is, the local sidereal time, the Greenwich sidereal time of 
 the event is S L. With this sidereal time the Greenwich 
 mean time may be calculated by the process indicated in the 
 preceding paragraph. Or, if with the local sidereal time S the 
 mean time be calculated just as for Greenwich, the correction for 
 difference of meridians is pL, fj, being the factor (the recipro- 
 cal of 1,002737909) for converting sidereal time into mean time. 
 For the Cambridge Observatory, the longitude is 22 S ,75 East, 
 and the correction pL is equal to 22 S ,69. 
 
 261. It will thus be seen that in order to deduce from the 
 sidereal times of events, as obtained by observations made at 
 any Observatory, the epochs of the same events in Greenwich 
 
252 PRACTICAL ASTRONOMY. 
 
 mean time, it is necessary to know the Longitude of the 
 Observatory with reference to the meridian of Greenwich. The 
 same element, together with Latitude, is necessary for indi- 
 cating the locality of the Observatory on the eajth's surface, 
 which the practical astronomer is required to do. Having 
 already had under consideration (in Arts. 219220) a method 
 of obtaining exactly the Latitude of an Observatory, we have 
 now to enquire by what means its Longitude may be accurately 
 determined. Terrestrial Longitude is the arc between the 
 meridian of any place and that of Greenwich measured on the 
 earth's Equator, and is called East or West Longitude, or is 
 designated by the signs or +, according as the place is east- 
 ward or westward from Greenwich. This arc is measured 
 astronomically by the interval between the transits of a given 
 star across the two meridians, because the star is apparently 
 carried from one meridian to the other in that interval by the 
 diurnal motion of the heavens due to the earth's rotation about its 
 axis, to which rotation, assumed to be uniform 1 , all astronomical 
 measures of time are referred. The sidereal day, which is the 
 interval from the passage of a star, or the first point of Aries, 
 across a meridian to its return to the same, being supposed to 
 consist of 24 hours, it follows that the astronomical measure of 
 longitude has to 24 hours the same ratio as the arc of terres- 
 trial longitude has to 360, or that one hour corresponds to 15. 
 Hence it may be seen that the longitude of a place is measured 
 by the difference between the sidereal time at the place, and 
 the sidereal time at Greenwich, of the same event, because the 
 difference between such times measures the arc between the 
 two meridians. This is the general principle on which terres- 
 trial longitude is ascertained by astronomical means. 
 
 262. There are various ways of comparing the simul- 
 
 1 It was suggested by M. Delaunay, late Astronomer Imperial at the Paris 
 Observatory, aiid lias since been maintained by other physicists, that the uni- 
 formity of the earth's rotation might be disturbed to a sensible amount by the 
 effect of the attractions of the Sun and Moon on the tidal waves. If this be the 
 case, the calculation of all time-intervals and epochs will be in some degree 
 altered. I am not aware that the amount, or even the actuality, of disturbance 
 from this cause has been strictly deduced from hydrodynamical principles. 
 
THE TRANSIT CIRCLE. 253 
 
 taneous sidereal times of two Observatories. The simplest 
 appears to be that by transfer of Chronometers, according to 
 which, when several chronometers are used, each is first timed 
 and rated by the transit-clock of one of the Observatories, is 
 then transferred to the other Observatory, and after being 
 timed, and rated by a clock and transit-instrument there, is 
 conveyed back to the first for another timing and rating, and so 
 on as often as may be thought desirable. In this way, after 
 taking account of the meridional adjustments of the instru- 
 ments, and the recorded clock and chronometer times and the 
 calculated rates, the simultaneous sidereal times of the two 
 positions are repeatedly brought into comparison ; and by 
 taking the mean of the several differences given by a single 
 chronometer, and then the mean of the means given by the 
 several chronometers, the concluded difference of longitude is 
 obtained. By a process such as this, the details of which are 
 contained in the Cambridge Philosophical Transactions (Vol. ill. 
 p. 168), Professor Airy found the Longitude of the Cambridge 
 Observatory to be 23 S ,54 eastward from Greenwich. 
 
 263. Since the invention of the Electric Telegraph, astro- 
 nomers have taken advantage of the transmission of galvanic 
 signals for the determination of the difference of longitude 
 between two Observatories. When it can be so arranged that 
 the signals can be sent and received at both Observatories, and 
 the operator can also take his time from the transit clock as in 
 an ordinary transit observation, the sender notes the time of 
 sending a signal and the receiver that of receiving it, according 
 to the indications of the respective clocks. The signal is usually 
 the movement of a galvanic needle produced by contact being 
 made by the sender for completing the galvanic circuit, which 
 movement the receiver has to watch for. The signals are 
 transmitted a certain number of times, previously agreed upon, 
 alternately from the two stations. After applying corrections 
 for the errors of position of the transit-instruments, and the 
 errors of the clocks on the true sidereal times of the Observa- 
 tories, and also eliminating as far as may be errors which might 
 be due to difference of personal equations of the observers, 
 
254 PRACTICAL ASTRONOMY. 
 
 whether as regards the taking of transits or noting the signal- 
 times, the mean of all the values of the difference of the 
 sidereal times measures the longitude, if the transmission of 
 the signals be absolutely instantaneous. If there should be a 
 sensible rate of transmission, its effect on the determination of 
 the longitude is got rid of by sending signals in opposite 
 directions, as may be thus shewn. Supposing the passage of a 
 signal between the two stations to occupy the interval r, the 
 time recorded at the receiving station will be too late by this 
 interval. Hence, since the measure of longitude is the excess 
 of the sidereal time at the east station above the simultaneous 
 sidereal time at the west station, according as the receiving 
 station is to the east or the west of the other the measure of 
 longitude will be too great or too small by T. Hence the true 
 longitude is the mean between two measures obtained by 
 signalling from the two stations in opposite directions. Conse- 
 quently to eliminate this source of error the number of signals 
 should be the same from both stations. In that case a very 
 probable value of 2r would result from subtracting the mean of 
 all the measures deduced from signals made at the west station 
 from the mean of those deduced from the signals made at the 
 east station. 
 
 264. If the signal- times, whether of sending or receiving, 
 are not immediately taken from the transit-clock, but are 
 recorded by means of transportable chronometers at any 
 position more or less distant from the clock, the noted times 
 have to be reduced to sidereal times of the Observatory by 
 means of comparisons of the chronometers with the clock, made 
 both before and after the series of signals, in order to have data 
 for making allowance for the rates of the chronometers. The 
 comparison of a sidereal chronometer with the transit- clock 
 should, for greater certainty, be made by the intervention of a 
 mean-time chronometer, because this time-piece may be ac- 
 curately compared with each of the other two by coincidence of 
 beats. In making, according to this process, a determination of 
 the longitude of the Cambridge Observatory by galvanic signals 
 on May 17 and 18, 1853 (the particulars of which are given in 
 
THE TRANSIT CIRCLE. 255 
 
 the Transactions of the Cambridge Philosophical Society ,Vol. ix., 
 pages 487 514), the Greenwich signals were received and the 
 Cambridge signals transmitted at the Cambridge Telegraph 
 Office of the Great Eastern Railway, and the Observatory 
 sidereal times were inferred in the manner just stated by 
 comparisons of chronometers. The mean result, deduced from 
 281 signals, gives for the longitude of the Cambridge Observa- 
 tory, 22 S ,75 East of Greenwich. Also from all the signal-times, 
 reduced to sidereal times of the Observatories, the interval (T) 
 occupied by the transmission of the signals was found to be 
 8 ,020. On account of the comparatively small length of the 
 conducting wire between the two stations (about 55 miles), this 
 value is not trustworthy for calculating the rate of transmission, 
 but may, I think, be regarded as a limit which the interval of 
 transmission did not exceed. 
 
 265. Having now gone through the explanation of the 
 means actually employed at the present time for determining 
 with as much accuracy as possible by meridian instruments the 
 places in Right Ascension and Polar Distance of fixed stars and 
 of the centres of the moving bodies, and also the processes for 
 calculating the epochs at which they have the assigned places, 
 we may be said to have taken account of all that is of 
 fundamental importance in practical astronomy. Accurate 
 determinations of celestial positions by instruments adapted 
 for observations out of the meridian, depend ultimately on the 
 results of observations made on the meridian. Again, as 
 respects physical astronomy, suppose the elements of the orbit 
 of a moving body to have been deduced from observations (by 
 processes such as those given in Gauss's Theoria Motus Corp. 
 CeL), till they are known with sufficient accuracy for being em- 
 ployed by the theoretical astronomer for calculating Tables where- 
 by the body's place at a given time may be computed. Then 
 the farther corrections which the elements may require are 
 ascertained by means of continued observations of the body 
 made either actually on the meridian, or by the intervention of 
 places of stars determined by meridian observations, the Right 
 Ascensions and Polar Distances given by such observations 
 
256 PRACTICAL ASTRONOMY. 
 
 being subsequently compared with theoretical values computed 
 with the assumed elements. Suppose, for illustration, A and A 
 to be the tabular R A. and P.D. of a moving body at the assigned 
 epoch of an observation from which the values, as observed, are 
 found to be a and 8. The practical astronomer, after obtaining 
 such observational values with all attainable accuracy, gives 
 values of A a and A 8, as deduced from a series of observations, 
 to the theoretical computer, calling them Errors of the Tables on 
 the assumption that the observations are exact. Each such error 
 is put on one side of an equation, the other side of which con- 
 sists of a variation of the function from which A, or A, was 
 computed, with respect only to the letters it contains represent- 
 ing the elements, and these variations of the elements are con- 
 sequently to be taken as algebraic excesses of the values of the 
 elements employed in constructing the Tables above their true 
 values. From a large number of such equations, divided into 
 groups and treated according to the rules of the method of least 
 squares, values of the variations of the elements are obtained, 
 which, with signs changed, are the corrections to be applied to 
 the assumed elements. In proportion as the number of the 
 equations is larger, these corrections may be considered to be 
 more free from the effects of incidental errors of observation. 
 An exemplification in part of the investigation above indicated 
 is given by the process described in Arts. 247 252, whereby the 
 errors of the assumed obliquity of the Ecliptic and the assumed 
 R.A. of the fundamental stars are corrected. I conclude this 
 first and principal Part of the Lectures with making the remark, 
 which the foregoing statement of the relation between practical 
 and physical astronomy justifies, that however far the theorist 
 may carry his analytical investigations, the accuracy thereby 
 attainable is limited by the accuracy of the data furnished by 
 instrumental observation. 
 
 The remaining portion of the Lectures will treat of (1) the 
 construction and use of instruments employed for obtaining 
 celestial places by observations out of the meridian, especially 
 
THE EQUATORIAL. 257 
 
 the Equatorial and the Altazimuth Instrument ; (2) other fixed 
 instruments, particularly those adapted to finding exact values 
 of the constants contained in the formulae for calculating star- 
 corrections given in Arts. 140 147 ; (3) uses of movable and 
 portable instruments, as the Zenith Sector and the Theodolite, 
 for determining geographical positions, and of the Sextant for 
 nautical purposes ; (4) various applications of instruments for 
 special purposes, such as finding longitudes by land and sea, 
 measures of small differences of R. A. and P. D. between the 
 places of two objects, measures of the relative positions of the 
 components of double and multiple stars, &c., together with 
 miscellaneous subjects not treated of in the previous sections. 
 
 THE EQUATORIAL. 
 
 266. An Equatorial Instrument is made suitable by its 
 construction and mounting for determining Right Ascensions 
 and Polar Distances of objects at whatever part of the heavens 
 they are visible. For this purpose it has an axis of revolution 
 placed parallel to the earth's axis. Like the Transit-circle, the 
 Equatorial is a complete instrument for determining celestial 
 places; but as being liable to unsteadiness on account of the 
 oblique position of its axis, it is not as trustworthy for deter- 
 mining absolute R. A. and P. D. as a meridian instrument. If> 
 however, it be used only for differential observations, that is, for 
 comparing the place of an unknown object with that' of a known 
 star, or one whose exact place is afterwards ascertained by 
 meridian observations, very accurate results are obtainable. 
 The Equatorial is employed in observing Comets, newly dis- 
 covered small Planets, Eclipses of the Sun, Moon, and Jupiter's 
 Satellites, occultations of stars and Planets by the Moon, double 
 and multiple stars, and star-groups. It is also much used for 
 securing approximate places of unknown stars, and of stars 
 selected for comparison in differential observations, in prepara- 
 tion for afterwards obtaining their exact places by meridian 
 observations. As in the instances of the Transit Instrument 
 
 c. 17 
 
258 
 
 PRACTICAL ASTKONOMY. 
 
 and Mural Circle, the Equatorial will be treated of under four 
 Divisions. 
 
 I. DESCRIPTION OF THE PARTS AND MODE OF MOUNTING. 
 
 267. Figure 40 delineates a wooden model exhibited in 
 my Lectures for indicating the parts, mounting, and uses of an 
 Equatorial. The different parts correspond as to position, and 
 in most respects as to proportion, to those of the five-feet 
 Equatorial of the Cambridge Observatory, to which instrument 
 the following descriptions will more especially apply. Q and R 
 
 Fig. 40. 
 
 are solid stone piers placed, the former southward and the other 
 northward, so as to be cut symmetrically by the plane of the 
 meridian, on a massive stone pillar rising about 20 feet from 
 the level of the ground, on which also a sidereal clock (Graham 
 
THE EQUATORIAL. 259 
 
 with Harrison's gridiron pendulum) is posited for convenience 
 in taking observations. I is an iron frame attached at H to the 
 support R, and having at its upper extremity a Y in which the 
 pivot P at the higher end of the axis of motion rests. The 
 plane faces of the frame / and of the bracing pieces i, i, are 
 made to be parallel to the axis of the Telescope when pointed 
 towards them, so that they only intercept a quantity of light 
 corresponding to their narrow edges. The lower pivot of the 
 polar axis turns in a cup fixed in a plate which is adjustible 
 in the plane of the meridian by two opposing screws, one of 
 which is represented at Z>, and in the transverse direction by 
 the screw represented at a and one opposite. Above this plate 
 there is another which is pushed from beneath by strong springs 
 against four small wheels at the ends of four arms joined to the 
 polar frame, by which means the weight of the instrument is in 
 great part taken off, and the friction in the cup diminished, 
 without impeding the movement of rotation about the axis. 
 
 268. The two terminals of the polar frame (represented in 
 the Figure as circular plates) are in great part open frames, the 
 upper one being such for the admission of light from objects 
 near the Pole. The lower one carries the hour-circle gradua- 
 tion. These terminals are joined by the columns pp, pp, and 
 the opposite columns qq, qq, so that the Telescope EA, with the 
 connected circle of declination dd, may be turned between the 
 two sets of columns about a transverse axis supported by them. 
 The pivot at one end of this axis (that not seen in the figure) 
 rests in a Y fixed in a cross-piece joining the columns qq, qq 
 and that at the other end in a Y adjustible by being moved 
 transversely to the cross-piece x, and fixed in position by the 
 screw s. E is the eye-end and A the object-end of the Tele- 
 scope ; m, m are the positions of two opposite micrometer-micro- 
 scopes by which the declination-circle is read off, which are of 
 the same construction as the microscopes of the Mural Circle ; 
 and n, n are the positions of two opposite microscopes for 
 reading off the Hour-circle. To one of the microscopes of each 
 set a pointer is attached for setting the instrument in declina- 
 tion and hour-angle. Slow motion is given to the declination- 
 
 172 
 
260 PKACTICAL ASTRONOMY. 
 
 circle by tangent-screw-and-clamp apparatus attached to the 
 polar frame at T y and worked by a handle at the command of 
 the observer. Analogous arrangements are made for slow 
 motion of the Hour-circle. 
 
 269. The graduation-lines of the declination-circle are 5' 
 apart, and, for convenience, the pointer is so placed with 
 reference to the zero of graduation, that its reading is the 
 approximate declination or polar distance of the object. The 
 upper face of the Hour-circle is graduated in hours, which 
 are subdivided into minutes, and the graduation-lines on the 
 edge, which are looked at with the microscopes, are 20 s apart, 
 one micrometer revolution is 4 s , or one-fifth of a graduation- 
 interval, and each second is divided on the micrometer-head 
 into 10 parts. Thus the hour-circle reading, which gives the 
 hour-angle of the object from the meridian, consists of a pointer 
 reading suitably adjusted, the number of integral revolutions of 
 4 s indicated by indentures of the comb, and the fraction of a 
 revolution in seconds and tenths of a second, read from the 
 micrometer-head. In the eye-piece of the Telescope, which is 
 an achromatic refractor of the usual construction, are a double- 
 wire micrometer, one equatorial fixed wire, and five vertical 
 wires. The required adjustments of the wires and wire-frame 
 are effected by screws nearly as in the case of a Transit-circle. 
 [For additional particulars respecting the Cambridge five-feet 
 Equatorial, see p. xxi. of the Introduction to Vol. v. of the 
 Cambridge Observations (for 1832)]. 
 
 270. To assist in pointing the Telescope to objects, an 
 Equatorial, especially if it be of large size, requires to be 
 furnished with a Finder, which is a telescope of small aperture 
 and large field, having its axis parallel to that of the large one, 
 and attached either to the telescope-tube, or to the connected 
 declination-circle. The Finder is sufficiently adjusted if an 
 object seen at the centre of its field is seen at the same time at 
 or near the centre of the field of the telescope. 
 
 271. The Illumination of the Telescope's field of view is 
 effected by suspending a lamp in gimbals opposite one end of the 
 transverse axis, at which a lens is fixed for throwing light from 
 
THE EQUATOKIAL. 
 
 261 
 
 the lamp on a ring reflector placed at the middle of this axis, 
 and inclined by an angle of 45 to the axis of the telescope. 
 Before incidence on the lens the light passes through an ad- 
 justable opening for regulating the quantity. 
 
 272. In case there should not be room for the observer's 
 head when looking through the telescope at an object in the 
 neighbourhood of the Pole, a diagonal eye piece is substituted 
 for the ordinary Rasmden eye-piece, to give the means of looking 
 at the image in a direction at right angles to the axis of the 
 telescope. The simplest form of the diagonal eye-piece is that 
 with two lenses described in Art. 23. The form usually adopted 
 for an Equatorial has four glasses, the shapes and arrangement 
 of which are the same as those of the four-glass erecting eye- 
 piece mentioned in Art. 22. As no description of the erecting 
 eye-piece was there given, I shall now describe it in detail, before 
 proceeding to give an account of the four-glass diagonal eye- 
 piece. Fig 41 exhibits the course, through the four glasses, of a 
 
 Fig. 41. 
 
 pencil of rays having its origin at r, which may either be a 
 point of an image formed at the geometrical focus of the object- 
 glass of the telescope, or any point of a wire at the same focal 
 distance. The pencil, after passage through the first lens, 
 crosses the axis of the eye-piece at ss, where it is limited by 
 having to pass through a small circular hole in a diaphragm in 
 that position. Then after passing through the second lens, by 
 which it is made a convergent pencil, it is refracted by the 
 third lens, so as to form an image of r at the point / of a field 
 which is bounded by the diaphragm ss. This image is seen by 
 the eye at ee by rays that are made suitable for vision by 
 passing through the fourth glass. As the image of the point r 
 below the axis is formed at the point r above the axis, the 
 inverted image formed at the focus of the object-glass is made 
 erect by this eye-piece. 
 
262 
 
 PRACTICAL ASTRONOMY. 
 
 273. The construction of the four-glass diagonal eye-piece 
 may be explained as follows by reference to Figure 42. The 
 first lens is at that end of the broken tube GFE which is 
 inserted in an outer tube terminating at G in a screw by means 
 
 Fig. 42. 
 
 of which the eye-piece is attached to the telescope. At the 
 position ss (corresponding to that marked by the same letter in 
 Fig. 41), there is a diaphragm with a small circular hole for 
 limiting all the pencils, its centre being on the axis. A pencil 
 of rays from any point of an image at the focus of the object- 
 glass, after passing through the first lens, the aperture in the 
 diaphragm at ss, and the second lens a little further on (see 
 Fig. 41), is incident on the reflector rr inclined at an angle of 
 45 to the axis, and being thereby turned through 90 from its 
 original direction, passes through the two-glass eye-piece FE, 
 forming an image between the glasses which is viewed through 
 the eye-glass at E. The adjustment to distinct vision of the 
 telescope-wires is effected by shifting the eye-piece bodily from 
 or towards the wire-frame by a movement of the inner tube 
 within the outer tube with reference to the fixed position G, 
 and by moving the eye-piece FE inwards or outwards in the 
 direction of the axis. 
 
 Ramsden's Refraction-piece, an example of which is attached 
 to the eye-end of the Telescope of the Cambridge five-feet 
 Equatorial, will be most suitably described when the corrections 
 for refraction out of the meridian are under consideration. 
 
THE EQUATORIAL. 263 
 
 II. THE ADJUSTMENTS OF AN EQUATORIAL. 
 
 274. If an Equatorial be only used for taking differential 
 observations (Art. 266), it will generally suffice to make an 
 initial determination of the instrumental errors, and thereby 
 correct them mechanically, without repeating the operations 
 unless there be reason to suppose that the residual errors have 
 considerably changed in course of time, or through incidental 
 causes. If the object and the star with which it is compared do 
 not differ greatly in R.A. and P.D., it may be presumed that 
 the positions of both are for a long time affected to the same 
 amount by changes of the instrumental errors, if the errors be 
 originally small, and the instrument be steadily mounted. It 
 is, however, an essential condition that it should never be 
 largely out of adjustment ; on which account the values of the 
 errors ought to be determined from time to time. Such deter- 
 minations can all be made by means of observations of stars, 
 and the errors so ascertained might be corrected by mechanical 
 means. If small instrumental errors should be allowed to re- 
 main un corrected, their effects on the observations of KA. and 
 P.D. will have to be got rid of by applying corrections calculated 
 according to appropriate formulae. In this way celestial places 
 may be independently determined by Equatorial observations. 
 I propose to obtain formula proper for thus correcting instru- 
 mental errors by calculation, as well as to shew how to correct 
 them mechanically. I shall also take occasion to indicate 
 processes by which certain of the errors may be corrected 
 exclusively by mechanical means. 
 
 275. The adjustments of an Equatorial by means of stars 
 will first be considered, and the different processes will have to 
 be indicated in a certain logical order. 
 
 (1) The first operation is to correct the Index Error of 
 the declination-circle. In Figure 43 (1), (711 represents the 
 position of the axis of motion of the instrument, and C is a 
 point through which the axis of the declination-circle may be 
 
264 PRACTICAL ASTRONOMY. 
 
 supposed to pass ; for if the two axes do not actually intersect, 
 the reasoning will hold good if the declination-circle be conceived 
 to be transferred bodily to a position in which its axis will pass 
 through C. Also this axis may be assumed to be at right 
 
 Fig. 43. 
 
 12) 
 
 angles to CH on the principle that a small uncorrected de- 
 viation from that direction will not affect the determination of 
 the index error. This being understood, suppose the circular 
 arc o-OIl, which is in the plane containing CH and the axis 
 Co- of the Telescope when directed to a certain star, to be a 
 portion of a complete circle marked by exactly the same 
 graduation as that of the declination-circle. Then, if the mean 
 of the two microscope readings be either zero, or 180 from 
 zero, when the Telescope points in the direction CO, the index 
 error to be determined is the arc OH, supposing that the circle- 
 reading is required to be 0, or 180, when the pointing is in 
 the direction CH parallel to the axis of motion. In the case 
 of the Cambridge instrument, the graduation proceeds from 
 to 360, and the Circle-reading is 180 when the graduated face 
 of the declination-circle is westward and the Telescope points 
 in CO, the reading increasing as the Telescope is moved from 
 that position southward. To obtain a measure of OH the 
 Telescope is first directed to a star (which should be near the 
 meridian), and the circle is read off; and then, after turning 
 the instrument through half a revolution by the hour-circle- 
 graduation in order to reverse the declination axis, the Telescope 
 is pointed to the same star, and the circle is again read off. 
 
THE EQUATORIAL. 265 
 
 Let 0' be the position to which is brought by the reversion, 
 so that Oil = O'll ; and let the measure of the arc a-0, inferred 
 from the first reading, be D, and that of the arc aO', inferred 
 from the second reading be D r . Then the measure of the index 
 
 error Oil is ^ (D' J9). By this quantity both microscope 
 
 readings require to- be corrected to get rid of index error, the 
 sign of the correction being determined by the relative mag- 
 nitudes of D and D'. The correction is made mechanically by 
 altering the readings of the microscopes to the required amount 
 by screws provided for that purpose (see Arts. 175 and 176). 
 The microscope readings are farther to be corrected so that in 
 some one position of the instrument they are made to bisect 
 opposite lines of graduation. To do this, take the microscope 
 readings for the selected position, and supposing 180 + e to be 
 the algebraic excess of the reading of that microscope which in 
 the order of the graduation is in advance of the other, correct 
 
 /> 
 
 (by the screws above-mentioned) the former reading by ^, 
 
 and the latter by + | . Then as the mean microscope reading 
 
 will not thereby be changed, the index error will remain 
 corrected as before, and the microscope readings will be just 180 
 apart. This will generally not be the case for other positions 
 of the instrument ; bat as the effect of a displacement of the 
 circle is quam proxime to increase one microscope reading just 
 as much as it diminishes the other, the mean microscope- 
 reading will always be nearly free from instrumental error. 
 
 When the graduation of the Circle proceeds (as in the 
 instance of the Cambridge Equatorial) from to 360, the 
 index error might be left uncorrected and of any amount, and 
 be corrected in the reduction of the observations as in the case 
 of a Transit-circle. For the meridian instrument such index 
 errors are eliminated by means of determinations of the " zenith 
 point," whereas in the use of an Equatorial, they would have to 
 be corrected by observations of known stars. It is usual, how- 
 ever, for greater convenience, to correct the index error me- 
 
266 PRACTICAL ASTEONOMY. 
 
 chanically, so that a Pointer attached to one of the microscopes, 
 and agreeing in its indication with the reading of that microscope 
 to the accuracy of 1' or 2', may be used for setting immediately 
 for an object according either to declination or polar distance. 
 
 (2) The next step is to make the angular elevation of 
 the axis of motion equal to the altitude of the Pole of the 
 heavens. For simplicity it will be supposed that after actual 
 or virtual correction of index error, the mean Circle-reading is 
 when the Telescope is pointed in a direction parallel to that 
 axis. Then, if a known star be bisected when it is on, or very 
 near, the meridian, as indicated by the sidereal clock (the error 
 of which has been ascertained by comparisons with the transit- 
 clock), the mean Circle-reading will be the apparent polar 
 distance of the star affected only by error of the angular 
 elevation of the polar axis, and the amount of refraction. The 
 true polar distance of the star and the colatitude of the place of 
 observation being known, the refraction can be calculated (as sub- 
 sequently shewn) with true zenith distance for argument, and the 
 Circle-reading be corrected accordingly. The algebraic excess 
 of the true polar distance above the result thus obtained, which 
 may be called the instrumental polar distance, is the correction 
 to be applied to the latter for error of elevation of the polar 
 axis. For greater certainty, the adopted correction might be 
 the mean of values obtained by observations of several known 
 stars. The mechanical correction would be effected by turning 
 the lower screw b (Fig. 40) on which the instrument bears, in 
 the direction indicated by the sign of the correction, and to the 
 required amount, and then fixing the position of the pivot by 
 bringing the opposite screw into bearing. The amount of 
 revolution to be given to the screw for correcting a given error 
 might be calculated by a simple proportion after measuring the 
 interval, parallel to the axis of the screw, occupied by an 
 integral number of threads, and the distance between the two 
 pivots. (See Art. 171.) 
 
 (3) For finding the deviation of the axis of revolution 
 from the plane of the meridian, the polar distance of a known 
 star is to be observed when the R.A. differs from the corrected 
 
THE EQUATORIAL, 267 
 
 sidereal time of the clock by 6 h , or its hour-angle from the 
 meridian is 90. In that case the observation gives the instru- 
 mental polar distance with sufficient exactness whether or not 
 the error of position of the axis in altitude be mechanically 
 corrected. But the measurement is affected by the apparent 
 displacement of the star in a vertical plane caused by refraction, 
 and the correction in polar distance required on that account 
 has to be calculated according to a formula the investigation of 
 which will be given in a subsequent article. The algebraic 
 excess of the true above the instrumental polar distance thus 
 corrected is the quantity which measures the angular deviation 
 of the axis of revolution from the plane of the meridian, the 
 sign of the excess indicating the direction of the deviation. 
 The error in amount and direction being thus known, the 
 mechanical process by which the polar axis is brought into the 
 plane of the meridian and fixed by means of the screw a 
 (Fig. 40) and its antagonist, is exactly analogous to that 
 indicated above for the correction in altitude. 
 
 (4) The next operation is to determine the error of 
 collimation of the Telescope. For this purpose equatorial stars 
 are to be selected, as the determination would otherwise be 
 affected by error of position of the declination-axis, the mode 
 of correcting which has yet to be indicated. The transit of an 
 equatorial star having been taken and recorded for an arbitrary 
 position of the plane of collimation (not far from the meridian), 
 the Hour Circle is to be read off. Then after reversing the 
 Telescope by changing the graduated face of the declination 
 circle with respect to eastward and westward, and clamping the 
 instrument, a transit of the same star is to be taken and the 
 Hour Circle again read off. Let T^ be the interval between the 
 transit-times, and T^ the difference of the Hour Circle readings. 
 Then supposing the plane of collimation to deviate too much 
 towards the east in the first position, the times recorded before 
 the reversion will be too early by a certain quantity, and that 
 recorded after the reversion will be as much too late. Hence T t 
 will be greater than I 7 /, the latter being the hour-angle through 
 which the instrument is turned from one position to the other, 
 
268 PRACTICAL ASTRONOMY. 
 
 and the difference T : I 7 / will measure double the error of 
 collimation. Hence the required correction in arc is 
 
 whether T t be greater or less than T^ if the upper sign be 
 taken to apply to a position the same as that which the instru- 
 ment had before the reversion, and the lower sign to the reverse 
 position. This error is to be corrected by moving the wire- 
 frame. 
 
 (5) The method of correcting by star-observations the 
 position of .the declination-axis, may be explained by reference 
 to Figure 43 (2). Let C be the intersection of the corrected 
 plane of collimation with the axis of motion of the declination- 
 circle ; through C conceive to be drawn a plane parallel to the 
 earth's equator, cutting the plane of collimation in A CA' ; and 
 through the same point C draw CP parallel to the earth's axis. 
 Then the plane of collimation, being required to pass through 
 the pole of the heavens, should coincide with the plane of the 
 arc APA. Suppose that it actually coincides with the plane 
 of the arc ApA. Then the error of position of the declination- 
 circle which has to be corrected is the small spherical angle 
 at A ; or, if a plane through C at right angles to AC A cut the 
 plane ApA' in Cp, it is the small angle PCp, and is measured 
 by the arc Pp. The value of the angle is obtained by observing 
 as follows. Transits of a star are taken, and the Hour Circle is 
 read off, just as for determination of collimation-error. (See (4) 
 above.) Let T a be the interval between the two transits, and 
 T the difference of the Hour Circle readings. Then o-n repre- 
 senting a small portion of the diurnal path of the star, the angle 
 crPn, due to the error of position of the declination-axis, is equal 
 
 in arc to (T 2 T). Let the required arc Pp be a, and the 
 star's polar distance Per be S. Then 
 
 L aPn an cosec 8 = Pp cos 8 x cosec 8 = a. cot S. 
 
 Hence a = L aPn x tan S = (7; - T 9 ") tan 8. 
 
THE EQUATORIAL. 269 
 
 A star should be selected for which the factor tan 8 is not large, 
 otherwise the result may be injuriously affected by errors in the 
 observed values of T 2 and T 2 '. In general the value of S should 
 be less than the colatitude of the Observatory, or the transits be 
 taken between the zenith Z and the pole P. Just as in the 
 case of correction of collimation error, the required correction 
 
 is + -Q- (T a T') tan 8, according as the position of the instru- 
 tt 
 
 ment is the same, or not the same, as that it had before the 
 reversion. The amount and sign of the correction being thus 
 found, the correction may be effected mechanically by means of 
 a screw acting transversely to the declination-axis at one end of 
 it, as indicated by the screw s in Figure 40. 
 
 (6) The effect of the foregoing adjustments being to 
 make the plane of collimation pass through the pole of the 
 heavens, the last adjustment is to correct the index-error of the 
 Hour Circle. The two microscopes (n, n in Fig. 40, p. 258) 
 point to the opposite ends of the horizontal diameter of the 
 Circle, the graduation proceeds from O h to 24 h , and the micro- 
 scope-readings are supposed to increase as the instrument is 
 turned eastward. Let the reading of the East microscope be 
 set to its zero, as indicated by the hole of the comb and the 
 graduated micrometer-head, and let the instrument be turned 
 about the polar axis by the slow-motion handle till the gradu- 
 ation-line corresponding to O h is bisected by the cross-wires of 
 the microscope. It will be assumed that the position and 
 pointing of the microscope has already been so adjusted, that the 
 plane of collimation is thus made to be nearly coincident with 
 the plane of the meridian. Under these circumstances, let the 
 transit of a known equatorial star be taken, and after obtaining 
 the error of the clock by means of chronometer reference to the 
 transit- clock, let the sidereal time of the transit be calculated. 
 The difference between this time and the apparent RA. of the 
 star measures the distance of the plane of collimation from the 
 meridian. For instance, if the sidereal time be less than the 
 star's RA. by the interval /3 s , the plane of collimation requires 
 to be turned westward through /3 s to be placed in coincidence 
 
270 PRACTICAL ASTEONOMY. 
 
 with the meridian. This may readily be done with either 
 microscope by first bisecting a graduation-mark by the cross- 
 wires, and after diminishing the reading by turning the micro- 
 meter-head through ft, causing by the slow-motion handle the 
 same mark to be again bisected. Then, supposing the graduation 
 of the declination-circle to be westward, after setting each 
 microscope to its zero-reading, the east one is to be pointed 
 exactly to O h , and the west one exactly to 12 h , by means of the 
 screws provided for adjusting the pointings of the microscopes. 
 By these processes the index error is corrected, and at the same 
 time the microscope-readings are adjusted. After this the mean 
 microscope-reading measures the deviation of the plane of 
 collimation from the meridian eastward. 
 
 Mechanical Adjustments of an Equatorial. 
 
 276. The adjustment of the direction of the axis of revolu- 
 tion of the instrument, as having reference to the Pole of the 
 heavens, is necessarily performed by means of observations of 
 stars. The other adjustments have reference either to the 
 earth or the instrument, and can consequently be effected 
 either by stars, as already shewn, or by mechanical means. It 
 is convenient to be furnished with means of applying the latter 
 method in order to be readily able to secure that the instrument 
 shall at no time be largely out of adjustment. 
 
 (1) The axis of the Declination-circle has to be placed 
 so as to be coincident with, or parallel to, a straight line cutting 
 the polar axis at right angles. 
 
 (a) To satisfy this condition use is made of a swinging, 
 or hanging, spirit-level, which is held in position by two parallel 
 supports fixed to the nave of the declination-circle (see Fig. 40, 
 p. 258). Hitherto we have only had under consideration the 
 striding level (Arts. 67 79); it is now required to describe the 
 swinging spirit-level, and the arrangements whereby it is ap- 
 plied for adjusting the declination-axis of an equatorial. This I 
 proceed to do by reference to Figure 44, (1), (2), (3). In (1) is 
 
THE EQUATOKIAL. 
 
 271 
 
 represented the brass tube ab, containing the level and movable 
 about two cylindrical pivots e and e ', which are fixed to the 
 two supports so as to have their axes in coincident directions. 
 The tube has at its two circular ends two Y's, which rest on the 
 pivots in the manner shewn in (3). The positions of these Y's 
 
 Fig. 44. 
 
 are eccentric with respect to the ends of the tube, so that the 
 part ba is always downwards. A scale for reading off the 
 bubble-ends is attached to the brass tube by means of screws ff t 
 which fix the scale in position after it has been shifted in the 
 direction of the axis of the tube for adjusting the level-readings. 
 For making this adjustment it is required to be able to reverse 
 the level. This can be done after drawing out the rod h (3), 
 which detains the pivot e within the limits of the aperture 
 when the instrument is being turned about its axis. The tube 
 of the level may then be reversed, and the readings of the 
 bubble-ends for an arbitrary position of the scale may be 
 determined for the case in which the axis of motion of the tube 
 is horizontal, in the manner described in Arts. 69 and 70. It 
 will be convenient to adjust the scale in that case so as to make 
 the two readings equal, care having been previously taken to 
 place the instrument so that the bubble is not far from the 
 mid-position relative to the exposed part of the glass. 
 
 (ft) After this adjustment, turn the Telescope till the 
 Level is immediately above the declination-axis, move the 
 Polar Frame till the axis of motion of the Level is placed 
 horizontally by means of the scale-readings of the bubble-ends, 
 and clamp the Hour Circle. That axis being now in the 
 
272 PRACTICAL ASTRONOMY. 
 
 position AB [Figure 45 (1)], bring it to A'B' by turning the 
 Telescope in declination through 180. Then AB and A'B are 
 in a vertical plane and make equal angles with the declination- 
 axis Oc, and the movement of the bubble measures the angle 
 A OA' t which is double the angle of deviation of Oc from the 
 horizontal direction. Hence the bubble must be brought half- 
 way back by the screw s [Figure 44 (1) and (2)], which changes 
 the height of the pivot e [as shewn in (2)], by which means the 
 
 level-axis is placed parallel to the declination -axis. The aper- 
 ture in which the pivot is caused to move is covered by a 
 movable plate d. 
 
 (7) The Level-axis and Declination-axis being now 
 parallel, place the latter in a horizontal position by the Level, 
 and read off the Hour Circle. Turn the Polar Frame half 
 round till the declination-axis is again horizontal, read off the 
 Hour Circle again, and then make the Hour Circle reading to 
 differ just 12 h from the first reading. Suppose that by these 
 operations the declination-axis is moved from its original posi- 
 tion ab to a'b' [Figure 45 (2)]. Then since the two positions are 
 thus made to incline equally in opposite directions to the polar 
 axis OP, the dotted line ad joining the two places of any point 
 a of the declination-axis is perpendicular to OP, and the error 
 of position of the axis in the second position is the angle aa'b' ', 
 which is the half of the angle aob'. Hence the requisite correc- 
 tion is made by placing the instrument in a position inter- 
 mediate to the two last, as inferred from the Hour Circle 
 readings for these positions, and then putting the bubble in 
 mid-position by the adjusting screw s (Fig. 40) of the declina- 
 tion-axis. Any future want of adjustment will then be indicated 
 by the scale-readings. 
 
THE EQUATORIAL. 273 
 
 (2) The Collimation Error may be corrected by means 
 of a mark near the meridian as follows. The declination-axis 
 being adjusted, bisect the mark by the middle wire, and read 
 off the Hour Circle. Reverse the Telescope, by reversing the 
 instrument from graduation West to graduation East or the 
 contrary, bisect the mark again, and read off the Hour Circle. 
 Then by means of the difference of the Hour Circle readings, 
 point the Telescope in a direction intermediate to the two 
 former directions, and by the wire-frame adjustment make the 
 middle wire bisect the mark. The error of collimation of the 
 middle wire is then corrected. 
 
 (3) Lastly, the Index Error of the Hour Circle may be 
 corrected, after all the foregoing adjustments have been per- 
 formed, by first placing the declination-axis horizontal by the 
 swinging Level, whereby the plane of collimation is brought 
 into the plane of the meridian, and then by the proper ad- 
 justing screws (Art. 272 (6)) making the Hour Circle reading 
 zero. 
 
 277. Supposing all the adjustments to have been made as 
 exactly as possible by means of stars, if the plane of collimation 
 be placed in the meridian, and the Telescope be pointed by the 
 declination- circle to a polar distance equal to the colatitude 
 + 90, the line of collimation will be directed due" southward, 
 and will be horizontal. The Cambridge Five-feet Equatorial 
 has a swinging Level attached to the Telescope-tube, with its 
 axis of motion approximately parallel to the axis of the tube. 
 As this level, like the one pertaining to the declination-circle, is 
 capable of reversion (Art. 276 (1) (a)), let the adjustment of its 
 axis to horizontality be performed when the Telescope's line of 
 collimation points, as just mentioned, horizontally southward. 
 The level-axis will thus be placed parallel to the line of 
 collimation. After making this arrangement, the level-scale 
 might be adjusted so that the readings for the bubble-ends are 
 equal, or the bubble is put in mid-position. Then if at any 
 future time the line of collimation be pointed horizontally 
 southward by the graduation of the declination-circle, and the 
 bubble be seen to have shifted from its mid-position, this will 
 C- 18 
 
274 PRACTICAL ASTRONOMY. 
 
 indicate that the angular elevation of the polar axis has 
 changed, and may require re-adjustment. 
 
 278. The determination of the position of the polar axis by 
 the method given in Art. 275 (2) (3) requires the use of a 
 graduated declination-circle. If the Equatorial has no declina- 
 tion-circle, and only a graduated sector (which is the case of the 
 Northumberland Equatorial, as there will be occasion to men- 
 tion subsequently), the error of position of the polar axis may 
 be found by the following process, by means of an equatorially 
 adjusted micrometer-wire. In Figure 46, Z is the zenith, P the 
 
 pole of the heavens, II the instrumental pole, a the place of a 
 star, ZPv(6) its hour-angle reckoned positive towards the west, 
 and IIw, Hq are respectively perpendicular to ZP and Per. It 
 is required to find the coordinates Pn(x) and TLn(y) of II. Let 
 PII = /o and tnPn = a. Then Pq = Pft cos ^ UPq = p cos 
 (0 __ a ) = p cos a cos + p sin a sin 6 = a cos 6 + y sin 0. For an- 
 other hour-angle (6') of the same star, the Telescope being 
 clamped so as to prevent change of instrumental polar distance 
 Pq = x cos 6' + y sin #', and for a third hour-angle (0"), Pq" = 
 x cos 0" + y sin 0". Then, Hq being drawn perpendicular to Per, 
 qcr is very approximately the instrumental polar distance, and 
 Pq' _ Pq is the difference of the instrumental polar distances in 
 the first and second positions. This difference, which would 
 vanish (but for refraction) if II coincided with P, is measured 
 by the difference of the micrometer-readings for those positions. 
 Let, therefore, the values of Pq' Pq and Pq" - Pq when con- 
 
- 
 
 THE EQUATORIAL. Z / O 
 
 verted into arc, and corrected for refraction (by a process to be 
 given subsequently), be respectively a. and /3. Then we have 
 
 a = x (cos 0' - cos 0)+y (sin 0' - sin 0), 
 P = a; (cos 0" - cos 0)+y (sin 0" - sin 0). 
 
 These two equations give for calculating x and y, 
 
 15 cos i (0' + 0) sin i (0' - 0) - a cos | (0" + 0) sin | (0" - 0) 
 
 Z Z 2i 
 
 2 sin i sin (0" - 0) sin i (0" - 0') sin | (0' - 0) 
 
 sin | (0' + 0) sin \ (O f - 0) - a sin | sin (0" + 0) sin \ (0" - 0) 
 Z . 25 Z ^ 
 
 y= T =1 ? > 
 
 2 sin ^ sin (0" - 0) sin ^ (0" - 0') sin (0' - 0) 
 
 a? being the angular deviation of the pole of the instrument from 
 the Pole towards the south, and y that towards the west. 
 
 279. For taking observations of Right Ascension and Polar 
 Distance with an Equatorial, the following adjustments and 
 determinations are also required. (1) To adjust the fixed trans- 
 verse wire equatorially, small stars near the equator are made, 
 by screw-adjustment, to traverse the wire so as to be bisected by 
 it from end to end. (2) Readings of the two micrometer-wires 
 for coincidence with the fixed wire are taken at the five vertical 
 wires to correct for defect of their parallelism to that wire. 
 
 (3) The equatorial intervals between the vertical wires are 
 determined by transits of Polaris, after bringing the plane of 
 collimation to coincide nearly with the plane of the meridian. 
 
 (4) The value in arc of one revolution of the eye-piece double 
 micrometer is found by means of transits of Polaris. The modes 
 of performing these operations need not be stated here, since 
 they have been given in full detail in treating of the Transit 
 and Mural Circle, or the Transit- Circle. (For examples of 
 numerical calculations of equatorial adjustments, and determina- 
 tions of micrometer-revolutions, see the Cambridge Observa- 
 tions, Vol. XI. pp. Ixxiii Ixxvi. and Yol. xn. pp. xxix xxxvi.) 
 
 182 
 
276 PRACTICAL ASTRONOMY. 
 
 III. METHODS OF OBSERVING WITH THE EQUATORIAL, AND 
 ELIMINATION OF INSTRUMENTAL ERRORS. 
 
 280. The Apparent Eight Ascensions and Polar Distances 
 of celestial objects may be obtained by the use of an Equatorial 
 either directly, or by the intervention of stars of comparison. I 
 propose to adduce both methods, giving, first, the processes for 
 determining the places of objects independently of known 
 places of stars. First, the instrument has to be set for observing 
 the given object. It will be supposed that the right ascension 
 and polar distance of the object are, at least, approximately 
 known; which will be the case except under circumstances that 
 will be subsequently mentioned. The setting for observing in 
 P.D. is effected by the graduation of the declination-circle and 
 the index coincident in position with A, one of the two micro- 
 scopes. The setting for an observation in E.A. to be taken at a 
 chosen epoch of mean time is performed by means of the eastern 
 microscope of the hour-circle, and the known error of the clock 
 on sidereal time. By subtracting the calculated sidereal time 
 of epoch from the approximate E.A. of the object, its approxi- 
 mate hour-angle eastward at that time is obtained. If the 
 hour-circle index be set to this hour-angle, the setting in 
 declination secures that the object shall pass nearly across the 
 middle of the field, and the observation can be taken both in 
 E.A. and P.D. if the observer has been careful to set the instru- 
 ment a convenient interval before the object enters the field. 
 The process of taking the observation is just the same as that 
 for an observation with a transit-circle. The times of transit 
 across the vertical wires are recorded, and the object is bisected 
 by the fixed wire, by slow movement of the declination-circle, or 
 optionally by one of the two micrometer-wires. By use of the 
 three wires several objects following in quick succession, and of 
 approximately the same P.D., may be observed without move- 
 
THE EQUATORIAL. 277 
 
 ment of the declination-circle for a fresh setting. The readings 
 of the two declination-microscopes are then recorded, just as in 
 Circle-observations, as are also the micrometer readings (if any), 
 and the readings of the two hour-circle microscopes. 
 
 281. The Runs of the declination-microscopes have to be 
 taken from time to time, as well as, on the same principles, those 
 of the hour-circle microscopes, in order to apply in both cases 
 corrections for Runs in the reduction of the observations. Also 
 errors of hour-angle and polar distance, due to ascertained error 
 of position of the polar axis, have to be corrected in the 
 reduction of the observations. Formulae proper for calculating 
 the amounts of these corrections may be obtained as follows. 
 In Figure 46, cr is the place of the object, Pq is error of position 
 of the polar axis in P.D., being equal to Pa- Her ; and if Iler 
 and Pa- be produced so as to cut the Equator A Q in m' and m, 
 mm is the error of hour-angle. As in Art. 278, 
 
 Pq = x cos 6 -f y sin 0, 
 
 x and y being the ascertained errors of the position of II in 
 altitude and azimuth. The value of Pq given by this formula 
 is the quantity to be added to the observed P.D. to obtain the 
 true P.D. Also 
 
 II<7 = p sin (d a) = p cos a sin 6 p sin a cos 6 x sin 6 y cos 0. 
 
 Let Pa = S. Then the error in hour-angle, mm', is equal to 
 L mam' cos S, or Ilq cot 8. Hence mm (in time) 
 
 = -T - (x sin 6 y cos 0). 
 Xo 
 
 Since when x and y are both positive, and the hour-angle is 
 positive, or cr is on the west side of the meridian, the point m 
 is more westward than m, it follows that the above quantity is 
 to be added to the R.A. by the observation to obtain the 
 true R.A. 
 
 282. After applying the above corrections for the error of 
 position of the polar axis, in case there may still be residual 
 error requiring correction, it is proper in every instance to 
 
278 PEACTICAL ASTRONOMY. 
 
 observe the object in reverse positions of the declination-circle, 
 that is, with graduation westward and graduation eastward; 
 and one observation should be separated from the other by as 
 small an interval of time as possible. As this method of 
 observing in reverse positions tends to correct, by mean results, 
 instrumental errors in various ways, it ought always to be 
 adopted when the observer proposes to determine the places of 
 objects by direct instrumental means. 
 
 283. In like manner formulae might be obtained for 
 calculating corrections of observed R.A. and P.D. for a given 
 error of position of the axis of the declination-circle. But it 
 is unnecessary to do this, because if this error of position be 
 corrected mechanically in the manner described in Art. 275 (5), 
 and if the rule of observing in reversed positions of the declina- 
 tion-circle be also. attended to, a degree of accuracy is attained 
 not inferior to that which would result from applying the above- 
 mentioned corrections. So for a given error of collimation, 
 obtained as indicated in Art. 275 (4), corrections might be 
 calculated and applied just as in meridian-transits. But if the 
 mean of the RA. obtained, within a small interval, in reverse 
 positions of the Telescope, be adopted, the effect of collimation- 
 error will be very nearly eliminated. The only other instru- 
 mental corrections are those for reduction to the mean of the 
 wires for omitted wires in taking the transits, and the correc- 
 tions for Runs both in the readings off of the declination- 
 microscopes and the hour-circle microscopes. The principles 
 and the methods of applying the corrections of both kinds may 
 be gathered from the explanations given in Arts. 116 126, and 
 in Arts. 177 179, and need not be repeated here. When all 
 the foregoing corrections have been carefully determined and 
 applied, it is found that celestial places may be obtained inde- 
 pendently by an Equatorial with a great degree of accuracy. 
 
 284. It remains to give account of the method of observing 
 apparent RA. and P.D. with an Equatorial by differential 
 observations, that is, by comparisons with the RA. and P.D. of 
 a known star. This is the usual method of obtaining celestial 
 places by an Equatorial, being, in fact, more accurate than the 
 
THE EQUATORIAL. 279 
 
 independent method, inasmuch as it is more free from errors 
 incident to the oblique position of the polar axis, and the place 
 of the comparison-star may be ascertained at any time with 
 as much accuracy as may be desired by means of meridian 
 observations. The difference between the R.A. of the selected 
 comparison-star, and the approximately known R.A. of the 
 object compared, may be of any convenient amount ; but the 
 difference of their P.D. should be small, not, in general, ex- 
 ceeding two or three degrees. For observing the unknown 
 object at or about a proposed mean, or sidereal, time, the 
 instrument is set in the manner described in Art. 280, regard 
 being had to the difference between its R.A. and that of the 
 star. The hour-circle is then clamped to prevent movement 
 about the polar axis, and the difference of the R.A. of the two 
 objects, the difference of their P.D. being small, is considered 
 to be measured with sufficient accuracy by the difference of 
 the times of their transits across the mean of the wires, when 
 bisected, one after the other, by the fixed wire. 
 
 285. The difference of their P.D. is obtained either by 
 reading off the declination-circle and correcting for Runs, or 
 by a measurement made by a micrometer-wire. In the former 
 case the preceding object is bisected, as it crosses the mid- wire, 
 by the fixed wire by means of the slow-movement in declina- 
 tion, the graduation-lines are then bisected by the cross- wires 
 of the two microscopes (as in transit-circle observations), after 
 which the declination-circle is moved for taking a like observa- 
 tion in R.A. and P.D. of the following object. The microscope- 
 bisections are then read off, and new bisections are made and 
 read off for the second observation. The observation is com- 
 pleted by recording the position of the graduation of the 
 declination-circle as to eastward or westward, it being unneces- 
 sary to read off the hour-circle, because the apparent hour- 
 angle, being the difference between the known R.A. of the 
 comparison-star and the sidereal time of its transit across the 
 mean of the wires, is always derivable from the observation. 
 In the other case, that of the difference of P.D. being so small 
 as to allow of its being measured by one of the micrometer- 
 
280 
 
 PKACTICAL ASTRONOMY. 
 
 wires, the preceding object is bisected, as before, by the fixed 
 wire, and the following object by the micrometer- wire as it 
 crosses the mid-wire. The difference between the micrometer- 
 reading for the bisection, and that for coincidence with the 
 fixed wire at the mid-wire, gives, after conversion into arc, the 
 apparent difference of P.D. This method, as requiring no 
 movement of the declination-circle, is likely to be more accurate 
 than the other, and should be employed whenever the difference 
 of P.D. is not too large for safe measurement with the micro- 
 meter. Even in cases of determining places of objects by 
 differential observations, it is advisable to take observations in 
 reverse positions of the instrument, as residual errors are 
 thereby likely to be eliminated. The above described differ- 
 ential observations are much used for determining initial places 
 of newly discovered planets and comets. 
 
 IV. FINAL CALCULATION OF RIGHT ASCENSIONS AND POLAR 
 DISTANCES FROM EQUATORIAL OBSERVATIONS. 
 
 286. After getting rid, as accurately as possible, of all 
 effects of instrumental errors by the processes described in 
 Section III., it is required to apply in all instances corrections 
 for refraction in R.A. and P.D., and in the case of moving 
 bodies to correct also for local parallax in KA. and P.D. 
 
 Fig. 47. 
 
 Formulae for calculating the corrections for refraction in inde- 
 pendent observations may be obtained as follows by reference 
 to Figure 47 (1). Z is the zenith, P the pole of the heavens, 
 8 the place of object, PS its true polar distance, and ZQ is an 
 
THE EQUATORIAL. 281 
 
 arc cutting PS at right angles. Let S' be the place of the 
 object as altered by refraction, so that SS' is the vertical refrac- 
 tion. The observations, freed from instrumental errors, give 
 PS', and the hour-angle ZPS' ; and since PZ, the colatitude 
 (I) is known, the apparent zenith distance ZS' may be calculated 
 with these data from the triangle ZPS'. Let SS' = r, and 
 ZS' = z'. Then if we put r = A' tan z' t the factor A' may be 
 calculated exactly by Bessel's Refraction Tables according to 
 the example given in page LXIII. of the Tabulce Eegiomontance^ 
 or according to that given in page (iii) of Appendix I. to 
 the Greenwich Observations of 1836. The vertical refraction, 
 A tan z' t may then be obtained with as much precision as in 
 the case of a meridian observation. 
 
 287. In Figure 47 (1), let PS' = X, PQ'=A', ZQ' being 
 perpendicular to P$', and let L PS'Z = S f . Then the refraction 
 in P.D. = Sn, which without sensible error is equal to SS' cos $', 
 or A' tan z cos S'. But because cos S' = tan Q'S 1 cot /, refraction 
 in R.A. = A' tan Q'S' = A' tan (' A'), and A 7 is to be obtained 
 from the equation tan A' = tan I cos ZPQ'. The refraction in 
 RA. = L SPS' = S'n cosec S'= SS' sin S' cosec S' = A tan / sin S' 
 x cosec 8'. But by the triangle ZQ'S', sin Q'S' = cot S' tan ZQ', and 
 cos S' = tan Q'S' cot z. Hence tan z sin S' = tan ^Q' sec Q'/Sf . 
 Consequently, refraction in RA. = J/tan ZQ cosec S' sec (S' A'). 
 To obtain A' from the Table, the argument z is first to be 
 found from the equation cos z cos ZQ cos (8' A'). It will be 
 seen that the calculations might be shortened by means of 
 tabulated values of A 7 , tan ZQ', and cos ZQ', the argument being 
 the hour-angle. 
 
 288. Before proceeding to obtain formulae for the refrac- 
 tion-corrections in differential observations, it will be supposed 
 that values of A, and also of A' tan /, have been tabulated for 
 a series of apparent zenith distances separated by 5 or other 
 convenient interval, with the view of deducing their values by 
 interpolation for any intermediate zenith distance. In forming 
 these tables it suffices to employ mean values of the readings of 
 the barometer and thermometer as known from observations 
 extending over one or more years, because as the difference of 
 
282 PRACTICAL ASTRONOMY. 
 
 the zenith distances of the two objects is small, the errors 
 arising from employing mean instead of actual values may be 
 assumed to be the same for the refraction- corrections of both, 
 so as to disappear in the differences of KA. and P.D. In the 
 refraction-formulae for differential observations the most con- 
 venient argument is the true zenith ZS (Figure 47 (1)), 
 because, as the co-latitude PZ and the common hour-angle 
 ZPS are known, this zenith distance can be calculated for the 
 star from its known polar distance PS, and for the other object, 
 with sufficient exactness, from its approximate polar distance 
 inferred from the observed difference of P.D. uncorrected 
 for refraction. Putting z for ZS, let us suppose that the ver- 
 tical refraction is A i&nz. Then if its amount be r, we have 
 r = A tan z = A tan z = A tan (z r). Hence it will be found, 
 terms of the second order with respect to r being included, that 
 
 A' A' 
 
 r ~ i A' a tan z, and /. A = 
 
 If now r be obtained by interpolation from the above-mentioned 
 table of values of A' tan /, the argument pro hdc vice being z 
 (which is near enough), and then, with argument zr, Abe 
 interpolated from the other table, the value of A may be 
 calculated from the foregoing equation, and that of r from the 
 equation r = A tan z. By the process indicated the vertical 
 refraction may be obtained with the utmost precision ; but 
 generally it would be sufficiently accurate, after obtaining A 
 in the manner above stated, to take A = A. On the foregoing 
 principles a table of values of A, separated by suitable intervals 
 of true zenith distance, might be calculated, and its value for 
 any other true zenith distance be thence derived by interpola- 
 tion. After thus obtaining A tan z, the formulas for the refrac- 
 tions in RA. and P.D. become the same as those in Art. 287, 
 after removing the dashes from A, 8, A, and Q. (See Fig. 
 47 (1).) 
 
 289. The following formulas for calculating very small 
 differences of refraction in R.A. and P.D., the hour-angle being 
 given, are deduced by differentiation from the last-mentioned 
 
THE EQUATORIAL. 283 
 
 formulae for refractions in R.A. and P.D. Let a be the appa- 
 rent difference of P.D. of the two objects, as affected by refrac- 
 tion, and a be the true difference, for the same hour-angle. 
 Then a = a' (1 + A sec 2 (8 - A)). Let ff be the observed differ- 
 ence of transits at the given hour-angle, and /3 the true 
 difference. Then 
 
 /3 = j3' - ^ tan ZQ sec' 2 (8 - A) cosec 2 8 cos (28 - A). 
 
 For greater accuracy the mean between the apparent and true 
 P.D., viz. 8 + ^ , should be substituted for 8. 
 
 290. It will be seen from the foregoing investigations that 
 the calculations of the refractions in R.A. and P.D. would be 
 facilitated by any ready means of obtaining with sufficient 
 accuracy the zenith distance Z8 r and the angle $'. This 
 purpose is intended to be answered by Ramsderis Refraction- 
 piece (already referred to in Art. 273), the construction and use 
 of which I propose now to explain by means of Figure 47 (2) 
 and (3). In (2) ZPz is a circular plate at the eye-end of the 
 Telescope-tube, intersected by the axis of the tube perpen- 
 dicularly at S, and graduated at its rim from to 360. The 
 graduation is made to revolve about that axis by turning a 
 milled-head-and- screw, and is read off by an adjustible index 
 fixed to the tube. (The apparatus is, in fact, nearly the same 
 as that of a position-circle.) A small spirit level ee is attached 
 to the revolving plate, and may be placed, by turning the 
 plate, so that the bubble is in mid-position when the Telescope 
 is directed to any point of the heavens. By construction the 
 plane ZSz passing through the zero of graduation is made to 
 divide the level symmetrically in such manner that when the 
 bubble is in mid-position the line SZ is, as nearly as may be, 
 vertical. Accordingly the angle 8' is obtained as follows. The 
 Telescope being first pointed horizontally southward, the index 
 at Z on the tube is adjusted to exactly. Conceive, now, 
 the Telescope to be directed to any projection of the point $ 
 on the heavens, and let PS be in the new plane of collima- 
 
284 PRACTICAL ASTRONOMY. 
 
 tion. Then to put the bubble again in mid-position * the 
 circular plate has to be turned through an arc equal to the 
 change of position of the plane of collimation, which is exactly 
 measured by the reading of the index at P on the tube. This 
 determines the inclination of the plane of collimation to a 
 vertical plane through the object, that is, the angle S'. 
 
 The construction for measuring the apparent zenith distance 
 ZS' consists of a small graduated quadrant, represented by 
 Fig. 47 (3), which is fixed by the mechanist with its plane per- 
 pendicular to the circular plate and intersecting it in a line 
 parallel, as near as may be, to the fixed line ZSz, so that when 
 the bubble of ee is in mid-position the quadrant is in a vertical 
 plane. Attached to the quadrant is a small spirit-level Hh, 
 movable about a pivot at H } and having an index at h for 
 reading off. When the Telescope is pointed in the meridian 
 horizontally southward, and the bubble of ee is in mid-position, 
 the index at h should be adjusted to 90. Then for any other 
 pointing, when the bubble is in mid-position, the reading of the 
 index gives the zenith distance ZS'. 
 
 291. The reductions of equatorial observations of the RA. 
 and P.D. of Limbs to observations of Centres, by applying 
 corrections for apparent semi-diameters, do not require explana- 
 tions different from those given in treating of meridian observa- 
 tions. (See Arts. 113 and 209213.) 
 
 292. Lastly, we have to obtain formulae for calculating the 
 corrections of equatorial observations of the RA. and P.D. of 
 bodies of the solar system for Parallax. In Figure 48, E is 
 the centre of the Earth, A the place of observation, and <r the 
 position in space of the observed body. The angular points of 
 the spherical triangle PZ'S' are situated on AP the direction 
 of the Pole of the heavens, AZ' the prolongation of EA, and 
 AS' the direction in which the body at <r is seen from A. Draw 
 AS parallel to Ecr. Then the angle SAS' is equal to the 
 parallactic angle at <r, and AS is in the plane Z'E<r, which is 
 the same as, the plane Z'AS'. Hence AS cuts the arc Z S\ 
 and the parallax is measured by SS' '. Now let Z'Q' and Sn be 
 drawn at right angles to PS'. Then taking EQ, the radius of 
 
THE EQUATORIAL. 
 
 285 
 
 the Earth's Equator, to be unity, let EA = p, L Z'Az the 
 angle of the vertex = e, L PAz the co-latitude = I, the arc 
 l + = l', and PS' the apparent P.D. of the object = S'. 
 
 Fig. 48. 
 
 Also if P be the equatorial horizontal parallax, we have 
 
 F O 
 sin P = -Jz ; and if P' be the horizontal parallax at A, 
 
 &<T 
 
 sm 
 
 P' = 
 
 EA^EA EQ 
 Ea- 
 
 Putting p for the angle at <r, 
 
 we have, by the rectilinear triangle AEcr, si 
 
 AE . 
 j-r~ sin 2> AG 
 
 = p sin P sin "'$'. Then, the parallactic angle at <r being 
 supposed small, the angle SPff, which is the Parallax in RA., 
 is very nearly = Sn cosec & = 88' sin 8' cosec S' = p sin P sin Z'S' 
 x sin $' cosec B' = p sin P sin Z' Q' cosec 8' ; and the Parallax in 
 P.D. = Sn = p sin P sin Z'tf cos ' = p sin P sin Z'Q' cot S (v sin>Sf' 
 x sin Z"j8T = sin ^Q') = p sin P cos ^Q' sin Q'S' (/ sin Q'fiT = cot ' 
 x tan Z'Q') =p sin P cos ^'.Q'sin (8' - PQ'). The calculations might 
 be shortened by tabulating values of sin Z'Q' y cos Z'Q', and P Q', 
 the argument being the hour-angle. (For the Cambridge values 
 of p and e, see p. 227.) 
 
 293. The foregoing formulas apply with sufficient accuracy 
 to all the moving bodies except the Moon, the large parallax 
 of which requires the investigation of more exact formulae. 
 Although it is not likely that an astronomer will have occasion 
 
286 PRACTICAL ASTRONOMY. 
 
 to employ an Equatorial for independent determinations of the 
 Moon's place, I propose to obtain formulae for the parallax- 
 corrections in such observations, because the account of the 
 determination of the places of all celestial bodies by an Equa- 
 torial is otherwise incomplete, and also because the same 
 formulae are available in calculations relating to occultations 
 of bodies by the Moon, as will be subsequently shewn. In order 
 to investigate the formulae required for the present purpose, in 
 addition to the designations by letters in the preceding Article 
 of parts of Figure 48 it will be supposed that Z'S = z, Z'S' = z, 
 SS = Z'-Z=p, / Z'PS = 0, L Z'PS' = 0', PS = 8, and PS' (as 
 before) = 8'. Also, as in Art. 292, sin p = p sin P sin z'. Now 
 by the spherical triangles PSS' and PZ'S, 
 
 sin (ff - 0) = sin PSS' = sin 6 sin V 
 sin p sin 8' sin 8' sin z ' 
 
 TT ,/y m -r, sin z sin T sin 6 
 
 Hence sm (& 0) = p sm P . - - & . 
 
 sm z sm 8 
 
 But since sin PZ'S = ^^^1 = 8m e ' si f 3/ it follows that 
 
 sm z sm / 
 
 sin / sin ff sin 8' ~ 
 
 = ~ z~~= *" Consequently, by substitution m the ex- 
 pression for sin (ff ff), 
 
 sin (ff 6} = p sin P sin I cosec 8 sin 6' (1). 
 
 A T>ira cos 8 cos ^ cos I' cos 8' cos / cos I 
 
 Again, cos PZ S = : : = - : ~ 
 
 sm z sm I sm z sm I 
 
 ~ sin 6 sin 8 . _ 
 = cos 8 . -; -^. K7 + p sm P cos I . 
 sm sin 8 
 
 ~ cos 8' sin z cos I sin (z z] 
 
 /. cos 8 = : ; 1 : -, & 
 
 sm z sm z 
 
 Hence cot 8- cot y = cot 8' fgg* - l] + p ****** . 
 
 Vsm ff J *^ sin 8 
 
 T} j . ^ , *> sin f8' 8) 
 
 But cot 8 - cot 8 = . \ . J 
 
 sm 8 sm o 
 
THE EQUATORIAL. 287 
 
 . * sin (0 0') cos TT 
 
 and Sm ^ i 2 
 
 ^~0'~ . a 0-0' ' 
 
 sin v cos - 
 
 **//) /y N 6 -\- 6 
 
 sin o sin o sm (u u) cos 
 
 .'. sin (8' - 8) = -r ^ + p sin P cos Fsin 8'. 
 
 sin 0' cos 
 it 
 
 Consequently, after substituting the value of sin (0' 0) from 
 (1), it will be found that 
 
 /M rs\ T^f . . r ' v -{ , 7/\ ^ 
 
 sin (o o) = p sm P( cot o sm I cos ^ sec 5 h cos ) sm o . 
 
 z 2 
 
 Hence, < being a subsidiary angle calculated from the 
 equation 
 
 . W V -\- V 00 /ON 
 
 tan (j>= cot o cos ^ sec ^ (2), 
 
 we shall have 
 
 sin (S' S) = p sin P sec </> cos (I + <) sin 8' (3). 
 
 The equations (1), (2), (3) serve for calculating exactly & 0, 
 the parallax in R. A., and S' 8 the parallax in P.D. The 
 results may be obtained conveniently by successive approxi- 
 mations, the known quantities 0' and 8' being put for and 8 
 on the right-hand sides of the equations for a first approxi- 
 mation, and then with the consequent values of and 8 pro- 
 ceeding to a second approximation; and so on 1 . It is evident 
 that if the geocentric values and 8 were given, the local values 
 0' and 8' might be calculated by an analogous use of the same 
 formulae. 
 
 294. If $=the Moon's geocentric semi-diameter, and $' = 
 the apparent or measured semi-diameter, we have 
 $ _ ACT _ sin z _ sin sin 8 
 ~S f ~~ Ea- ~ sin z' ~ sin & sin 8' ' 
 
 1 The investigation here gone through shews that the formulas in p. xxxiii. of 
 Vol. xin. of the Cambridge Observations, which are the same under another 
 form as those in p. 23 of the Syllabus of my Astronomical Lectures, are not 
 sufficiently accurate. The formulae I have given (p. 298) in the Article which 
 constitutes the Appendix to the Nautical Almanac for 1854 are correct. 
 
288 PRACTICAL ASTRONOMY. 
 
 After the foregoing calculation of 6 and S, the value of S 
 may be calculated from this formula when S', 6', and 8 are 
 given. Means of obtaining S' by equatorial observations have 
 been indicated in Arts. 110 and 125. Accordingly, an Equa- 
 torial may be used for obtaining by means of observations of 
 the Moon, when known to be full, values of her geocentric 
 semi-diameter. The same formula will give S' when S, 0, and 
 S are known from Tables, and the local values of & and S' have 
 been calculated as above stated. [The local values &', 8, and 8' 
 are required in the Calculation of Occultations.] 
 
 I have now gone through all that relates to the determina- 
 tion of celestial positions by an Equatorial. There are various 
 other purposes for which this instrument may be employed, the 
 consideration of which will be most conveniently entered upon 
 after giving descriptions of it under other forms of construction 
 and modes of mounting. 
 
 THE NORTHUMBERLAND EQUATORIAL OF THE CAMBRIDGE 
 OBSERVATORY. 
 
 295. This powerful instrument, which at the time it was 
 constructed was one of the largest of its kind, was presented to 
 the Cambridge Observatory by the Duke of Northumberland, 
 High Steward of the University 1834 1840. The erection, as 
 regards both the Mounting and the Dome, was begun by 
 Mr Airy in 1835, and completed under his superintendence 
 after his appointment at the close of that year to the Green- 
 wich Observatory. In 1844 he produced, as a separate publica- 
 tion, an account, illustrated by engravings, of the parts and 
 appendages of the Instrument and Dome, which also forms an 
 Appendix to Vol. XV. of the Cambridge Observations (1843). 
 Any one desirous of acquiring full information respecting the 
 details of this undertaking, which was, I believe, the first 
 instance of the successful mounting of so large an Equatorial, 
 
THE EQUATORIAL. 289 
 
 must have recourse to that "Account." For the purpose of the 
 present work it will suffice to state the following particulars. 
 
 296. The object-glass is one of Cauchoix's : its effective 
 aperture is 11 J inches. All the brass- work, the graduations, 
 and the eye-pieces, were executed by Simms. The focal length 
 of the Telescope is 19f feet. The mounting is similar to that 
 of the Five-feet Equatorial. The Polar Axis consists of six 
 stout deal poles, the ends of which are fastened to two six-sided 
 cast-iron frames, at the centers of which are the upper and 
 lower pivots. The poles are braced across their middle by 
 transverse iron bands, counter to which there are 24 deal spars, 
 crossing each other two and two, and abutting near the middle 
 of the poles so as to thrust them obliquely outwards, being 
 made to act thus by means of screws which turn in shoulders 
 fixed to their opposite ends and press against the iron frames. 
 This apparatus answers the double purpose of giving stiffness 
 to the Polar Axis, and adjusting the iron frames so as to be 
 perpendicular to the axis of revolution of the instrument. The 
 support of the upper pivot consists of two strong wooden beams 
 connected by two cross iron bars, and surmounted by a tri- 
 angular iron frame, at the apex of which is the Y for the 
 pivot to rest in. The beams are inclined towards each other 
 in a plane perpendicular to that of the meridian and deviating 
 a little from the vertical towards the south : their lower 
 extremities, which are armed with iron off-sets, are firmly 
 embedded in massive brick -work, and their narrow faces are 
 turned towards the middle point of the axis of revolution. By 
 this construction the view of no part of the heavens is materially 
 obstructed ; Polaris and its companion can be well seen through 
 the upper iron frame. The support of the lower pivot is a 
 large stone slab resting on a deep mass of brick-work rising 
 to a small elevation above the floor of the Dome. The pivot 
 turns in a socket carried by a square mass of iron which is 
 movable by adjusting screws to allow of bringing in the usual 
 way (Art. 275 (2) and (3)) the axis of the instrument into 
 coincidence with the axis of revolution of the heavens. This 
 mounting is found to be remarkably steady. 
 
 c. 19 
 
290 PRACTICAL ASTRONOMY. 
 
 297. The tube of the Telescope is made of well-seasoned 
 deal. The pivots of the declination-axis, which turn in two 
 hollow cylinders each formed of two brass pieces screwed 
 together, are supported by two opposite poles of the Polar 
 Frame in such manner that one of the pivots admits of being 
 adjusted by screws so as to place the declination -axis perpen- 
 dicular to the polar axis. (See Arts. 275 (5) and 276 (1)). The 
 Telescope-tube just passes between the other four poles. 
 Attached to one side of this tube is a flat brass bar nearly 
 6 feet in length, carrying at one end a small graduated arc 
 perpendicular to its length, and turning at the other about a 
 pin fixed to the Telescope-tube at the distance of 2| feet from 
 the declination-axis. This is called the Declination-Sector. 
 The graduation of the Sector is read off by a micrometer- 
 microscope fixed to the Telescope-tube. For directing the 
 Telescope to any required P.D., graduated brass rods of dif- 
 ferent lengths are provided, the graduation of which was 
 effected by the intervention of an adjusted setting-circle 
 attached to the tube, the Telescope being moved in the plane 
 of the meridian, and marks being engraved on the rods 
 (clamped as below stated) corresponding to different selected 
 readings of the setting-circle. When each mark was drawn, 
 the microscope pointed to a fixed reading about the middle 
 of the sector-graduation. The marks were drawn at the edge 
 of a cylindrical sliding-piece, capable of being clamped to the 
 rod, which edge, consequently, afterwards served as index for 
 setting according to this graduation. The rod proper to be 
 used for a given P.D. was attached at its extremity, by means 
 of a clamp, to a pin fixed in one of the poles of the Polar Axis, 
 and by the above mentioned sliding-piece, previously clamped 
 to the rod with its edge at the given P.D., it was similarly 
 attached by a pin to the declination- sector. Near this pin, and 
 carried by the sector, is a combination of one toothed wheel 
 and two pinions turned by a milled head and acting on rack- 
 work fastened to the Telescope-tube. As the sector is 
 made stationary by the rod, the effect of turning the milled 
 head is to cause the Telescope to move in declination. In 
 
THE EQUATORIAL. 291 
 
 setting for an object the sector- reading has always to be 
 brought by this movement to the above mentioned fixed 
 reading; which is readily done, and with sufficient accuracy, 
 by bringing a pointer fixed to the sector to the adjusted position 
 of a zero attached to the tube. The rods also answer the 
 additional purpose of steadying the Telescope when used for 
 observing. 
 
 298. The construction above described is evidently adapted 
 for measuring differences of P. D. which, although small (not 
 exceeding for instance 1-J ), are too large for micrometer- 
 measures. For this purpose I frequently employed it. The 
 value in arc of the interval between the sector-divisions was 
 obtained by measuring with the sector the difference of P. D. of 
 two stars, and comparing the result with the difference as 
 determined by reduced meridian observations, stars being 
 selected whose difference of P. D. was not far short of the whole 
 range of the sector. (See infra Art. 303.) 
 
 299. The Hour Circle is 5J feet in diameter. It is not 
 permanently attached to the Polar Axis, but can be clamped to 
 the lower iron frame at pleasure. There are two adjustible 
 indices with verniers, pointing to the graduation of the hour- 
 circle, one supported by the pier in or near the meridian on the 
 south side of the polar axis, and the other fixed to the iron 
 frame. The position of the latter may be adjusted by first 
 setting the former to true sidereal time while a regulated clock 
 is turning the detached hour-circle, and after pointing the 
 Telescope to a known bright star so that the star is seen in the 
 middle of the field, clamping the hour-circle to the instrument, 
 after which process the hour-circle and Telescope are both 
 carried by the clock (as explained below) according to the rate 
 of sidereal time. The movable index should then point to the 
 right ascension of the star. Accordingly, it has to be placed and 
 adjusted so as to point exactly to that K A. Or, supposing 
 the movable index to be fixed approximately in position, the 
 instrument detached from the hour-circle might be moved so 
 that this index points to the R. A. of a known star, and then 
 after clamping the hour-circle to the frame, pointing to the star, 
 
 192 
 
292 PRACTICAL ASTRONOMY. 
 
 and setting the clock going, the fixed index might be adjusted 
 to true sidereal time. After adjusting the indices, the pointing 
 of the Telescope in R A. for any star is conveniently effected 
 by first setting the fixed index to sidereal time, and then set- 
 ting the movable index to the E. A. of the star. 
 
 300. The graduation of the Circle has been performed with 
 great care, and by the aid of the verniers can be read off with the 
 naked eye to 1 s of time 1 . The outer rim of the Circle is cut 
 with teeth to which an endless screw, connected by a brass rod 
 with the clock, can be applied at pleasure, for the purpose of 
 giving movement to the instrument about the polar axis. The 
 clock is kept in motion by a heavy weight, the force of which is 
 made to maintain the motion during the act of winding up. The 
 clock's rate is under command, and can be regulated to sidereal 
 time nearly. The clamping of the Hour-Circle to the iron frame 
 is effected by a tangent-screw clamp fixed to the frame, by 
 means of which, with the aid of a handle extending to the place 
 of the observer, he can, when the endless screw is applied, give 
 motion to the instrument, through a limited space, upon the 
 hour-circle. The rate of motion given to the Circle by the 
 clock is not affected by this movement. Hence supposing the 
 Circle to be moved exactly according to sidereal time, small 
 differences of R. A. can by this contrivance be measured by 
 reading off the pointings of the movable index before and after 
 the change of position. In any case correct measures may be 
 obtained by allowing for the rate of motion of the Hour-Circle 
 as inferred from the difference of hour-circle readings for two 
 bisections of a star and the sidereal interval between. It will 
 thus be seen that this Equatorial is adapted for measuring small 
 differences both of R A. and P.D. in differential observations. It 
 is advisable, in every instance in which it can be done, to begin 
 and end the series of measures with a bisection of the compa- 
 rison star. 
 
 301. The foregoing descriptions will be made clearer by 
 
 1 For greater precision in measuring differences of R.A., I added a graduated 
 sector, analogous to the declination-sector, and arranged so that it could be 
 clamped at pleasure to the hour-circle and be read off by a microscope. 
 
1 
 1 
 
rit 
 
 
 U 
 
THE EQUATORIAL. 293 
 
 referring to the annexed Plates IV. and V. (which are the same 
 as XVI. and XIX. in the "Account, &c."), and to the following 
 explanations of the instrumental arrangements. 
 
 Plate IV, Figure 1, represents the eye-end of the Telescope 
 with its double- wire micrometer mounted, as viewed from the 
 side opposite to that to which the brass bar is attached. A is 
 the wooden tube, and B a brass breech-piece. C is a position- 
 circle, D its slow-motion-screw and clamp, and EE are micro- 
 scopes for reading its divisions. F is the head of a pinion for 
 adjustment to focal length, and G- is the tube which is slid 
 inwards and outwards by the action of F. H is an eye-piece 
 inserted in G, carrying the double-wire micrometer S, and 
 held by friction only. All the different eye-pieces are thus 
 inserted in G. I is a declination-rod. K is the graduated arc 
 of the brass bar, and L the micrometer-microscope for reading 
 it. (See Art. 297.) M is the lamp for illuminating the field of 
 view. JVis a circular plate, turning on a screw at its center, 
 and having a snail-shaped hole for limiting the aperture through 
 which the light of M enters the side of the Telescope. The 
 light is incident at an angle of 45 on a gilded face of an 
 elliptical ring, through which the beam of light from the 
 object-glass passes without interference. is a milled head 
 for turning a pinion whose teeth work in the teeth of the 
 wheel N. PPPP are the square ends of the rods by which 
 the screws effecting the adjustments of the object-glass are 
 turned. (See Arts. 17 19.) Small keys are provided which 
 fit on these square ends ; but as they are seldom required, they 
 are not usually mounted. Q is a half-seconds chronometer 
 fixed by screws in a cell ; its winding-up-key may be seen- 
 projecting from one side. By the use of this chronometer there 
 is no necessity for reference to a clock, which, with a Telescope 
 of such a length, would be inconvenient ; and consequently no 
 clock is provided in the room 1 . B is the finder ; its focal length 
 is 28J inches, and aperture 2| inches. 
 
 1 In order to avoid exposing the eye to variations of light in observing very 
 faint objects, I adopted the plan of having the seconds counted audibly from a 
 chronometer by an assistant shut up in an adjoining closet. 
 
294 PRACTICAL ASTRONOMY. 
 
 302. Plate V. is a north-and-south section of the building 
 and instrument, intended to give a general view of the principal 
 parts of every kind in combination, excepting the machinery of 
 the shutters (for explanation of which see the Plate VII., and the 
 descriptions in pp. 15 18, contained in the before mentioned 
 "Account of the Equatorial and Dome"). It exhibits the 
 interior appearance of the enclosing octagonal wall, of which 
 WW and TFare north and south vertical sections; the interior 
 appearance of the Dome (NN)\ the fixed south steps (Z7), for 
 observations of objects- near the Pole northward ; the clock- 
 work box (ra, m) to the east of the steps ; the support of the 
 upper pivot, consisting of two wooden beams, surmounted by 
 an iron triangle (see Art. 296), one beam being represented 
 by T, and one side of the triangle by S; the adjustible support 
 (p) of the lower pivot, with a friction-wheel acting by a strong 
 spring to diminish the wear of the pivot ; the graduated hour- 
 circle hh (traced by dotted lines) ; the general structure of the 
 Polar Frame (see Art. 296) ; the bearing-piece for fixing the 
 position of the declination-axis (Art. 297) ; the Telescope as 
 mounted with its Finder ; a declination-rod (a) in position, with 
 its movable clamp attached (Art. 297) ' T the brass bar with 
 sector fixed at its end, together with the microscope for reading 
 the sector's graduation, and the rack-work fastened to the tube, 
 by means of which motion is given to the Telescope in decli- 
 nation by turning a milled-head (see Art. 297) ; the chair- 
 frame and chair, movable in circular grooves by a windlass at 
 the back of the frame, n, n being additional supports of the 
 observer's head when the object is near the zenith and the 
 chair cannot be used, and a separate seat being provided 
 suitable for supporting the observer when he has to take a 
 position between the poles of the polar frame. The Plate also 
 exhibits the observer in the upper chair holding in his hand 
 the long handle (b) by which slow motion in hour-angle is 
 given to the polar frame relatively to the hour-circle (Art. 300), 
 and having at his command the handle (e), by which motion is 
 communicated to a wheel (f) whereby rotation is given to the 
 chair-frame about a central pivot (o), as well as the handle (J), 
 
North and South sectwn 
 
THE EQUATORIAL. 295 
 
 which by means of the apparatus at g gives motion to the 
 Dome. (7, G are two of six holdfasts by which the Dome is 
 retained in any required position; at r are pulley-lines sus- 
 taining the weight, which is the moving power (Art. 300), 
 hanging in a well below Q ; y is a spring which being pressed 
 by the key in winding up, moves a click whereby the action of 
 the weight is continued during the process (Art. 300) ; w is a 
 small weight, which by machinery connecting it with the end- 
 less screw acting on the hour-circle, was occasionally applied to 
 supplement the moving power of the qlock. The Dome rests on 
 six cannon balls in a circular groove. For turning it tbrough 
 large spaces a lever is provided with fulcrum fixed to the wall, the 
 upper part of which is made by force of hand to act on pirns t, t, t 
 fixed to the curb of the Dome. 
 
 It will thus be seen that all the movements- during the 
 continuation of a series of observations are undfer the control 
 of the observer, without the necessity of his leaving his position 
 in the observing chair. As, however, it is hardly possible for 
 the observer by himself to perform with desirable ease and 
 rapidity all the operations required in the manipulation of 
 so large an instrument, I found it preferable to arrange that 
 some one should always be present to assist in turning the 
 Dome with the lever, rotating by hand the chair-frame about its 
 pivot, shifting the observer's seat by the windlass,, and recording, 
 generally, the observations 1 . 
 
 303. The value in arc of one revolution of the double- 
 wire micrometer mentioned in Art. 301, (the value being 
 assumed to be the same for both wires), was determined by 
 means of transits of 8 Ursae Minoris and Polaris in the manner 
 described in Art. 59 relatively to the micrometer of the Transit- 
 instrument, the transits being taken at positions of each wire 
 separated by a large number of integral revolutions. The 
 adopted result was, l r = 16",970. The value in arc of the 
 
 1 The large Equatorial of the Greenwich Observatory, the mounting of 
 which is of the same kind as that of the Northumberland Equatorial, is 
 described in full detail in an Appendix to the Volume of Greenwich Observations 
 for 1868. 
 
296 PRACTICAL ASTRONOMY. 
 
 interval between consecutive divisions of the sector-arc (l tf ) and 
 the value of one revolution (I**) of its microscope-micrometer (see 
 Arts. 297 and 298) were obtained by measuring with the sector 
 the difference of P.D. of two stars, A, Pegasi and //. Pegasi, 
 eliminating the effect of refraction, and comparing the result 
 with the difference of P.D. of the stars accurately known by 
 meridian observations. (The difference of P.D., about 62', is 
 considerably greater than any range of the arc ever employed.) 
 The measure was obtained in integral division-intervals, together 
 with integral micrometer revolutions and decimal parts of a 
 revolution, and the value of one division-interval in micrometer 
 revolutions was found by taking the mean of repeated measures 
 of that interval by the micrometer. Thus the difference of P.D. 
 of the stars could be expressed in integral division-intervals and 
 a fraction of such interval, and be compared with the known 
 difference of P.D. in seconds of arc. The means of numerous 
 measures taken in this way gave l d = 204",258 and l r ' = 10",178, 
 which values are adopted in the reduction of differential equa- 
 torial observations taken by means of the sector. (The details 
 of the measures for determining the values of F, l d , and I* 1 ', are 
 contained in pp. xxxi xxxv of the Introduction to Vol. xn._ 
 (1839) of the Cambridge Observations.) 
 
 The Counterpoise Mounting of an Equatorial. 
 
 304. Figure 49 represents a wooden model, used in my 
 Astronomical Lectures, which, although somewhat rude, may 
 serve for describing the counterpoise mode of mounting an 
 Equatorial. The main principle of this mounting consists 
 in placing the Telescope on one side of the polar axis and 
 counterbalancing it by a weight on the other side. The frame/ 
 represents the pier, or other kind of construction, on which the 
 instrument rests. The position of the polar axis is indicated by 
 the cylinder a, having an upper and a lower support, and 
 capable of being turned about its axis of figure, whereby 
 rotation is given to the Telescope and the Declination-axis. 
 Near the lower end of the polar axis is the hour-circle h, which 
 
THE EQUATORIAL. 
 
 297 
 
 is read by two opposite microscopes nn. The Declination-axis, 
 the direction of which is indicated by 66, is fixed transversely 
 to the polar axis, and carries at one end the Telescope AE, and 
 towards the other the counterpoise 0. The declination-circle 
 
 Fig. 49. 
 
 d is read off by the two opposite microscopes mm. The attach- 
 ment of the Telescope to the declination-axis is nearer to E, the 
 eye-end, than to A, the object- end, on which account two 
 counterpoises cc, fixed to the ends of rods, are made to counter- 
 poise the superior weight of the object-end. 
 
 305. This form of mounting has the advantage of giving 
 to the eye-end of the Telescope a range of motion of less radius 
 than that in the case of the Telescope being supported at its 
 middle point, and on this account, as well as because it does not 
 require the movement of a large polar frame for setting the 
 Telescope, it is less laborious to the observer than the other 
 mounting. At the same time it demands a Dome of larger 
 capacity to secure that the object-glass shall be under cover in 
 all positions of the Telescope, and, probably, a large telescope 
 thus mounted is more liable to tremors than is the case with 
 
298 PRACTICAL ASTRONOMY. 
 
 the mounting of the large Equatorials of the Greenwich and 
 Cambridge Observatories. Also with the counterpoise mounting 
 it is not possible to have a large hour-circle without unduly 
 weakening the support of the polar axis ; and there is,, besides, the 
 disadvantage that for some movements of the Telescope across 
 the meridian, especially when the pointing is between the Pole 
 and the Zenith, the eye-end of the Telescope comes into contact 
 with the support of the instrument. In that case, in order 
 to point the Telescope continuously, it is necessary to reverse 
 it so as to bring it to- the opposite side of the polar axis. Means 
 of adjusting by screws the positions of the axes of motion and 
 the microscope readings are, of course, provided in this as in 
 the other kind of mounting 1 . 
 
 306. Figure 50 represents another mode of counterpoise 
 mounting, the peculiarity of which is, that a cylindrical sup- 
 port PP, the axis of figure of which is directed to the pole 
 of the heavens, is firmly attached to a vertical wooden support 
 in such manner that the Telescope AE and its connected 
 apparatus, together with the counterpoise C, may be made to 
 revolve about the polar axis, as well as about the declination- 
 axis, which is the axis of figure of the cylinder DD, without the 
 possibility of encountering any obstacle, to whatever part of the 
 heavens the Telescope be pointed. This construction is, there- 
 fore, especially adapted to sweeping for objects whose RA. and 
 P.D. are only very roughly known (these circumstances are 
 alluded to by a remark in Art. 280, p. 276). A is the object-end, 
 E the eye-end, and F the finder, of the Telescope ; dd is the 
 declination-circle, which, by means of a hook's joint and handle, 
 may be fixed by the clamping-piece at I ; hh is the hour-circle, 
 which in like manner may be fixed by the clamping-piece at k ; 
 c is a leaden weight which, by means of the slit and screw at #, 
 
 1 The counterpoise mounting of Fraunhofer's 14-feet refractor at Dorpat is 
 represented in Plate VII. of Pearson's "Practical Astronomy," and the same 
 plate is given in Coddington's Optics, Part II. The mounting of the Sheep- 
 shanks Equatorial of the Greenwich Observatory, \vhich for the most part is the 
 same as the counterpoise mounting, is described in the Introductions of the 
 Greenwich Volumes of Observations, as, for instance, in pages xviii and xix of 
 the volume for 1875. 
 
THE EQUATORIAL. 
 
 299 
 
 Fig. 50. 
 
300 PRACTICAL ASTRONOMY. 
 
 may be shifted and clamped so as to counterpoise the Telescope- 
 tube. The upright portion of the support is attached at its 
 base to a triangular frame, which is adjustible by vertical 
 screws at p, p, r, and by horizontal screws at q, q. The vertical 
 screws are turned by spikes inserted at m and a corresponding 
 position (not seen in the figure), and at n. The screws q, q, 
 acting against the capstan heads p, p, turn the frame about the 
 vertical axis rn. After the adjustment of the polar axis is 
 effected, the three capstan heads p, p, r are firmly screwed down. 
 By having at command the vertical movements at p, p, r and 
 the horizontal movements at q, q, the polar axis may be 
 adjusted so as to be in the plane of the meridian and point to 
 the Pole. The box B covers two small adjusted spirit-levels, 
 one for levelling in the plane of the meridian, and the other for 
 levelling in the transverse direction. The bubbles, supposed to 
 be put in mid-positions after adjusting the polar axis, will 
 afterwards indicate by change of their positions, whether a 
 re-adjustment of the polar axis be required 1 . 
 
 307. In making use of an Equatorial in sweeping for an 
 object recognisable by its physical aspect, such as a comet, or 
 a nebula, the Telescope should be clamped in R.A. at an 
 arbitrary epoch so as to point considerably westward of the 
 most probable hour-angle at that epoch of the object sought for, 
 and be moved up and down on each side of the most probable 
 circle of declination, till the object is detected in the field 
 of view by its appearance. If the object be a small planet 
 not distinguishable in appearance from a minute star, and a 
 probable place be given, the sweeping may be effected by 
 noting down the places of all likely objects in contiguous zones 
 
 1 The stand represented in Fig, 50 was placed on a substantial brick wall, 
 raised to a convenient height above a flat area at the top of the south-west 
 angle of the Cambridge Museum Buildings. This wall, for the sake of steadi- 
 ness, stands on a separate base, so as to be unconnected with the structural 
 supports of the building. The mounting is so arranged that the Telescope 
 (which is the 46-inch Dollond of the Cambridge Observatory), can, with its 
 attachments, be readily dismounted when not required for use, and be deposited 
 in an adjoining apparatus room, or, if left on its stand exposed to the open air, 
 be protected by a suitable waterproof covering. 
 
THE EQUATORIAL. 301 
 
 of declination of a certain length, the central point of the 
 aggregate zone being the given place, and after a time 
 sweeping over the same area, to see whether any star in it 
 has changed position, or any new object has entered within its 
 limits. To render this mode of sweeping sure and rapid, the 
 instrument should be carried by clock-movement. On these 
 principles I searched for the Planet Neptune in the year 1843 
 with the Northumberland Equatorial. (See Appendix No. 1 to 
 Vol. xv. of the Cambridge Observations.) 
 
 Other Observations made with an Equatorial. 
 
 308. In addition to the use of an Equatorial for deter- 
 mining celestial positions, there are various kinds of observa- 
 tion for which this instrument is adapted, which we may now 
 proceed to take into consideration. One kind, of special interest, 
 consists of observations of the planet Mars in opposition, made 
 for the purpose of determining his parallax, and thence inferring 
 the parallax of the Sun. The determination of Mars's parallax 
 is made by means of measures of small differences of declination 
 between the planet and selected stars, corresponding measures 
 being taken at places differing little in longitude, but considerably 
 in latitude, such as Greenwich in the northern hemisphere and 
 the Cape of Good Hope in the southern hemisphere. From a 
 comparison of the differences of declinations for the same stars 
 as observed at the two places at noted times, and reduced to the 
 same epochs by taking account of the Earth's diurnal motion and 
 the apparent motion of Mars, we may calculate the differences 
 in P.D. of simultaneous directions of the two lines joining the 
 North or South Limb of the planet and the places of observation, 
 and hence infer, with sufficient approximation, the angle these 
 lines make with each other at a given time. This angle is sub- 
 tended by the line joining the two places, the length of which, 
 together with its angles of inclination to the above two directions, 
 may be inferred from the known dimensions of the earth, the 
 known latitudes and longitudes of the places of observation, and 
 
302 PRACTICAL ASTRONOMY. 
 
 from the data furnished by the observations. The same angle 
 is subtended by the perpendicular from one place of observation 
 on the visual direction of the observed Limb from the other, the 
 length of which is readily inferred from the previous results. 
 Hence the angle subtended at Mars by the earth's equatorial 
 radius, that is, his parallax at the time, may be at once calcu- 
 lated. It is desirable that as often as possible the differences of 
 declination should be micrometer measures, and that the N. and 
 S. Limbs of the planet should be taken alternately. A list of 
 stars proper for use in this kind of observation is given in suc- 
 cessive Nautical Almanacs. After thus obtaining Mars's parallax 
 at a certain time, and allowing for the effects of the eccentricities 
 of his and the earth's orbits, the Solar Parallax may be calculated 
 from the known relation between the planetary mean distances 
 and periods. 
 
 309. There is another method of taking observations of 
 Mars in opposition for determining his parallax; namely, ob- 
 serving his displacement in R. A. when he is far east of the 
 meridian, and far west of the meridian, as seen at a single Ob- 
 servatory. Conceive that by the diurnal and orbital motions of 
 the earth an Observatory is carried from the position A to the 
 position A', in space, in a certain time, and that the E. or W. 
 Limb of the planet is compared with a suitably situated star in 
 the first position, and with the same star in the other position. 
 The observations give the means of determining the astro- 
 nomical directions of the Limb from the Observatory in the two 
 cases. The known effect of the apparent motion of Mars in 
 producing the difference between these directions being elimi- 
 nated, there remains the angle subtended by the straight line 
 joining two positions the distance between which is wholly due 
 to the earth's rotation; and since the length and position of 
 this line can be calculated, the solar parallax may be inferred 
 in a manner analogous to that stated in Art. 308 \ 
 
 1 In an Article contained in the Monthly Notices of the Astronomical 
 Society, Vol. xvn. No. 7, p. 220, the Astronomer Royal expresses the opinion 
 that this is the best of all the methods of determining the solar parallax. The 
 same Article contains a discussion of various modes of calculating this funda- 
 
THE EQUATORIAL. 303 
 
 310. By observations with an Equatorial of the elongatioas 
 of a satellite from its primary, the mass of the latter may, ac- 
 cording to the principles of physical astronomy, be calculated. 
 In this way, by observations with the Cambridge Five-feet 
 Equatorial of the elongations of Jupiter's fourth Satellite, Mr 
 Airy obtained a correction of the previously adopted mass of 
 the planet. The details of the observations are given in Vols. 
 V. vili. of the Cambridge Observations (for the years 1832 
 1835), and the calculation of the results in the Memoirs of the 
 Astronomical Society, Yols. VI., VIIL, IX., and X. 
 
 311. Measures of the Moon's Diameter by transits of the 
 First and Second Limbs, when the Moon is exactly or nearly 
 full, have already been under consideration in Arts. 110 and 125. 
 The use of an Equatorial for this purpose has the advantage, 
 generally, of giving the means of taking a series of such measures 
 while the Moon is so nearly full that no calculations of correc- 
 tions for defect of illumination are required. 
 
 312. An Equatorial is suitable for observing an Eclipse of a 
 star, or a planet, or the Sun, by the Moon, these phenomena 
 requiring, with very rare exceptions, observations out of the 
 meridian. The observation of an Eclipse or Occultation of a 
 Star consists in simply recording, with as much accuracy as the 
 circumstances permit, the instant, as shewn by a time-piece, of 
 the disappearance, or reappearance, of the star at the Moon's 
 dark or bright Limb. The noted time has to be reduced to the 
 true sidereal time of the place of observation ; whence, on an 
 assumed value of the Longitude of the place, the corresponding 
 Greenwich Mean Time is inferred. The calculations required 
 for obtaining results from observations of occultations of stars 
 may be conducted as follows. 
 
 313. From data furnished by the Nautical Almanac, the 
 Moon's Geocentric R.A. and P.D., the equatorial horizontal 
 parallax, the geocentric semidiameter, and the star's apparent 
 R.A. and P.D., may be obtained for the given Greenwich Mean 
 
 mental quantity, more especially that by means of observations of transits of 
 Venus across the sun's disk. 
 
304 PRACTICAL ASTRONOMY. 
 
 Time. By means of these results the Moon's apparent KA., 
 P.D., and semidiameter have to be obtained by allowing for 
 parallax according to rules which have been already investi- 
 gated (see Arts. 292 294), and the apparent distance of the star 
 from the Moon's centre has also to be calculated. Each of the 
 quantities on which the calculations are based is supposed to 
 require correction. Let the corrections of the Moon's tabular 
 R.A. and P.D. be respectively x and y, and the Moon's true 
 horizontal parallax at the place, and her true geocentric semi- 
 diameter, be equal to the calculated values multiplied respectively 
 by 1 4- 0,001ra and 1 + 0,001^. Also let e and / be the correc- 
 tions of the star's RA. and P.D., t a correction of the noted time 
 of observation, T the correction of the assumed longitude, and 
 v a correction which the assumed geocentric colatitude of 
 the Observatory may require. Then if we call the Moon's 
 calculated apparent R.A., P.D., semidiameter, and distance 
 of the star from her centre respectively E, X, S, and D, and 
 their true values E + &R, X + SX, S+8S, D + SD, the values 
 of SB, B\, SS, &Z) admit of being expressed in linear functions 
 involving the unknown corrections x, y, e, /, m, n, t, T, and v. 
 A process for obtaining these expressions, and formulse for 
 calculating the numerical values of the coefficients of the cor- 
 rections they severally involve, are given in the Appendix to 
 the Nautical Almanac for 1854. By equating the true appa- 
 rent distance of the star from the Moon's centre to the Moon's 
 true apparent semidiameter (Art. 294), an equation of condition is 
 obtained containing all the unknown quantities. Every obser- 
 vation of a disappearance, or reappearance, furnishes one such 
 equation, and from the mean results of the calculation of a 
 large number of observations, exact values of the corrections 
 may be inferred. It may generally be assumed that x, y, e, 
 and f are known from meridian observations. Supposing also 
 m, n, and v to have been sufficiently determined, and the 
 recorded time to be so exact as to allow of assuming the cor- 
 rection t to be zero, the final equation gives the correction r of 
 the assumed longitude of the Observatory. 
 
 314. In observations of the eclipse of a Planet by the 
 
THE EQUATORIAL. 305 
 
 Moon, four notes of time may generally be taken, namely, the 
 times of external and internal contacts of limbs at disappear- 
 ance, and those of internal and external contacts at reappearance. 
 In observing the eclipse of either a star or a Planet, the observer 
 should take the precaution of putting an equatorially adjusted 
 wire on, or very near, the object before the disappearance, that 
 he may have some guide as to where to look for it at reappear- 
 ance, otherwise the instant of its first becoming visible might 
 be missed. The phenomena at the Moon's dark limb generally 
 admit of more accurate notes of time than those at the bright 
 limb, the brightness of the limb causing some uncertainty. In 
 forming the final equations derivable from the observations, 
 e and / will represent the errors in R A. and P. D. of the 
 Planet's tabular place, and in addition to the terms containing 
 m and n relating to the Moon, there will be terms containing 
 m and ri for corrections of the Planet's assumed horizontal 
 parallax and semidiameter. 
 
 315. The observations and calculations applicable to an 
 eclipse of the Sun, are analogous to those above mentioned 
 relative to a planet, as far as regards the times of external and 
 internal contacts; but the large disk of the Sun generally allows 
 of taking in addition several series of differential transits and 
 differential micrometer-measures. The differential transits may 
 be taken between cusp and cusp, or between a cusp and the first 
 or second limb of Sun or Moon, or between a limb of the Sun 
 and a limb of the Moon. The micrometer-measures of differences 
 of P. D. may be taken between cusp and cusp, or between a cusp 
 and the N. or S. Limb of Sun or Moon, or between either the two 
 N. limbs, or the two S. limbs, each micrometer-measure being 
 accompanied by a note of the time. Each such differential 
 quantity admits of being calculated from data furnished by the 
 Nautical Almanac, and supposing that these data may all 
 require certain unknown corrections, the true differential quan- 
 tity may be expressed in a linear function involving these 
 corrections, by equating which function to the observed measure, 
 there results an equation of condition between the corrections. 
 In this manner, when numerous differential measures have 
 o. 20 
 
306 
 
 PRACTICAL ASTRONOMY. 
 
 been observed, a large number of equations of condition may be 
 obtained, which, being treated by the method of least squares, 
 may serve to give accurate values of the unknown quantities 1 . 
 
 316. Another class of observations, of much interest, namely, 
 determinations of the relative positions of the components of 
 double, triple, and multiple stars, are greatly facilitated by the 
 use of an equatorially mounted telescope, carried by clock- 
 movement. (The arrangements for this purpose pertaining to 
 the Northumberland Equatorial may be gathered from Plates 
 IV. and V., and the accompanying descriptions in Arts. 301 and 
 302.) These observations are made by means of a position- 
 circle ( C in Plate IY.), which is a small graduated circle fixed 
 near the eye-end of the telescope, and capable of being turned 
 by a tangent-screw, or by hand, about the telescope- axis. The 
 graduation proceeds from to 360 and can be read off to the 
 accuracy of 1' by a single vernier, or, if great accuracy be 
 desired, by two opposite verniers. The angular and linear 
 measures may be taken either by means of a double-wire 
 micrometer (Arts. 31 and 32), or a double-image micrometer 
 (to be presently described), movable with the position-circle. 
 The mode of using the former will be first stated. 
 
 317. In Figure 51 (1) and (2), AB and ab represent 
 positions of the two parallel micrometer-wires. The line BbBb 
 
 Fig. 51. 
 
 in (1), and BbbB in (2), indicate the equatorial direction, which 
 is determined by turning the position-circle till one of the 
 component stars, S or cr, travels, the instrument being fixed, 
 
 1 The Appendix to the Naut. Aim. for 1854 contains the numerical details of 
 the calculation of an Occultation by the Moon in illustration of the method I 
 have here proposed. For Bessel's processes of computing Eclipses of the Sun, 
 and Occultations of Stars, with examples numerically worked out, see Loomis's 
 Practical Astronomy, Sections rv. and v. of Chapter XI. 
 
THE EQUATORIAL. 307 
 
 along one of the wires so as to be bisected both shortly after 
 entering, and shortly before leaving, the field of view, and then 
 reading off the position-circle. This gives the reading for the 
 equatorial reference-direction for the angular measures. The 
 angle of position may be defined to be the angle which the line 
 drawn from the brighter to the smaller of the two stars makes 
 with the circle of declination passing through the centre of the 
 field and cutting the reference-direction at right angles. The 
 telescope being supposed to invert images, and pro hdc vice to 
 be pointed southward on the meridian, the lowest point of the 
 field is called the north point (n), the extreme point to the 
 right hand the following point (/), that uppermost the south 
 point (s), and the extreme point to the left hand the preceding 
 point (p). It has been agreed by modern astronomers to reckon 
 position angles from at n, through /, s, p in succession to 360 
 at n again. The circle is thus divided into four quadrants 
 named in order nf, sf } sp, np, and the angles of position in these 
 quadrants range respectively from to 90, from 90 to 180, 
 from 180 to 270, and from 270 to 360. In the case of two 
 components the angle of position is measured by turning the 
 position-circle so as either to bisect both stars with the wire 
 used for recording the reference-position, or to place the two 
 stars, according to the judgment of the eye, exactly intermediate 
 to the two wires separated by a small interval, and then record- 
 ing the vernier reading. Both processes are exhibited in Fig. 
 51 (1), S representing the larger and cr the smaller component. 
 The second process is generally preferable for very close stars of 
 unequal magnitudes, the eye judging with accuracy of the 
 symmetry of the positions of the wires relative to the apparent 
 disks of the two stars. 
 
 318. After thus obtaining the reading of the circle for 
 angle of position, the position-circle is turned through 90 for 
 the purpose of measuring tlie distance between the stars. This 
 may be done by placing one of the wires, as AB, at a fixed 
 micrometer-reading (See Fig. 51 (2)), bisecting the stars simul- 
 taneously by the two wires by means of the slow movement of 
 the instrument combined with the micrometer movement of 
 
 202 
 
308 PRACTICAL ASTRONOMY. 
 
 the other wire ab, and then after passing this wire to the other 
 side of AB, bisecting the two stars again by the same means. 
 Let r l be the micrometer-reading for the first position of ab, and 
 r 2 that for the second position, and let c represent the reading 
 for coincidence of ab with AB. Then, supposing r\ to be 
 greater than r 2 , we shall have 
 
 So- = r t c = c r a . 
 
 Hence So- = - (r t - r 2 ), 
 
 which value, since c is eliminated, gets rid of error incident to 
 the determination of the coincidence reading. By means of the 
 reference-direction and the two measures of angle of position 
 and distance, the celestial position of one star relative to the 
 other at the epoch of the observations is ascertained. (When 
 the two stars are so nearly equal that it is hard to say which is 
 the brighter, it may happen that the relative positions obtained 
 on different occasions differ by 180.) By repetition of the 
 same processes the relative celestial positions of the components 
 of a triple, or quadruple, star, or of any limited portion of a 
 group of close stars, might be determined 1 . 
 
 319. For the sake of illustration, two examples of actual 
 observations of double stars, made with the double-wire micro- 
 meter of the Northumberland Equatorial, and of the calcula- 
 tions of the angles of position and distances, are here subjoined. 
 
 1842, Dec. 3, 7 h , after obtaining the reading 178 13' of the, 
 position-circle for the equatorial direction, four readings were recorded 
 for determining the angle of position of y Andromedae, the mean of 
 which gave 151 46'. By applying the reading for reference-direction 
 
 1 The method of observing double stars here described is the same as that 
 adopted by W. Struve for the composition of his great work, entitled Stellarum 
 Duplicium et Multiplicium Mensura Micrometricte. A large number of ob- 
 servations of double stars were made according to this method with the North- 
 umberland Telescope in the years 1839-1843, principally by Mr John Glaisher. 
 Subsequently the Equatorial was continually used in observing newly discovered 
 planets and comets, and much time was expended in provisional calculations 
 of their observed places, in consequence of which the observations of double 
 stars were not resumed, and the reductions of those taken were deferred. 
 Hitherto preparations have only been partially made for publishing the results 
 of these observations. 
 
THE EQUATORIAL. 309 
 
 (always subtractive), the result is - 26 27', to which is to be added 90 
 because the angle of position is measured from the north point. 
 Hence the required angle is 63 33', shewing that the position of the 
 small star relative to the other was north-following (nf). For 
 measuring the distance, the values found for r l and r 2 were respectively 
 10 r ,596 and 9 r ,375, each the mean of two readings. Hence the 
 distance (r l -r 2 ) = O r ,610- 10",35, the value of l r being 16",970 
 (Art. 303). The following notes were made in the observer's 
 memorandum book : The star is S 205 [i. e. No. 205 in Struve's 
 Catalogue of Double Stars] ; the small star followed [agreeing with 
 the result of the measures]; the estimated magnitudes 2 '3 and 5 '6; 
 the colour of the larger star, bright yellow, that of the other, a 
 beautiful blue; the night was bad for definition, and the stars 
 blurrish; the smaller star was not seen to be double [on other 
 occasions it was observed as a double star] j the magnifying power, 
 275 ; the observer, G (Glaisher). 
 
 1842, Dec. 3, 8 h , the star was 55 Piscium (=5 46); the reading 
 of the position-circle for the equatorial direction, 178 14' ; the mean 
 of four angular measures, 102 52'. Hence the angle of position 
 -102 52' -178 14' +90 =14 38', according to which the smaller 
 star followed. But the observer noted that the small star preceded; 
 from which it appears that the opposite vernier was inadvertently 
 read; and as the above angle is consequently to be increased by 180, 
 the angle of position by the observation was 194 38', which accords 
 with Struve's value. The value of r l was 10 r 362 and that of r a 
 9 r ,627, each by two micrometer-readings, consequently the distance 
 1 ( ri _ r a ) = O r ,368 - 6",25. The observer's notes were : Small star 
 preceding; magnitudes 6 '7 and 9; colours yellow and blue, very 
 decided ; sky misty ; magnifying power, 275 ; observer, G. 
 
 320. I proceed now to explain in what manner the double- 
 image micrometer is employed for observations of double stars. 
 Eye-pieces of various powers furnished with double-image 
 micrometers were executed by W. Simms, according to the 
 Astronomer Royal's designs, and adapted for use with the 
 Northumberland Equatorial. Their construction, and the mode 
 of applying them for taking measures, may be understood from 
 the following descriptions. In Plate IV., Fig. 2 represents the 
 form of the eye-piece, and Fig. 3 the action of the double-image 
 
310 PRACTICAL ASTRONOMY. 
 
 micrometer. The eye-piece is the same as that called the 
 erecting eye-piece in Art 272 (p. 261), where the disposition of 
 the glasses, and the course of a pencil of rays through them, are 
 shewn by Fig. 41. As the rays of each pencil fall centrically on 
 the second lens reckoned from the object-end, and are limited by 
 a small circular hole in a diaphragm close to the lens, advantage 
 is taken of these circumstances to form two equal images of an 
 object by dividing the lens through its centre into equal parts, 
 each half furnishing an image. The separation of the images is 
 effected either by a micrometer movement of one half lens rela- 
 tively to the other fixed (as shewn in Fig. 3 of Plate IV.), or by 
 equal movements of the two halves in opposite directions. The 
 eye-glass, or that nearest the eye, is contained in a cylindrical tube 
 prepared so that it carries a single wire crossing its axis at the 
 focal distance of that glass, and therefore distinctly visible 
 together with the image of any celestial object. This tube, 
 usually kept in position by friction (as is also the case of the 
 larger eye-piece tube), is furnished at its outer end with a 
 milled rim, whereby it can be turned by the fingers about its 
 axis, so as to be capable of adjusting the wire to any arbitrary 
 angular position. The same thing might be done without 
 moving the position-circle, by independently rotating the whole 
 eye-piece by hand. Such being the mechanical arrangements, the 
 observations of double stars are taken in the following manner. 
 321. After turning the micrometer-head for the purpose of 
 separating the images of the binary star to an arbitrary extent, 
 the images of one of the stars are placed by tentative movements 
 of the position-circle and telescope so that the line joining 
 them crosses the field near its centre in a direction judged 
 to be nearly equatorial. Then by slow-movement of the 
 telescope in R. A. and P. D. and by independent rotation 
 of the wire, both images of that star can be bisected by the 
 wire, the clock carrying the instrument. The clock being 
 now stopped, if the two images do not leave the wire, both 
 remaining well bisected, the required setting of the circle for 
 the separation of the images in the equatorial direction is 
 effected. Otherwise, by having independent rotations of the 
 
THE EQUATORIAL. 311 
 
 wire and the position-circle at command, together with slow- 
 movements of the telescope, it is possible by a few trials to 
 satisfy the condition that the central points of both images be 
 carried along the wire by the diurnal movement, as represented 
 by Figure 52 (1). The reading of the vernier when this is the 
 
 Fig. 52. 
 
 case, gives the equatorial reference-direction. The adopted 
 reference-reading should be the mean of several recorded 
 determinations. Next, the position-circle is turned till the four 
 images are all bisected by the wire, as shewn in (2) and (3) ; and 
 lastly, by turning the micrometer-head the three spaces inter- 
 mediate to the images are made, according to the judgment of 
 the eye, to be all equal. In (2) the separation of the images is 
 double the distance between the stars, and in (3) it is half that 
 distance. In the arrangement of (4) aS' is judged to be equal 
 
 SfSf' 
 
 and perpendicular to S<r, so that &7= =. The first arrange- 
 
 V 2i 
 
 ment, when practicable, is therefore to be preferred. 
 
 322. The reading of the position-circle for the new position 
 of the wire (in (2) and (3)), and the micrometer-reading, may 
 now be recorded. The former reading gives, by its excess above 
 that for reference-direction, the angle S'SE, and the latter the 
 distance between the stars multiplied, as the case may be, by 2, 
 or J, or, */2, and subject to index error. In all cases these 
 operations should be performed an equal number of times with 
 the images of the stars separated in opposite directions, whereby 
 probable errors incident to the determination of the angle S'SE 
 may be eliminated, and the necessity of obtaining the index error, 
 or micrometer-reading for coincidence of the separated images, 
 is avoided. The adopted angle of position is the mean of 
 
312 PRACTICAL ASTRONOMY. 
 
 several measures of tlie ^ S'SE increased by 90, and the 
 adopted distance is half the mean of several differences of pairs 
 of micrometer-readings taken with opposite separations of the 
 images. This measure is converted into arc by being multi- 
 plied by the value in arc of one micrometer-revolution obtained 
 as follows. The images of a star of known P. D. are separated 
 in the equatorial direction by an arbitrary large number of 
 integral revolutions, and by the judgment of the eye the wire is 
 placed at right angles to the line joining the images. The 
 clock-movement being stopped, the interval between the tran- 
 sits of the images across the wire is recorded, and the operation 
 is several times repeated. From the mean result in time, 
 and the known P. D. of the star, the interval in arc of a great 
 circle may be readily calculated, and from the given number of 
 integral revolutions, the value of V in arc be inferred. 
 
 323. We may next proceed to consider in what way the 
 double-wire micrometer is used for measuring the diameters of 
 Planets. For this purpose the two wires are placed simul- 
 taneously as tangents to the planet's disk at the extremities 
 of the diameter proposed to be measured, whether it be its 
 equatorial diameter, or polar diameter, or the major axis of the 
 elliptical boundary of a gibbous or horned phase. For these 
 positions of the wires let the reading of one micrometer (A) be 
 i\, and that of the other micrometer (B) be r 2 , the readings 
 increasing as usual towards the micrometer-heads. Then if the 
 reference reading of A be chosen to be a, and the reading of B 
 be a + c when its wire coincides with that of A at the reference 
 position, the distance of the limb touched by the wire of A from 
 the reference point is r { a measured in the direction towards 
 A, and that of the wire of B measured from the same point in 
 the opposite direction is r 2 a c. (By construction it is 
 provided that a may be an arbitrary number of integral revolu- 
 tions when the wire of A crosses the hole of the comb, and 
 that at the same time the coincidence reading a + c of B shall 
 differ very little from a, c being a small fraction of a revolu- 
 tion. Generally for the sake of convenience, a is assumed to 
 be 10 revolutions.) Hence it follows that the measure of the 
 
THE EQUATORIAL. 313 
 
 diameter is the positive value of + (r : + r 2 2a c). The co- 
 incidence reading a 4- c might be obtained in the usual manner 
 by calculating the mean of an even number of readings for 
 contacts of the wire of B alternately on opposite sides of the 
 wire of A at the fixed reading a (see Art. 51). But the 
 following process gets rid of the necessity of taking coincidence 
 readings. After recording the readings r t and r a , let the wires 
 be made to cross each other, and be transferred to the opposite 
 ends of the diameter, and let r/ and r 2 ' be respectively the new 
 readings of A and B. Then the measure of the diameter is the 
 positive value of + (r a ' + r^ 2a c), the sign being changed 
 because as respects the former measure the direction of this 
 measure is reversed. Hence 
 
 i\ + r z - 2a c = 2a + c r/ - r f ', 
 and therefore 2a + c \ (r l + r 2 + r[ + r 2 '). 
 
 Consequently, by substituting for 2a + c, the plus value of 
 ifo+r-j 2a c) i s tne same as tnat f i(i( r i+ r a)i( r i'+ r i))i 
 or i(( r i + r 2) ~ ( r i' + O)' If ^ ne number of sets of opposite 
 measures be n, and S, S' be respectively the sums of the 
 readings of A and B, it is easily seen from these results that 
 
 the mean value of 2a + c is _ - (S + $'), and the mean value of 
 
 the measure of the diameter is =- (8 ~ S'). (Data for numerical 
 
 applications of these formulae will be found in pp. 233 and 234 
 of Vol. xin. of the Cambridge Observations, year 1840.) 
 
 324. Also the double-image micrometer may be con- 
 veniently employed for measuring the apparent diameters of 
 Planets. In the instances of Jupiter, Mars in opposition, 
 Saturn's Rings, and occasionally his ball, the micrometer is read 
 for the external contacts of one image with the other at the 
 opposite ends of the diameter. According as one image is fixed, 
 or both move, the difference of the readings measures double 
 the diameter, or simply the diameter. When the planet's 
 phase is either gibbous or horned, the measures may be taken 
 in the same manner, but the separation of the images has to be 
 
314 PRACTICAL ASTRONOMY. 
 
 made in the direction of the major axis of the boundary of 
 illumination. (Examples of measurements of the diameters of 
 Jupiter and Venus with the double-image micrometer are 
 given in pages 234 and [142] [143], of Vol. xm. of the 
 Camb. Obs. 1 ) 
 
 325. The optical principle of the divided object-glass mi- 
 crometer, or Heliometer, is the same as that of the divided eye- 
 glass micrometer, inasmuch as the pencils of rays from points of 
 any celestial object fall centrically on the object-glass. The 
 Heliometer, so called because on account of the large range of its 
 micrometer-scale it is adapted for measurement of the Sun's 
 diameter, was employed in a special manner by Bessel for 
 determining the parallax of the double star 61 Cygni, this star 
 being selected because from its large annual proper motion (5") 
 it might be supposed to have sensible parallax. The measures 
 were taken between the middle point of the line joining the 
 component stars and each of two very small stars presumed to 
 have no sensible parallax, and distant from 61 Cygni about 
 12' and 8', approximately in rectangular directions. The 
 measures were continued through 2J years (in 1837 1840) 
 and from the variations of their values (the proper motion 
 of 61 Cygni being taken into account) it was concluded that 
 the star has an annual parallax of 0",35, a result which has 
 been confirmed by later observations 2 . 
 
 326. I propose to conclude the account of the uses of 
 the Equatorial by stating in what manner this instrument is 
 employed for discovering comets and small planets, and by 
 what means observations of these bodies are carried on after a 
 discovery has been made. Whether it be the purpose of the 
 observer to make an original discovery, or to find an object 
 when only a rough estimation of its place can be gathered from 
 an announcement of its having been discovered at a distant 
 
 1 For the mode of employing Rochon's Double Image Micrometer, for 
 measuring small angular spaces, see Godfray's Treatise on Astronomy, Arts. 
 113 and 114. 
 
 2 See Grant's History of Physical Astronomy, p. 551, and Vol. iv. of the 
 Monthly Notices of the Astronomical Society, pp. 152 160. 
 
THE EQUATORIAL. 
 
 TIN 
 
 315 
 
 observatory, the search would be conducted .by sweeping ac- 
 cording to one or the other of the methods described in Art. 
 307. After detecting the object the observer immediately 
 compares it by differential observations with a known star if 
 one happens to be in the neighbourhood, or with an unknown 
 star conveniently situated, the place of which is determined at 
 the time by comparisons with a known star. The original 
 discoverer, after thus obtaining two places, and deducing there- 
 from rates of motion of the body in K A. and P. D., com- 
 municates with as much despatch as possible these data to other 
 observers, for evidence of the originality of the discovery, and 
 for the purpose of receiving aid, under possible contingencies of 
 the weather, in securing additional observations. After three 
 places have been satisfactorily ascertained, whether by observa- 
 tions made at the observatory of the discoverer, or by a com- 
 bination of those made at several observatories, a first approxi- 
 mation to the orbit of the body can be deduced on the principles 
 of the theory of gravitation, and from the elements so obtained 
 a rough ephemeris of the apparent motion can be calculated. 
 For this purpose the three places need not be far separated in 
 time ; they might be used even if deduced from observations 
 made on three successive nights. This first ephemeris may 
 facilitate carrying on the observations till three other places 
 separated by larger intervals have been well ascertained, 
 whereby a more accurate ephemeris can be calculated 1 . By 
 
 1 The principles on which parabolic-orbits are to this day calculated from 
 three observed places are fully stated and exemplified in Newton's Principia, 
 Book in., Proposition XLI. In Proposition XLII. he enters upon considerations 
 relative to elliptical orbits, particularly with reference to that of Halley's Comet. 
 Various modes of applying Newton's principles, differing in the processes of 
 approximation, have been adopted by modern geometers. For instance, La- 
 place's investigation is conducted by analytical expansion, and in that respect 
 differs from the more geometrical process of Gauss. In the Memoirs of the 
 Royal Astronomical Society (Vol. xvn. pp. 59-77) I have produced and ex- 
 emplified a method, which, although it follows Laplace's in employing analytical 
 expansion, differs from it in the details of the numerical application. Subse- 
 quently I found that slight modifications in the analytical reasoning of the same 
 method gave rise to two other forms of expansion, one of which led to the well 
 known equation in. in Gauss's Theoria Hotits Corporum Ccelestium (Lib. n. 
 
3 1C PRACTICAL ASTRONOMY. 
 
 continuing this process the elements of the orbit are obtained 
 with increasing exactness, till finally an orbit is arrived at 
 which satisfies as nearly as possible all available observations 
 made at different observatories during this first apparition of 
 the object, and which may serve to determine, in the case of a 
 comet, whether the result is a parabolic orbit or one sensibly 
 elliptical. In any instance of an elliptical orbit, whether a 
 planet's or a comet's, the concluded elements are employed for 
 calculating an ephemeris which may serve for detecting the 
 body at the next apparition. Then by employing the results 
 of the observations of the two apparitions, more correct elements 
 might be obtained, and, it may be, of sufficient accuracy for 
 calculating the amounts of perturbations produced by the large 
 planets. By continuing this process, and combining the results 
 as to the elements, given by the first two apparitions after 
 taking account of the perturbations, with those similarly de- 
 rived from subsequent apparitions, mean elements for a given 
 epoch, which very approximately include the effects of planetary 
 perturbations, might be arrived at 1 . 
 
 OTHER FIXED INSTRUMENTS OF AN 
 OBSERVATORY. 
 
 The instruments which will be treated of under this heading 
 are, the Altitude and Azimuth Instrument, the Fixed Zenith 
 Sector, and the Transit in the Prime Vertical. 
 
 Sect, i., Art. 140), and of course to all the results deducible therefrom. As I 
 had reason to conclude that of the three methods I have referred to, Gauss's 
 was the most rapidly convergent, I have not thought it worth while to publish the 
 additional analytical investigations. 
 
 1 The Northumberland Equatorial was largely made use of in the years 
 1845-1858 for observations of newly discovered planets and comets, for the 
 purpose of furnishing data for calculating their orbits by the processes above 
 stated, but, excepting hi the instance of the Planet Neptune, no attempts were 
 made to obtain original places of these bodies. Provisionally reduced results of 
 the major part of the observations have been published in the Monthly Notices 
 of the Astronomical Society, and in the Astronomische Nachrichten. 
 
THE ALTAZIMUTH INSTRUMENT. 
 
 THE ALTITUDE AND AZIMUTH INSTRUMENT. 
 
 317 
 
 327. The main principle of this instrument (named also 
 for the sake of brevity an Altazimuth) may be indicated by 
 means of Figure 53, which represents a wooden model used in 
 
 Fig. 53. 
 
 my Lectures. As respects the mode of supporting the telescope 
 and two graduated circles, the arrangements resemble those of 
 an Equatorial, but because the axis about which the whole in- 
 strument revolves is vertical instead of being inclined, in point 
 of steadiness it is much superior to an Equatorial. The Figure 
 represents that the instrument is supported in the vertical posi- 
 tion by a frame, which practically might be constructed in any 
 manner that ensures the condition of steadiness. The vertical 
 axis of revolution, the direction of which coincides with a straight 
 line joining the centres of the circular holes 0, o, is the common 
 axis of a lower and an upper pivot. The lower pivot turns in 
 a perforation made in the fixed circular plate c, and the upper 
 pivot is held in close contact with a horizontal Y, which at 
 first should be so nearly adjusted, that the remaining errors of 
 position of the axis may admit of being allowed for in the 
 reduction of the observations (see next Art.). A is the object- 
 
318 PKACTICAL ASTRONOMY. 
 
 end, and E the eye-end, of the telescope ; dd is the altitude 
 circle, and ee the azimuth circle, read off respectively by pairs 
 of opposite microscopes, placed as indicated by m, m and n, n. 
 The columns pp, pp and qq, qq, with their circular terminals, 
 represent the material connection between the upper and lower 
 pivots, which, whatever form be given to it, should especially 
 be rigid. The axis of motion of the altitude circle, which is 
 required to be in a direction at right angles to the axis of 
 revolution of the instrument, should be mechanically adjustible 
 to such position, unless the deviation be allowed for in the 
 reduction of "the observations (see next Art.). The Figure 
 indicates, as a possible mode of adjusting the altitude-axis, that 
 one of the pivots (y) rests in a support which is movable in the 
 vertical direction relatively to a fixed transverse bar (x\ and 
 may be fastened to the bar by a nut and screw (s] in any re- 
 quired position. 
 
 328. A Fixed Altitude and Azimuth Instrument, the con- 
 struction and mounting of which were planned by the present 
 Astronomer Koyal, has been made use of at the Greenwich 
 Observatory since the year 1847, for the important purpose of 
 increasing the number of exact observations of the Moon avail- 
 able for application in the Lunar Theory. I propose to give 
 here some account of the methods of observing with this 
 instrument, the special means employed for obtaining accurate 
 results, and the processes of the calculation by which the 
 observations are reduced 1 . It will, first, be proper to state that 
 by means of a pier of brickwork forming the basis of a tri- 
 angular arrangement of the parts of the frame which serve for 
 fixing the position of the Y for the upper pivot, the condition 
 of steadiness of the vertical axis is secured, and that for the 
 sake of rigidity of the instrument, as many of the moving parts 
 
 1 A full description of the Greenwich Altazimuth Instrument is given in the 
 Introduction to the volume of Greenwich Observations for 1847, pp. iv. xxvii., 
 and in an appendix to that for 1867, illustrated by diagrams, and accompanied 
 by a perspective view of the Instrument as mounted (previously published in the 
 Illustrated London News, September, 1847). The methods of observing with it 
 and reducing the observations are explained in successive volumes, for instance 
 in pp. xiii. xvii. and xci. cxi. of the volume for 1875. 
 
THE ALTAZIMUTH INSTRUMENT. 319 
 
 as possible are formed in one cast of metal, no small screws 
 are used in the union of parts, and no power of adjustment 
 is left in any part, it being intended that the observations shall 
 be so arranged that every instrumental error shall be deduced 
 from the ordinary observations, and that numerical corrections 
 shall be applied in the reduction of the observations. Also 
 the friction on the lower pivot is diminished by a weight 
 acting by means of a lever from below in the vertical direc- 
 tion. 
 
 329. It is, in the first place, to be remarked that the 
 methods of taking observations of azimuth and zenith distance 
 are independent of each other, and may be described separately. 
 For an observation of azimuth, the instrument, after being suit- 
 ably set, as to time and pointing of the telescope, for catching 
 the object, is clamped in azimuth, and in making the observa- 
 tion a moderately-slow-motion screw is used to cause the object 
 to cross the middle of each vertical wire. The transit-times 
 are recorded either by the chronographic method (Arts. 129 
 130), or by eye-and-ear observation. The mean of the times, 
 corrected for personal equation of the observer, and for ascer- 
 tained clock-error, gives the sidereal time of the observation, 
 whether the transits be taken at all the wires, or omitted at 
 some. The readings of the micrometer-microscopes (four in 
 number) of the horizontal azimuth circle are then recorded. 
 These are corrected for error of Runs on the principles stated 
 in Art. 180 relatively to the Mural Circle. The formula used 
 at Greenwich for this purpose is investigated as follows. Sup- 
 pose that by the usual measurements for Runs an arc of 5' is 
 found to be equivalent to ap^b q^ CyS d micrometer 
 revolutions of the respective microscopes, a, 6, c, d being the 
 measures to the nearest tenth of a revolution, and p, q, r, s very 
 small fractions of a revolution. Then if w, oc, y, z be the micro- 
 meter readings in any observation, the concluded circle-reading 
 corrected for Runs is the Pointer Reading increased by 
 
 5/ [ w , . * , V , z } . 
 4* (a p b q c-r d sy 
 
320 PRACTICAL ASTRONOMY. 
 
 or, very nearly, on account of the smallness of p : q, r, s, and 
 
 w x 
 
 near equality of , 7- , &c 
 
 4 
 which form is convenient for calculation. 
 
 330. In the case of omitted wires another correction is 
 required, the amount of which is equal to the angular distance 
 between the mean of the wires observed and the mean of all 
 the wires (six), multiplied by the cosecant of zenith distance 
 to obtain the corresponding arc of the horizon. The calculation 
 of this correction requires, therefore, a previous determination 
 of the angular distances of the several wires from the mean 
 of all, which is effected by tho following process. From the 
 recorded times of horizontal transit of a star across the vertical 
 wires, taken as above stated in an observation for azimuth, the 
 distance in time of each wire from the mean of all is inferred. 
 Like inferences are drawn from the vertical transit of a star 
 across the set of horizontal wires (to be presently mentioned). 
 
 Putting X for the latitude of the Observatory, a. for the azi- 
 muth of the star, calculated for the sidereal time of observation, 
 (see Art. 329), and reckoned from the North or South point, 
 and T for the interval in seconds of time for any wire as 
 inferred from a vertical transit, it is easily shewn that the 
 required vertical interval in arc = 15 T cos X sin a. Hence, 
 T being similarly derived for a vertical wire from a horizontal 
 transit of a star whose P. D. is $ across the vertical wires, the 
 required horizontal interval in arc = 15 T (sin 2 S cos 2 X sin 2 c)-, 
 the whole movement of the star in the time T' being 152 7 sin S. 
 The calculations indicated by the two formula having been 
 gone through for each of a large number of transits, (say 50 
 vertical and 50 horizontal), the means of the several results 
 give the intervals in arc of the respective wires from the mean 
 of all for both sets of wires. After applying, in the case of an 
 observation for azimuth the corrections calculated by the second 
 formula for omitted wires, the concluded azimuth reading per- 
 tains in every instance to the mean of all the wires. 
 
THE ALTAZIMUTH INSTRUMENT. 321 
 
 331. This concluded reading has next to be corrected for 
 error of collimation, the amount and direction of which are 
 obtained most readily by making use, in the usual way, of a 
 mark on a level with the instrument. (At Greenwich another 
 method is also employed, depending upon the concluded azi- 
 muth readings for a high star and a low star in reverse positions 
 of the telescope. See vol. for 1875, p. cvi.) As the mark is 
 necessarily bisected by a single wire, the immediate result of 
 the operation is the horizontal error of collimation of the 
 selected wire, from which that of the mean of the wires is 
 derived by applying a correction for the already ascertained 
 distance of the wire from the mean of all. The correction of 
 the azimuth reading for error of collimation is evidently equal 
 to the Horizontal Error (with the appropriate sign) multiplied 
 by the cosecant of approximate Zenith Distance. A new deter- 
 mination of the constant of this correction is made at the end 
 of each lunation from the aggregate of the values obtained in 
 the course of the lunation. 
 
 332. The reading of the azimuth circle is also affected by 
 any error of position of the altitude-axis, "which is required to 
 be exactly horizontal. For calculating such error, four levels, 
 two upper and two lower, are attached to the instrument with 
 their axes in directions parallel to the axis of the altitude-circle, 
 and for each observation the two ends of each bubble are read 
 off. From the eight data, regard being had to the order of the 
 scale-graduations, and the values of the scale-intervals, a con- 
 cluded Level-indication may be obtained by the calculation 
 indicated in Arts. 68 70. The rule adopted for the Greenwich 
 instrument is to add all the scale-readings, subtract 1000 from 
 the sum, and divide the remainder by 8, the mean of the values 
 of the scale-intervals being found to be as nearly as possible 1". 
 From this concluded level-indication, that for the horizontal 
 position of the altitude-axis (obtained as shewn in the next Art.) 
 has to be subtracted, and the remainder, which, with the proper 
 sign, represents the angular elevation of one end of the axis, 
 gives, after being multiplied by the cotangent of Z. D., the 
 correction for level-error to be applied to the azimuth reading, 
 
 c. 21 
 
322 PRACTICAL ASTRONOMY. 
 
 that factor being required on account of the displacement of the 
 vertical axis in a plane at right angles to the plane of collima- 
 tion of the altitude-circle. By subtracting from this finally 
 corrected reading the zero, or index error, of the azimuth circle 
 (see next Art.), the concluded azimuth of the object at the time 
 of observation is obtained. 
 
 333. It only remains now to find the level-indication for 
 the horizontal position of the axis of the altitude-circle, and 
 the zero of azimuth. After applying corrections for collimation 
 error, calculated with the mean value of the constant of the 
 correction obtained as stated in Art. 331, let O r and O t be the 
 readings of the azimuth circle for observations of a known star 
 with graduated face of the altitude-circle to the right-hand and 
 to the left-hand, C r and C t its azimuths computed for the times 
 of observation, L r and L l the level-indications, D r and D t ap- 
 proximate computed zenith distances. Then putting y for the 
 level-indication corresponding to the horizontal position of the 
 altitude-axis, and z for the zero or index error of azimuth, we 
 have, from what is shewn in Art. 332, 
 
 from which equations it will be found that 
 O r -C r - (Ot - Oj) + Z r cot 
 
 ~ cot D r + cot D t 
 
 = (Q r - Q r ) cot D t +(0,- <?,) cot D r + (L r - L,} cot D r cot D t 
 
 cot D r + cot D t 
 
 For the determination of y it suffices to use a single star, 
 but a high star is preferable to a low one. The zero of azimuth 
 adopted for any day's observations is either deduced from the 
 star-observations of that day, or, if the zero appears to be 
 steady, by combining with them the observations of other days. 
 334. Each observation of the Moon's First or Second Limb 
 for azimuth, as often as it is possible, is made with the gradua- 
 tion of the altitude-circle Right and Left, and is accompanied 
 by observations of azimuths of known stars, also made with 
 graduation Right and Left. Measures for determining the 
 coefficient of collimation by a mark or collimator are generally 
 
THE ALTAZIMUTH INSTRUMENT. 323 
 
 grouped with the observations on any day on which the Moon 
 is observed, and not unfrequently are taken on other days. 
 The calculations of G r and C t for the given sidereal times are 
 founded on places of the stars derived from a star-catalogue 
 formed from observations made with the Greenwich Transit- 
 Circle. 
 
 335. We may now proceed to consider the method of 
 taking independently (see Art. 329) observations of zenith 
 distance. In an observation for zenith distance vertical transits 
 are observed across the set of horizontal wires by a process 
 analogous to that for the horizontal transits (described in Art. 
 329), the Altitude-Circle being clamped, and the moderately- 
 slow-motion of the azimuth-circle being used to make the 
 transit of the object take place across the middle of each 
 horizontal wire. It is, however, to be noticed that the mean 
 of the times so taken does not give the time at which the 
 object was at the mean of the wires unless its vertical motion 
 were uniform, which is not strictly the case, especially for zenith 
 distances observed near the meridian. Hence a correction of 
 the mean of the times (generally a very small quantity) is re- 
 quired on this account, formulae for the calculation of which 
 are investigated in pp. xcv. xcvii. of the Introduction to the 
 Volume of Greenwich Observations of 1875. By applying this 
 correction, together with corrections for personal equation and 
 error of the clock, to the mean of the observed times, the 
 sidereal time of the observation is obtained. 
 
 336. Next, the readings of the four microscopes of the 
 Altitude-Circle are recorded. These readings are in micrometer 
 revolutions, and the mean of their sum in arc, corrected for 
 Kuns, is calculated by a formula exactly analogous to that 
 obtained in Art. 329 relatively to the azimuth circle. In case 
 of the omission of any wires in observing the transits, the 
 resulting sidereal time applies to a direction of collimation 
 passing through the mean of the observed wires. Hence, as it 
 is requisite that the direction of collimation should in every 
 instance be made to pass through the mean of all the wires, a 
 correction is required equal in amount to the interval in arc 
 
 212 
 
324 PRACTICAL ASTRONOMY. 
 
 between the mean of the observed wires and the mean of all. 
 A formula has already been obtained in Art. 330 for deriving 
 from vertical transits the interval for any single wire, viz. 
 interval = 15I 7 cos A, sin a. Hence the required correction for 
 the interval between the mean of the omitted wires and the 
 mean of all may be readily inferred. When a mark, or colli- 
 mator, is observed, the amount of correction is simply the 
 interval in arc of the horizontal wire used for bisection from 
 the mean of the wires. After calculating, as above stated, the 
 corrected mean, in arc, of the microscope readings, the result 
 added to the Pointer Reading gives the concluded reading of 
 the altitude-circle. 
 
 337. There are two levels parallel to the plane of the 
 altitude-circle, one vertically above the other, and in any ob- 
 servation the scale-readings of the two ends of each bubble 
 are recorded. The directions of the graduations are so arranged 
 that the mean of the four readings multiplied by 1",075, which 
 is the mean value of one scale interval for both levels, gives a 
 Level-indication inclusive of a certain constant depending on 
 the assumed zeros of the scales (see Art. 332), which constant 
 is, in fact, equal to the level-indication when the axis of the 
 azimuthal rotation is vertical. The correction for refraction is 
 calculated in the usual way (Art. 215) for an approximate 
 zenith distance by employing an approximate zenith point. 
 By applying the level-indication derived from the four scale- 
 readings, and the correction for refraction, to the Pointer 
 Reading, a corrected reading of the altitude- circle is obtained, 
 which, however, involves the above-mentioned constant. The 
 difference between such reading, and that for the zenith direc- 
 tion (named the Zenith Point), is the concluded zenith distance, 
 supposing the Zenith Point to be obtained as follows. Let 
 O r and O l be the corrected circle-readings in two successive 
 observations of the same star in reverse positions of the gradua- 
 tion, C r and GI the corresponding zenith distances computed 
 from an adopted place of the star, and Z the Zenith Point. 
 Then assuming the computed values to be exact, we have 
 (Q r - Z] = G r and +(0,-Z) = C h 
 
THE ALTAZIMUTH INSTRUMENT. 325 
 
 according as O r or O l is greater than Z, the quantities on both 
 sides of the equations being necessarily positive. Hence it will 
 be found that 
 
 according as O r is greater or less than O t . If instead of a star, 
 a mark, or collimator, be bisected, C r and C t will be equal, and 
 the formula is simply 
 
 . 
 
 As either of these values of Z involves, in the same manner as 
 the corrected altitude-reading, the constant depending on the 
 level zeros, it will be seen that the concluded zenith distance, 
 being the difference between that reading and Z, is free from 
 the constant. 
 
 338. Thus the observations of the Moon consist of deter- 
 minations, at ascertained sidereal times, of azimuths of the 
 First or Second Limb, and of zenith distances of the Upper 
 or Lower Limb, by the processes above described, the zero of 
 azimuth being derived from accompanying star-observations of 
 azimuth, and the zenith point from accompanying star-observa- 
 tions of zenith distance, excepting so far as instead of the latter 
 a collimator is used. With this exception the lunar observa- 
 tions depend upon star-places obtained by means of observations 
 on the meridian. (The star-places in the Greenwich reductions 
 are deduced fundamentally from the " New Seven-year Cata- 
 logue of 2760 stars for 1864, 0, founded on observations with 
 the Transit-Circle," and forming Appendix II. to the Volume 
 of Observations for 1868.) Since it is possible to observe the 
 Moon much nearer to her conjunction with the Sun with the 
 Altazimuth than with the Transit-Circle, and in other respects 
 the number of available opportunities as respects the state of 
 the sky is greater, it is found practicable to obtain about twice 
 as many places with the former instrument as with the other. 
 In a Report of the Astronomer Royal to the Greenwich Board 
 of Visitors dated June 3, 1876, the respective numbers per 
 lunation are stated to be 180 and 88. 
 
326 PEACTICAL ASTRONOMY. 
 
 339. Having gone through all the processes of observation 
 and calculation whereby places of the Moon sufficiently trust- 
 worthy for use in the Lunar Theory are obtainable with an 
 Altazimuth, we may now consider by what processes of calcula- 
 tion such places are compared with theoretical places. As, 
 however, this is a matter of such length and complexity as to 
 be unsuitable for treatment in detail in a work like the present, 
 I shall only undertake to give an abstract of the procedure 
 adopted at Greenwich, and must refer for fuller information to 
 the contents of Sections 9 and 10 (pp. c. cxi.) of the Introduc- 
 tion to the Greenwich Observations of 1875. First, the Mean 
 Solar Times of the observations both of azimuth and altitude 
 are calculated in the usual way from the sidereal times obtained 
 as shewn in Arts. 329 and 335. Next, the Moon's Tabular 
 Geocentric E.A, P.D., Parallax, and Semidiameter, for the 
 mean times of the observations of her Limbs, are interpolated 
 to second differences from the data in the Nautical Almanac. 
 (The Astronomer Royal has proposed a mode of interpolating 
 to second differences, described in pp. ci. ciii. of the above- 
 named Introduction, whereby errors incident to changes of 
 sign are avoided.) The Moon's place is then transferred from 
 the Earth's centre as origin to the point of intersection of the 
 Earth's axis with the normal to the elliptical meridian drawn at 
 the position of the Observatory. A correction has to be applied 
 on that account to the tabular P.D., the formula for which is 
 obtained as follows. The semi-axis major and the semi-axis 
 minor of the terrestrial ellipse being a and 6, and the astro- 
 nomical colatitude of the Observatory being I, it is known from 
 Conic Sections that the distance of the foot of the normal from 
 centre is (a 2 b 2 ) (6 2 + a 2 tan 2 &)**. Putting ae for this quantity, 
 we have, with sufficient approximation, ae x sin of Moon's P.D. 
 for the distance of centre from the normal, and for the angle 
 subtended at the Moon by this small line 
 
 x seconds of Moon's Eq. Hor. Parallax x sin Moon's P.D. 
 
 This is the correction to be applied, always subtractively, to 
 the Moon's tabular P.D. for transfer of origin. (For the lati- 
 
THE ALTAZIMUTH INSTRUMENT. 327 
 
 tude of Greenwich e = [7,7174788], the ratio of the Earth's 
 axes being taken to be that of 300 to 299.) From the three 
 data, the Hour Angle (equal to the difference between the 
 calculated R.A., whether of star or Moon's centre, and the 
 sidereal time of observation), the calculated P.D. of the object, 
 corrected for the Moon as just stated, and the colatitude of the 
 Observatory ( = 38. 31'. 21",89 for Greenwich), it is simply a 
 problem in Spherical Trigonometry to infer the values of the 
 arcs which may be termed the normal-centric azimuth and 
 normal- centric Z.D. of the Star or the Moon's centre. (Formulas 
 for the calculations are given in p. civ. of the same Introduction.) 
 340. For comparing an observed azimuth of the Moon's 
 first or second Limb with the corresponding tabular value, the 
 calculated azimuth of centre is corrected for semi-diameter by 
 the following process. To the interpolated semi-diameter 
 (Hansen's, corrected by + 0",49 from results of Greenwich 
 observations) a correction is applied for excess of the Moon's 
 distance from the foot of the normal above her distance from 
 the earth's centre, which excess is equal to ae x cos ])'s P. D. 
 (see Art. 339). Hence if 8 = the semi-diameter in seconds, 
 computed as above stated, 8' the required semi-diameter, and D 
 the Moon's distance from the earth's centre, we have 
 
 ~ COS ' 8 R D 
 
 JP + )-sP.D. 
 
 Thus the correction to be applied to 8 to obtain 8' is 
 
 -8x^ cost's P.D. 
 
 Since -^ is equal to the sine of the Moon's equatorial horizontal 
 
 parallax, if we use the mean value of the parallax, and assume 
 it to be 57', the correction becomes, with sufficient approxi- 
 mation, - 8 x [8,21958] 6 x cos ])'s P. D. 
 
 (For the latitude of Greenwich [8,21958] e = [5,93706], because 
 (Art. 339) e = [7,71748].) After correcting 8 by this quantity, 
 the correction to be applied to the tabular normal-centric 
 azimuth of centre for semi-diameter, is equal to 
 
 -f corrected semi-diameter x cosec of normal-centric Z. D. 
 
328 PRACTICAL ASTRONOMY. 
 
 By applying tins correction and comparing the result with the 
 observed normal-centric azimuth of Limb (Art. 339), which is 
 the same at the place of observation as at the foot of the 
 normal, the apparent error of Tabular Azimuth is obtained. 
 
 341. The correction applied to the tabular normal-centric 
 zenith distance (Art. 339) for semi-diameter, is the semi-dia- 
 meter interpolated for the mean time of the observation of 
 Z. D., corrected in this case by + 1",01 from results of Greenwich 
 observations, and reduced, as before, to the foot of the normal. 
 (The value of the last correction adopted at Greenwich is 
 
 - 930" x [5,93708] x cos D's P. D., 
 a mean value of the semi-diameter being used.) 
 
 342. The normal-centric Z. D. has also to be corrected for 
 Parallax. For this purpose it is necessary to obtain the paral- 
 lax depending on the normal-centric radius, or distance of the 
 foot of the normal from the place of observation, on which 
 account it is, first, required to multiply the equatorial horizontal 
 parallax interpolated from the Nautical Almanac by the ratio of 
 this distance to the semi-axis major. Using the same notation 
 as in Art. 339, this ratio is known from Conic Sections to be 
 
 ax(6 2 cos 2 Z-f a 2 sin 2 Z)-*. 
 
 (For Greenwich, the factor is [0,0008851], a being to 6 in the 
 ratio of 300 to 299.) Again, as the parallax must depend on 
 the Moon's distance from the foot of the normal, another factor 
 is required equal to the inverse of the ratio of this distance to 
 the Moon's distance from centre, which, as we have seen (Art. 
 
 340), is equal to 1 ~ cos D's P. D. (For Greenwich, taking, 
 
 as before, a = D sin 57', and e= [7,71748], the factor becomes 
 1 -[5,93706] cos D's P. D.) Supposing that P=the result of 
 multiplying the interpolated parallax by these two factors, 
 z the normal-centric Z. D. (Art. 339), and p the required 
 parallax in seconds, we have 
 
 sin^> = sin (P + a) sin z, 
 
 a being Airy's correction for parallax at Limb (see Arts. 124 and 
 232). The first approximation to the value of p being P sin z, 
 
THE FIXED ZENITH SECTOR. 329 
 
 if we substitute for siup and sin P the respective approxi- 
 mations P 3 sin 3 3 P 
 
 p ~ 
 
 (R being the number of seconds in an arc equal to radius), and 
 if we omit small quantities of an order that may be neglected, 
 the above equation becomes 
 
 ( P* 
 
 = p ~ cos2 * 
 
 Hence, putting P' for the result of correcting P by the two 
 small quantities within the brackets, the value of p is finally 
 P sin z. After applying to the normal- centric Z. D. of centre 
 the corrections (with their proper signs) for semi-diameter and 
 parallax, the tabular apparent Z. D. of Limb is formed, the 
 comparison of which with the observed Z. D. (Art. 337) gives 
 the tabular error in Z. D. So by comparing the calculated 
 azimuth of Limb (Art. 339) with the value deduced from 
 observation (Arts. 332 334), the tabular error in azimuth is 
 obtained. In order to make a direct comparison of the observed 
 places with the Lunar Tables, the tabular errors in azimuth and 
 zenith distance are converted into errors in R A. and P. D., 
 which is done by means of formula analytically investigated in 
 pages ex. and cxi. of the before -named Introduction. In the 
 application of these formulae several observations made in the 
 course of an evening may be grouped together, on the assump- 
 tion that in the interval occupied, the Moon's P. D. did not 
 vary much, and that the tabular errors in R. A. and P.D. were 
 invariable. A Table of double entry is appended to the. Intro- 
 duction of the Volume for 1854, with the arguments Hour 
 Angle and Polar Distance, whereby the calculations made by 
 means of the formulae are much facilitated. 
 
 THE FIXED ZENITH SECTOR. 
 
 343. The particular purpose to which the Fixed Zenith 
 Sector is applied is, the measurement with great accuracy of 
 small angular distances from the zenith on the meridian. In 
 
330 PRACTICAL ASTRONOMY. 
 
 principle it is a Mural Circle adapted exclusively to these 
 measures, and is generally reversible. Such an instrument, 
 under different forms, has long been employed at Greenwich in 
 observing zenith distances of the star 7 Draconis, for determining 
 . the constants of aberration and nutation, that star passing the 
 meridian very near the zenith. Bradley, who first made obser- 
 vations for this purpose, used a mounted Telescope of 12 \ feet 
 focal length, with plumb-line for the vertical direction, and a 
 screw-micrometer with graduated head for measuring variations 
 of Z. D.; but the telescope was not reversible. (This instrument, 
 which was constructed by Graham (1727), is represented in 
 Plate XXVII. of Pearson's Practical Astronomy, and a model of it 
 is shewn in my Astronomical Lectures.) Pond substituted for 
 Bradley's Zenith Sector, one which had a Telescope of 25 feet 
 focal length and admitted of reversion, of which he gave an 
 account in the Philosophical Transactions of 1834 and 1835. A 
 description of " The Great Zenith Sector" is also given in suc- 
 cessive Volumes of the Greenwich Observations to that for 1847, 
 after which (in the spring of 1848) it was dismounted, the 
 results of the observations made with it not appearing to justify 
 the expenditure of time and trouble it demanded. The failure 
 was considered to be due to its liability to tremors, and to 
 difference of the effects of temperature at the upper and lower 
 parts of an instrument of that great length. Accordingly the 
 present Astronomer Royal has replaced Pond's Zenith Sector by 
 one more compact as to the arrangement of essential'parts, and 
 less dependent on focal length. This instrument was constructed 
 by Mr Simms in 1851, and was prepared for immediate use in 
 the autumn of that year. A complete description of it under 
 the name of " The Reflex Zenith Tube," with illustrations by 
 Plates, forms Appendix I. of the Volume of Greenwich Obser- 
 vations for 1854. To this description recourse must be had for 
 the details of the construction. For the present purpose it will 
 suffice to state the following particulars. 
 
 344. The Zenith Tube is fixed, and the object-glass is 
 carried by a cell which is supported by the Tube, and at the 
 same time may be reversed in position by being turned about 
 
THE FIXED ZENITH SECTOR. 331 
 
 the vertical through 180. At a distance from the object-glass 
 somewhat less than half its focal length a trough of mercury, 
 independently supported, is placed so that the rays from a star 
 after passage through the object-glass and reflection at the mer- 
 cury, are brought to focus a little above the object-glass. (A special 
 mode of suspension of the trough by the intervention of vulcan- 
 ized caoutchouc is employed for diminishing tremors of the 
 mercury.) The star selected for observation at Greenwich is 
 7 Draconis, the mean north zenith distance of which on Jan. 1, 
 1875, was about 100". This is small enough to allow of the axis 
 of the zenith tube being fixed very approximately in a vertical 
 position. The reflected image of the star is formed beyond 
 the object-glass, and is looked at transversely by means of a 
 diagonal eye-piece, constructed nearly in the same manner as 
 that described in Art. 273. Naming the eye-glass of this eye- 
 piece the first glass, and the field-glass the second glass, a prism 
 just above the centre of the object-glass serves at the same time 
 to turn the pencil of rays through 90, and by the curvature of the 
 vertical face to act as third glass, the image of the star and the 
 wire for observation being as usual seen in coincidence through 
 the fourth glass. Between the field-glass and the prism there is 
 a vacant space, occupied only by a frame for carrying the eye- 
 piece when moved by the micrometer-head, in order that the light 
 from the star may be as little as possible intercepted. The eye- 
 piece is furnished with a micrometer and micrometer-head, by 
 which means the bisection of the star by a micrometer- wire may 
 be read off in micrometer revolutions. The horizontal ordinate of 
 the image depends on the star's zenith distance, and the distance 
 of the image from a vertical line through the optical centre of 
 the object-glass will be exactly that corresponding to the star's 
 zenith distance with radius equal to the focal length of the object- 
 glass. It is, therefore, required to measure the horizontal dis- 
 tance of the image from the vertical line through the centre 
 of the object-glass. 
 
 345. This is done by attaching a micrometer to the cell 
 of the object-glass at the distance at which the star's image is 
 formed, and by placing the diagonal eye-piece so as to receive 
 
332 PRACTICAL ASTRONOMY. 
 
 the pencils which have been reflected from the mercury and 
 obtain a distinct view of the micrometer wire. This apparatus 
 might be used to measure the changes of the star's zenith 
 distance from day to day : but it cannot be used to measure 
 the star's absolute zenith distance, because the vertical line 
 through the centre of the object-glass is not discoverable by 
 direct observation. The arrangement by which the Astronomer 
 Koyal effects the absolute measurement is described in page (iv) 
 of the above-mentioned "Description" in the following terms. 
 " Let the micrometer be attached to and carried by the cell of 
 the object-glass, and let the cell be reversible in azimuth, by 
 being turned horizontally round a vertical line parallel to that 
 which passes through the centre of the object-glass, that is, 
 any vertical line, and let the micrometer-wire be placed upon 
 the image of the star, first, when the object-glass cell is in a 
 certain original position, secondly, when the cell is reversed in 
 azimuth ; then it is evident that wherever the centre of the 
 object-glass may be, the star's image is in the two observations 
 formed at equal distances from the vertical passing through 
 that point, and in the same direction from the vertical as 
 referred to the earth, or in opposite directions from it as re- 
 ferred to the reversible object-glass-cell. And therefore the 
 distance between the positions of these two images upon the 
 micrometer-frame is double the distance of either from the 
 vertical line, or double the distance which gives the means of 
 computing the angular zenith distance. Thus the practical 
 problem of measuring the star's zenith distance is, in its most 
 important and fundamental point, completely solved." It is, 
 however, necessary to satisfy two conditions ; first, parts of the 
 micrometer, such as the appui of the micrometer-screw, must 
 be so firmly connected with the object-glass-cell as to retain 
 strictly the same relative positions when the object-glass-cell 
 is reversed, which is insured by the ordinary connections of 
 instrument- maker's work. Secondly, the object-glass must 
 rotate about a vertical line so as to preserve the same inclina- 
 tion to it in the reversed positions ; or, if it does not preserve 
 the same inclination, we must have the means of computing 
 
THE TRANSIT IN THE PRIME VERTICAL. 333 
 
 the effect of the difference of inclinations. The Astronomer 
 Royal then shews how each micrometer-reading may be cor- 
 rected for inclination of the object-glass-cell by the indications 
 of a spirit-level attached to the cell. 
 
 346'. The mean of the two micrometer-readings in the 
 reversed positions of the cell gives what may be called " the 
 zenith point," that is, the reading for the vertical direction 
 through the centre of the object-glass. Instead of using the 
 zenith point given by a single set of observations in reverse 
 positions of the cell, it is preferable to adopt for zenith point 
 the mean of a considerable number of such determinations, a.nd 
 then the measure of the star's zenith distance as given by a 
 particular bisection is the difference between the adopted 
 zenith point and the micrometer-reading for the bisection. 
 This measure is in micrometer revolutions. To convert it into 
 celestial arc, the diagonal eye-piece is replaced pro hdc vice by 
 one which might be represented by the figure in Art. 273 
 turned upside down. This new eye-piece is furnished with 
 cross-wires which by turning the micrometer may be made to 
 bisect the series of wires by which the observations are made. 
 The intervals between the wires are thus determined in micro- 
 meter revolutions. To obtain them in celestial arc the cell 
 is turned through 90, and transits of 7 Draconis across all the 
 wires are taken with the new eye-piece (without the cross- 
 wires), and hence the intervals in arc may be computed, and 
 by comparison of the resulting values with those expressed in 
 terms of micrometer revolutions, the value in arc of one division 
 of the micrometer is found. 
 
 By continuing such measures of the zenith distance of 
 7 Draconis through a whole year, and through a succession of 
 years, the constants of aberration and nutation are obtainable 
 with great precision by the method of computation first practi- 
 cally applied by Bradley. 
 
 THE TRANSIT IN THE PRIME VERTICAL. 
 
 347. The foregoing method of ascertaining the constants 
 of aberration and nutation is plainly of a local character de- 
 
334 PRACTICAL ASTRONOMY. 
 
 pending on the small difference between the polar distance of 
 7Draconis and the co-latitude of the Greenwich Observatory. 
 It will, therefore, be required to describe another of more 
 general application. With this view I have selected the 
 method adopted by Professor F. G. W. Struve at the Imperial 
 Observatory of Poulkova, which depends on observations made 
 with a transit-instrument in the Prime Vertical. A description 
 in detail of the large instrument constructed by Repsold for 
 this purpose is given in the Bulletin of the Imperial Academy 
 of St Petersbourg, Tome x. No. 14 16, and also in the Astro- 
 nomische Nachrichten, Nos. 468, &c. Respecting the process 
 of using this fine instrument for effecting the required deter- 
 minations it will suffice for my purpose to make the following 
 statements. 
 
 348. The Telescope is attached transversely at its middle to 
 one end of a horizontal axis of revolution, and is counterpoised 
 by a weight at the other end. The position of the axis, as 
 respects its deviation from horizontality, is ascertained by means 
 of a spirit-level of great sensibility, (see the note to Art. 67, 
 p. 57). which is permanently fixed to the axis. The plane 
 of collimation of the Telescope is adjusted, so as to be as 
 nearly as possible in the plane of the prime vertical, and 
 consequently the wires at which the transits are taken, are 
 perpendicular to the plane of the meridian when the Telescope 
 is pointed to the Zenith. The polar distance of any star used 
 for taking the transits exceeds by only a small arc the colatitude 
 of the Observatory, so that its path in the field of the Telescope 
 is very oblique relatively to the wires, and the interval between 
 its transit across any one wire and the next is considerable (in 
 one of Struve's instances, 44 s ). These arrangements being under- 
 stood, we may proceed to the explanation of the process of 
 observation by which the instrument effects its intended purpose. 
 
 349. In Figure 54 let P be the North Pole of the heavens, Z 
 the zenith of the place of observation, PZ the colatitude, SoS' 
 a portion of the diurnal path of the star, cutting the prime 
 vertical in S and $'. Let $ be the place of the first transit of 
 the star across the prime vertical, where, consequently, it 
 
TJIE TRANSIT IN THE PRIME VERTICAL. 335 
 
 passes from the north side to the south 
 side. In the field of view there are seven 
 wires parallel to the plane of collimation, 
 and so placed (regard being had to in- 
 version by the Telescope) that the image 
 of the star crosses them in the order from i. to VII., before it 
 crosses the plane of collimation ; the seventh being at a con- 
 venient small distance from that plane. By shifting the Tele- 
 scope about the axis of revolution the star is made to cross the 
 wires at their middle points. Quickly after the transit across the 
 seventh wire the instrument is lifted from its bearings and the 
 position of the Telescope is reversed, say, from southward to 
 northward. By this operation the set of wires is transferred to 
 the opposite side of the plane of collimation, and the star, pursu- 
 ing its course, crosses them again, but in the order from vii. to i. 
 The Telescope retaining its position, the second transit across the 
 prime vertical (that at 8') is observed in the same manner as 
 the first, the order in which the star's image crosses the wires 
 before crossing the plane of collimation being from I. to VII., 
 because the direction of crossing is now from the south to 
 the north side of the prime vertical. After recording the transits 
 at the seven wires, the instrument is again quickly reversed, 
 and the transits are taken in the order from vii. to I. The 
 spirit-level is read before and after each reversion for calculating 
 corrections for displacement of the axis from the horizontal 
 direction. It thus appears that transits of the star are taken 
 at each wire in both sets of transits, both before and after passing 
 the prime vertical. The apparatus devised by Eepsold for 
 lifting the instrument from its supports and reversing the 
 axis end for end, gives the means of performing the operation 
 within the considerable interval allowed by the oblique move- 
 ment of the star 1 . 
 
 350. The mean of the two times of transit of a given wire 
 in the first set of transits, is the time of its transit across the 
 
 1 In Struve's Description of the Central Astronomical Observatory of 
 Poulkova (St Petersbourg, 1845), in page 178, an example is given in detail of 
 the record of an observation such as that here explained. 
 
336 PRACTICAL ASTRONOMY. 
 
 prime vertical at S, and similarly the mean of the two transit- 
 times of the same wire in the second set, is the time of its transit 
 at &. The excess of the latter time above the other gives the 
 angle SPS'. The transits at each of the other wires in like 
 manner give a value of the same angle, and by the mean of 
 all the values the angle is determined with great precision. 
 The result is not sensibly affected by refraction, and it is con- 
 sidered that errors arising from want of exact adjustment of the 
 axis of revolution, and other incidental causes, are very approxi- 
 mately eliminated by the process of observing in reverse positions 
 of the Telescope. By the spherical triangle SPZ, 
 
 Hence since PZ is constant, 
 
 B . PS = i sin 2PS tan - x B . SPS'. 
 
 f4 
 
 By means of this formula the variation S . PS of the star's P.D. 
 due to aberration and nutation may be derived from the ob- 
 served values of B . SPS', and consequently the required con- 
 stants be inferred from continuous series of such observations, 
 as already indicated (Art. 346). The observational means above 
 described are those specially employed by W. Struve for deter- 
 mining the constant of aberration, 20",4451, cited in Art. 143 
 (3), p. 135. 
 
 351. As a corollary to the preceding results it may be 
 stated that the sum of all the times of transit of the star divided 
 by 14, half their number, gives with great exactness the time 
 of its transit across the meridian. Hence the difference between 
 this time and that of transit at S obtained as above shewn, is 
 the angle SPZ } from which, and PS, the P.D. of the star, 
 supposed to be known, PZ the colatitude of the place of 
 observation may be computed from the right-angled triangle 
 SPZ. Similarly PZ may be obtained from the triangle S'PZ, 
 and the mean of the two results may be regarded as the most 
 trustworthy determination of the colatitude derivable from one 
 set of observations made with the given star. By employing in 
 
OBSERVATIONS WITH TRANSPORTABLE INSTRUMENTS. 337 
 
 like manner repeated observations of several known stars, a final 
 result of great certainty might be arrived at. This is in 
 principle Bessel's method of determining the latitude of an 
 observatory by means of a transit-instrument set up in the 
 prime vertical. 
 
 I have now gone through, with, I think, as much completeness 
 as was required by the character of this publication, the descrip- 
 tions of the fixed instruments of a fixed observatory, and 
 the processes of using them for astronomical determinations. 
 What is effected by such means may be generally stated to 
 be fundamental with respect to the purposes to which Practical 
 Astronomy is applied in the use of transportable instruments, 
 such as those I am now about to treat of. Having already 
 several times adverted to the science as being essentially logical 
 with respect to the relation of its different parts (see, for instance, 
 Arts. 10, 102, and 137), I take occasion to remark here that 
 wherever in the subsequent portion of the work, places of stars, 
 or the places and apparent magnitudes at given times, of the 
 Sun, Moon, and planets, are cited as known, it is to be under- 
 stood that the data are inferred, either directly or by com- 
 putation, from observations made according to methods de- 
 scribed in the antecedent portion. 
 
 OBSERVATIONS WITH TKANSPORTABLE INSTRUMENTS. 
 
 352. Ramsderis Zenith Sector. This instrument, constructed 
 for use in the Trigonometrical Survey of England and Wales, 
 was applied from the beginning of the Survey to 1836, in 
 measuring by means of a plumb-line the meridian zenith 
 distances of stars culminating near the zenith of each place at 
 which it was temporarily set up, the purpose being to compare 
 the difference of the astronomical latitudes of two 'places on a 
 given meridian with the difference as inferred from actual 
 measurement on the earth's surface, and from theoretical com- 
 putation based on an assumed figure of the earth. Data were 
 c. 22 
 
338 PRACTICAL ASTRONOMY. 
 
 thus obtained for testing, or correcting, the constants adopted in 
 the theoretical calculation. It was also used for determining 
 the latitudes of certain stations fixed upon in the course of the 
 triangulation 1 . 
 
 353. Airy's Zenith Sector. This name is given to an 
 instrument the design, and superintendence of the construction 
 of which, were undertaken by the Astronomer Royal in con- 
 sequence of an application from Col. Colby, superintendent of 
 the Ordnance Survey of Great Britain and Ireland, who desired 
 to replace Ramsden's Zenith Sector by a new and more per- 
 fect instrument. In the "Account of the Ordnance Survey," 
 published in 1858 by Leiut.-Col. H. James, the principles 
 involved in the construction of "Airy's Zenith Sector" are 
 thus stated in page 57 : "The first principle was the arrange- 
 ment for making successive observations in two positions of 
 the instrument, face east and face west, at the same transit. 
 The second principle was the substitution of a level or system 
 of levels for the usual plumb-line. The third principle was 
 the casting in one piece, as far as practicable, of each of the 
 different parts of the instrument, in order to avoid the great 
 number of screws and fastenings with which most instruments 
 are hampered, and to secure, if possible, perfect rigidity." The 
 inferences of latitude from the observations of zenith distance 
 are all made by the intervention of known stars 2 . 
 
 (Examples of the Transportable Instruments that remain 
 to be described are exhibited in the Lecture-Room, with the 
 exception of the Altazimuth, the principle of which is included 
 in the construction of the Theodolite.) 
 
 354. The portable Altazimuth Instrument. Like the fixed 
 
 1 An account of Ramsden's Zenith Sector and the mode of using it is given 
 in pp. 533 546 of Pearson's Practical Astronomy, illustrated by Figures in 
 Plates XXVI. and XXVII., exhibiting its component parts, and the instrument 
 as a whole. 
 
 2 In his work entitled On the Theory of Errors of Observations and the Com- 
 bination of Observations (Macmillan and Co., 1875), Mr Airy has laid down 
 principles for calculating the probable errors of the results obtained in Trigono- 
 metrical Surveys by observations made with the Theodolite and the Zenith 
 Sector. See particularly instances in Sections 12 and 15. 
 
OBSERVATIONS WITH TRANSPORTABLE INSTRUMENTS. 339 
 
 Altazimuth, this instrument is furnished with graduated 
 altitude and azimuth circles, but it suffices for the purpose 
 to which it is applied to read off the graduation by Verniers. 
 It is supported by a tripod stand, and the height of each 
 foot is adjustible by a screw. The principal adjustment is 
 that of the axis of revolution to vertically, which by means 
 of two spirit-levels fixed to the supporting frame in rectangular 
 directions and having marks for the mid-positions of the bubbles, 
 is effected in the following manner. The adjusting screws being 
 supposed to rest on a firm basis, the instrument is turned about 
 its axis till the direction of the axis of figure of one of the 
 levels is parallel, as nearly as may be estimated by sight, to 
 the line joining two of the feet. The bubble of that level is 
 then adjusted to mid-position by turning either a screw of one 
 of the two feet or the screws of both. The screw of the third 
 foot is then turned till the bubble of the other level is in 
 mid-position. This completes the adjustment. A first trial, 
 ought, however, to be tested by turning the instrument quite 
 round to see whether the bubbles retain their positions ; for 
 if they do not a fresh adjustment has to be made. The axis 
 of the altitude-circle may be supposed to be sufficiently adjusted 
 by the instrument maker so as to be perpendicular to the axis 
 of revolution. The field of the Telescope has vertical wires 
 and a transverse wire which, after the adjustment of the vertical 
 axis, is readily adjusted to horizontally by a fixed land-mark. 
 The error of collimation of the intersection of the middle 
 vertical wire and the horizontal wire may be corrected in the 
 usual way by a mark, with reversion by means of the azimuth 
 movement. Instead of vertical wires, there might be two 
 wires crossing each other and the horizontal wire at the middle 
 of the field. 
 
 355. After these arrangements the use of a portable alt- 
 azimuth has the particular advantage of requiring, for finding 
 the time of day and latitude of a place in an unknown territory, 
 no other instrumental adjustment than that of the axis of 
 revolution to vertical ity in the manner above stated. This 
 adjustment being made, suppose a known star, or the N. or 
 
 22 2 
 
340 PRACTICAL ASTRONOMY. 
 
 S. limb of the Sun, to be bisected before culmination, by the 
 horizontal wire where it intersects the mid-wire or the cross 
 of wires, the Telescope being previously clamped to the altitude- 
 circle, and the bisection being effected by the azimuth move- 
 ment. The chronometer- time of bisection having been recorded, 
 and the Telescope remaining clamped, let the time of bisection of 
 the same object after culmination be similarly noted. When the 
 object is a star the mean of the two times is the chronometer- 
 time of its transit across the meridian. Also if the azimuth-circle 
 be read for the two positions, the mean of the two readings 
 is the azimuth-reading for meridian. Hence, supposing the 
 observer to be provided with a sidereal chronometer and the 
 Nautical Almanac for the year, a portable altazimuth might 
 be employed, in the same manner as a transit-instrument in the 
 meridian, for rating the chronometer by means of stars, and for 
 determining local sidereal time and local mean time. Again, 
 after correcting the index error of the altitude-circle by means, 
 for instance, of its readings for bisections of a land-mark 
 in reverse azimuths, if the instrument be then set to the 
 meridian according to a previous determination of the meridian 
 azimuth-reading, the meridian altitude of some known star 
 might be observed, and thence, after correcting for refraction, the 
 latitude of the place of observation be inferred. Lastly, if the 
 observer is able to ascertain, by any of the processes which will be 
 explained in the next Division, the Greenwich mean time contem- 
 poraneous with any local mean time obtained by the in- 
 strument, the difference of the two times gives the longitude of 
 the place. Thus the portable altazimuth is a very serviceable 
 instrument for enabling a voyager after landing on any coast, 
 or an inland explorer, to make approximate geographical deter- 
 minations of unknown localities. 
 
 356. After finding the azimuth-reading for meridian by 
 observing, as above stated, equal altitudes of a star, or rather 
 after obtaining the mean of several such determinations, and 
 thereby placing the plane of collimation in the plane of the 
 meridian, the chronometer- time of apparent noon may be inferred 
 by observing a transit of the Sun. This determination is, how- 
 
OBSERVATIONS WITH TRANSPORTABLE INSTRUMENTS. 341 
 
 ever, often made by noting the times of bisections of the Sun's 
 N. or S. limb at equal altitudes before and after culmination, in 
 which case a correction is required for change of the Sun's P.D. 
 in the interval between the bisections. The correction may be 
 calculated as follows, supposing the azimuth readings to be taken 
 at the two bisections, and the latitude to be known by the process 
 explained in Art. 355. Supposing P to be the North Pole of 
 the heavens, Z the zenith of the place of observation, and S 
 the place of the bisected limb, let PZ=\ ZS = z, PS = 8, and 
 the hour angle ZPS west of the meridian = h. Then we have 
 to find the change AA, of hour angle corresponding to the change 
 AS of the P.D. of the bisected point in the interval from the 
 first to the second bisection. Putting S for the angle PSZ, it 
 may be readily seen from the small variations of h and S, z and 
 X being constant, that 
 
 cosec S 
 
 15 
 
 in which expression AS may be supposed to be obtainable from 
 the Nautical Almanac ; but S and S have to be determined. 
 Now if A = the angle PZS, or the difference between the 
 azimuth-reading for meridian and that for the second bisection, 
 we have by the spherical triangle PZS, 
 
 ~ sin h . sin h sin X 
 
 cosec o = 
 
 j, : , 
 
 sin z sin A sin z 
 
 in which equations, without sensible error, h may be taken equal 
 to the excess of the noted time of the second bisection above 
 the mean of the two times, and A equal to the difference 
 between the second azimuth and the mean of the two azimuths. 
 Thus cosec S and S may be calculated from observed quantities, 
 and the value of A& be obtained from the foregoing formula. 
 The correction to be applied to the time of the second bisection 
 is A/t, and the mean between the corrected value and the 
 time of the first bisection is the chronometer-time of apparent 
 noon. (The correction for difference of the refractions at the 
 two positions is too small to be worth taking into account.) 
 
342 PRACTICAL ASTRONOMY. 
 
 The correction Ah has the same sign as A8, if S be less than 
 90, and consequently is positive or negative according as the 
 Sun's T.D. is increasing or decreasing. The opposite signs would 
 apply if S should be greater than 90. 
 
 357. The portable Transit-instrument 1 . The parts, mode of 
 mounting, and adjustments, of this instrument, being like those 
 of the fixed Transit, need not be described in detail. It will 
 suffice to mention the following particulars respecting an ex- 
 ample (exhibited) of Troughton's construction. The pivots 
 rest in Ys supported by two standards firmly attached to a 
 brass circular ring, which has three feet adjustable as to height 
 by screws, and requiring to be placed in contact with a solid 
 horizontal basis. One of the screws is vertically below one of 
 the Ys, and is used for the level-adjustment, the azimuth- 
 adjustment being performed by a horizontal screw-movement 
 of the other Y. The other two screws are so placed that a 
 line joining the feet is approximately in a direction parallel to 
 the telescope's plane of collimation. The setting-circle, which 
 is graduated for indication of altitude, is fixed to one end 
 of the axis of revolution, with which, consequently, it revolves 
 and is reversed, the support of a lamp at the other end for 
 illumination of the field of view being also reversible. The 
 indications of the setting-circle are read by a double vernier, 
 having an attached tongue whereby it may be adjusted and 
 clamped, the tongue being acted upon by one of two sets of 
 opposing screws fixed to the standards. Index error may be 
 got rid of by pointing the Telescope to a star of known altitude, 
 and then adjusting the vernier-reading to this altitude as altered 
 by refraction. After this correction the bubble of a spirit-level 
 attached to the vernier should be put in mid-position, in order 
 to give the means of adjusting the vernier by the tongue 
 after a reversion, so as to be again free from index error. 
 
 358. The adjustments of the plane of collimation of a 
 portable Transit, required to be set up in an unknown locality, 
 
 1 A full description of this instrument as constructed by Troughton (illus- 
 trated by a Figure), and of its adjustments and uses, is given in Simms's 
 Treatise on Mathematical Instruments, &c.," 8th edition, pp. 68 92. 
 
OBSERVATIONS WITH TRANSPORTABLE INSTRUMENTS. 343 
 
 may be performed as follows. The error of collimation of the 
 middle wire might be corrected by bisections of any suitable 
 fixed point (such as a bright point seen by reflection of sun- 
 light from a convex glass surface), and by reversing the instru- 
 ment. This correction is usually made by the instrument-maker. 
 Before correcting the errors of level and azimuth, the plane 
 of collimation has to be placed approximately in the plane 
 of the meridian. This could be done by employing a portable 
 altazimuth for drawing a meridian line by observations of equal 
 altitudes of a star (Art. 355), or, in default of such means, the 
 process described in Art. 10 might be adopted. The axis of 
 motion may then be adjusted to horizontality by applying 
 the feet of a corrected striding spirit-level to the pivots, and 
 turning the foot-screw for level-adjustment (mentioned in Art. 
 357), to put the bubble in mid-position. The adjustment to 
 the meridian may then be effected by transits of a circumpolar 
 star in the manner indicated in Arts. 88 and 89. Supposing 
 that, after making one such adjustment, it should be required 
 to place the instrument repeatedly in the same position, 
 this may be conveniently done by means of the following 
 expedient. The screw for level-adjustment, terminating in a 
 rounded surface, is applied to a horizontal plane surface, while 
 the other two screws have pointed feet, which fall into an 
 angular groove cut in the direction of the line joining their 
 extremities after the initial adjustment has been completed. 
 Thus the instrument may, at least approximately, be placed 
 in the adjusted position. A more exact adjustment, if thought 
 good, might then be made by the usual process. 
 
 359. The portable transit is principally used for rating 
 chronometers by means of transits of known stars, and for 
 calculating local sidereal time and local mean time. In many 
 cases, determinations of these quantities such as may be obtained 
 by a portable transit having a telescope of twenty inches focal 
 length, are sufficiently approximate. But certain observations, 
 for instance some that are available for calculating solar parallax 
 from a Transit of Venus, require to be accompanied by a 
 determination of local time with the precision that can only 
 
344 PRACTICAL ASTRONOMY. 
 
 be attained by stable mounting of transported transit-instruments 
 of considerable size. 
 
 360. The Theodolite. Figure 55 represents a Theodolite 
 made for me by Mr J. Simms for exhibition to the class in 
 my lecture-room. Excepting in being smaller, it differs very 
 little from the instruments furnished by Troughton and Simms 
 for the Great Trigonometrical Survey of India. In principle 
 it is a portable Altazimuth and a portable Transit combined, 
 and accordingly what has been said in Arts. 354 359 respecting 
 the parts and adjustments of those two instruments applies 
 in great measure to the Theodolite. As shewn in the Figure, 
 A is the object-end and E the eye-end of the Telescope. At 
 the focus of the object-glass there is a system of wires consisting 
 of two vertical wires, one on each side of the plane of collimation, 
 two wires equally inclined to it and intersecting at an acute angle 
 at the centre of the field, and a horizontal wire passing through 
 the point of intersection. For fixing the position of the wire- 
 frame, and correcting error of collimation, two sets of antagonist 
 screws with capstan heads are placed in the position indicated 
 by i, one for vertical, and the other for horizontal, movement. 
 By turning a milled head projecting from the telescope-tube 
 near the object-end (not seen in the figure), the distance of 
 the object-glass from the wire frame may be regulated so that 
 the image of a land-object can be seen distinctly with the eye- 
 piece at the same time as the wires. The mounting of the 
 Telescope on the supporting frame is the same as that of a 
 transit-instrument, one of the pivots resting in its Y being 
 shewn at e. The two screws g g hold down a plate, which is 
 moveable about the one with capstan-head on the left hand, 
 when both are unscrewed. After unscrewing them the plate 
 can be turned aside, and a like operation having been performed 
 with respect to the other pivot, the instrument can be lifted 
 from theYs for reversing it. The capstan-headed screw between 
 the two screws g has a piece of cork at its lower end, and is 
 made to press on the pivot for keeping it in position. The 
 vertical circle dd is graduated for reading off altitude by means 
 of a double vernier vv, the position of which relative to the 
 
OBSERVATIONS WITH TRANSPORTABLE INSTRUMENTS. 345 
 Fig. 55. 
 
^ 
 
OBSERVATIONS WITH TRANSPORTABLE INSTRUMENTS. 347 
 
 graduation is adjusted and fixed by the action of antagonist 
 screws (one of which is seen at z) on a projection from the 
 frame, the screws moving a vertical tongue rigidly connected 
 with the vernier and tapped for the action of the screws. The 
 graduation is from to 90 in the same direction in each of the 
 four quadrants, both indices pointing to when the telescope is 
 horizontal. Slow motion is produced by the tangent-screw and 
 clamp at b. The azimuth circular plate aa is graduated from 
 to 360 on a chamfered edge. Above this is another circular 
 plate opposite portions of which directly below the pivots, as 
 shewn at t, have chamfered edges, and are graduated to serve 
 for indices and verniers, the graduation of the lower plate being 
 read off by the microscope k and an opposite one. The vertical 
 and azimuthal circles are both graduated to a third of a degree, 
 and by the microscopes and verniers can be read off to 20". The 
 two above-mentioned plates have conical axes one within the 
 other, the outer one represented by 6r, being a hollow cone fixed 
 to the azimuth plate and the inner one fixed to the upper plate, 
 turning smoothly within the other. The two axles are held 
 together by a screw and circular plate H fixed to the lower end 
 of the interior one. The vernier plate and all the parts of the 
 instrument above the azimuth plate turn with that inner axle, 
 and may either be moved by hand through large angles, or 
 partake of slow motion by means of the tangent-screw and clamp 
 at m. Also after turning the handle o, so as to tighten the 
 collar about the axle G, slow motion of rotation may be given to 
 all parts of the instrument in common by means of the screw p t 
 which works in a tapped projection from the collar (not seen in 
 the Figure). 
 
 361. Such being the description of the parts of a Theodo- 
 lite, it is next required to give an account of its adjustments. 
 It will, at first, be supposed that its three feet s, s, s rest on a 
 firm plane approximately horizontal, the usual means of support 
 by a tripartite staff being considered afterwards. The first 
 adjustment is to set the common axis of revolution so as to 
 be exactly vertical ; which, after unclamping m and o, is effect- 
 ed by means of two corrected levels with marks, attached in 
 
348 PRACTICAL ASTRONOMY. 
 
 rectangular directions to the supporting frame, in positions 
 indicated by the letters c, c. The adjusting screws are q, q, q, 
 by which, and by the three legs when used, the azimuth plate is 
 first placed horizontally as nearly as the eye can judge. The 
 exact adjustment is then made by placing the axis of figure of 
 one of the levels in a position judged to be parallel to the line 
 joining two of the feet of the screws, putting the bubble of that 
 level in mid-position by means of one or both these screws, and 
 finally putting the bubble of the other level in mid-position by 
 the third screw. The accuracy of the adjustment should always 
 be tested by turning the instrument quite round to see whether 
 the bubbles retain their positions with so much exactness as to 
 shew that a fresh adjustment is not required. (See what is 
 said respecting this adjustment in Art. 354.) After the fore- 
 going process, the axle G should be clamped by the handle o, 
 and it is then advisable to repeat the operation by turning 
 independently the parts connected with the vernier plate. 
 Next, the error of collimation of the cross of wires is to be 
 corrected in the usual way by a fixed horizontal mark, and 
 reversing the horizontal axis end for end. The resulting plane 
 of collimation being supposed, at present, to be exactly vertical, 
 let a fixed horizontal mark be bisected by the horizontal wire at 
 the intersection of the cross-wires, and the altitude-circle be 
 read off by the verniers v, v. Then if the same process be gone 
 through after reversing the telescope by the azimuth movement, 
 the mean between the two readings gives the reading for the 
 direction of the vertical axis ; and if it be desired that this 
 direction should be indicated by 90, an index error has to be 
 corrected equal to the excess of that mean reading above 90. 
 This correction is effected by means of the screw z y which 
 regulates the position of the verniers on the limb of the circle. 
 After correcting the index error, if the vernier readings be set 
 to 0, the line of collimation of the telescope will point horizon- 
 tally. In that case the large level L, which has its axis of 
 figure parallel to the axis of the telescope, is to be adjusted 
 by putting and fixing the bubble in mid-position by means of 
 the two capstan-headed screws at h. Afterwards by means of 
 
OBSERVATIONS WITH TRANSPORTABLE INSTRUMENTS. 349 
 
 this level the telescope can be pointed horizontally in any 
 azimuth. 
 
 362. It is here to be remarked that the foregoing adjust- 
 ments will not be affected if the plane of collimation, instead of 
 being vertical as above supposed, be inclined by a small angle 
 from the vertical direction in consequence of a small deviation 
 of the axis of motion of the telescope from horizontality. It is, 
 however, necessary that that axis should be exactly horizontal, 
 and the deviation has consequently to be corrected ; which may 
 be done as follows. Bisect with the cross of wires some well- 
 defined distant point near the horizon, and record the azimuth 
 readings of the two verniers. Then by horizontal rotation of 
 the instrument set the vernier readings at exactly 180 from the 
 former positions. If the transverse axis be exactly horizontal, 
 the same distant point will be bisected by the cross of wires 
 after this reversion. But if it deviates from horizontality by 
 the small angle 6, its new position will make an angle 20 with 
 the first, and the plane of collimation will cut the horizon at 
 some point different from that originally bisected. Half the 
 distance between the two points is to be corrected by means of 
 the two capstan-headed screws //, whereby the end e of the 
 axis is moved and fixed. The axis will then be adjusted so as 
 to be, at least approximately, horizontal. For greater certainty 
 the operation should be repeated ; after which the index error 
 of the azimuth circle may be corrected by the following 
 process. 
 
 363. Below the telescope there is a compass-needle nn, 
 which may be clamped by turning a small exterior handle, and, 
 after adjusting the vertical axis, be undamped and allowed free 
 suspension, as occasion requires. The compass-card, which 
 rotates with the telescope, is so placed that the plane of colli- 
 mation passes nearly through its N. and S. points. Hence at a 
 locality where the magnetic declination is known, the plane of 
 collimation can be at once approximately adjusted to the 
 meridian. For instance at Cambridge, where the magnetic 
 declination is about 19 westward, the telescope maybe brought 
 nearly to the meridian by being turned, in common with the 
 
350 PRACTICAL ASTRONOMY. 
 
 compass-card, till the northward end of the needle points to 19 
 eastward, as marked on the card, from the N. point. If it be 
 desired to fix the azimuth reading at when the telescope 
 points northward in the meridian, the vernier-index is first to 
 be set to by turning the vernier-plate whilst the azimuth- 
 plate is clamped by the handle o, and then after connecting the 
 two plates by the clamp at m, and unclamping o, the plates are 
 to be moved together till the plane of collimation is shewn by 
 the compass to be nearly in the meridian, the telescope pointing 
 northward. The required adjustment of the zero of azimuth 
 may in this way be made, at least approximately. 
 
 364. For an exact adjustment recourse must be had to 
 astronomical observation, which, after the foregoing approxima- 
 tion, may be such as follows. With the instrument in its actual 
 state observe the altitude and azimuth of a known star at a 
 recorded time, and compare therewith the altitude (inclusive of 
 refraction) and azimuth of the star calculated for the given time 
 by the usual formulae from its known R.A. and P.D. The 
 excesses of the calculated above the observed values are the 
 corrections to be applied for index errors of the altitude and 
 azimuth circles. The result in the case of the altitude circle 
 will serve to test or rectify the correction already applied 
 (Art. 361) ; and th'e mechanical correction of the residual index 
 error of the azimuth circle may be effected by the following 
 process. Suppose the azimuth reading for bisection of a well- 
 defined terrestrial point to be A reckoned from the apparent N. 
 direction, and, the correction for index error having been found 
 by the foregoing method to be -f a, let the azimuth reading be 
 altered to A -4- a by the tangent-screw and clamp at m. The 
 cross of wires will then no longer bisect the same point, but may 
 be made to bisect it again by the slow-motion screw p, the axle 
 G being clamped. By this means the index error is corrected 
 so that the reading of the azimuth-circle will be for the N. 
 direction. It should be added that generally it would be 
 preferable to find the index errors of both circles by repeated 
 observations of stars, and apply the resulting mean corrections 
 in the reductions of the observations ; in which case, although 
 
OBSERVATIONS WITH TRANSPORTABLE INSTRUMENTS. 35 1 
 
 it is necessary to be provided with the means of executing these 
 corrections mechanically, it would only be occasionally required 
 to make use of them. 
 
 365. It remains to describe the mounting of the Theodolite 
 on a three-legged stand. In the Figure, uuu are upper portions 
 of the legs, connected in common at their upper ends by joints 
 with a large screw, on which a tapped three-armed plate 
 is screwed for support and adjustment of the instrument by the 
 three screws qqq. These screws have conical feet with flanges, 
 and the plate consists of two parts in contact, the upper of 
 which, after lifting the spring r, can be moved for bringing into 
 position three openings through which the conical feet have to 
 pass in order to rest in prepared angular grooves. The moved 
 plate is then restored to its first position so as to be in close 
 contact with the flanges, the spring r is let down, and the two 
 plates are screwed together by the screw x, by which means the 
 positions of the screws qqq are fixed, and the instrument is 
 firmly connected with the three-legged support, w is a small 
 plummet, suspended from a hook vertically under the centre of 
 the azimuth circle, and nearly reaching the ground, so that it 
 indicates the spot at which the instrument may be considered 
 to be set up. When it is required to move the instrument to 
 a new position, the parts uuu being turned on their hinges are 
 put together so as to form a single tapering staff, round which 
 three brass rings, temporarily held on by friction, keep the parts 
 together. In the foregoing description of the adjustments, men- 
 tion was several times made of two related screws, one for ad- 
 justing, and the other for fixing the adjustment. In consequence 
 of this precaution the Theodolite can be borne from place to 
 place with the staff resting on the observer's shoulder without 
 disarranging the connection of the parts 1 . 
 
 366. The account of the Theodolite may be appropriately 
 concluded by describing, as follows, an example of its use which 
 may serve to shew in some degree how it is employed in large 
 
 1 The adjustments of a Theodolite differing in several particulars from that 
 represented by Figure 55, are given in detail in Simms's Treatise on Mathe- 
 matical Instruments, pp. 1722. 
 
352 PRACTICAL ASTRONOMY. 
 
 trigonometrical surveys. In order to determine the difference 
 of longitude between the transit-instrument of the Cambridge 
 Observatory and the place of a clock in the Museums Building, 
 for the purpose of inferring true local mean time from compari- 
 sons of this clock with the transit-clock Hardy, the following 
 process was adopted. Two stations A and B were fixed upon in 
 the neighbourhood of the Observatory, from both of which the 
 N.E. pinnacle of King's College Chapel could be well seen. The 
 former station was nearly opposite the middle of the steps of 
 the Portico of the Observatory, and the differences between its 
 latitude and longitude, and those of a point T vertically under 
 the central point of the transit-instrument, were obtainable 
 simply by measures parallel and perpendicular to the meridional 
 direction through T. The Theodolite being set up at A, a 
 vertical adjustment of the axis of azimuthal motion was care- 
 fully made by the three adjusting screws and the two attached 
 levels (see Art. 361), the adjustment was next tested by turning 
 the large level L, supposing it to be already corrected for 
 indicating the horizontal pointing of the telescope, after which 
 the index error of the altitude-circle was corrected. The index 
 error of the azimuth circle was left of arbitrary amount, the 
 adopted method of observing being such as not to require its 
 correction. At B, a point nearly due south of A, and close to 
 the road to St Neots, a staff was set up vertically by a plumb- 
 line, and being furnished with an adjustible mark which could 
 be placed at the same height from the ground as the central 
 point of the telescope, the positions of that point and the mark 
 could be interchanged by means of the plummet w (in the Fig.). 
 The highest point P of the above-mentioned pinnacle being 
 visible both from A and B, the process of taking the observa- 
 tions was such as follows. 
 
 367. With the Theodolite at A, the instrumental altitudes 
 and azimuths of the point P, and of M, the position of the mark, 
 were recorded. The Theodolite was then transferred to B and 
 the staff to A, and the altitudes and azimuths of P and M were 
 observed from B. Without changing the position of the 
 Theodolite, an altitude of a limb of the Sun, and a transit of 
 
OBSERVATIONS WITH TRANSPORTABLE INSTRUMENTS. 353 
 
 both vertical limbs (see Art. 329), were taken at times indicated 
 by a sidereal chronometer whose error on true sidereal time was 
 ascertained. The distance between A and B, which corresponds 
 to the base-line of a trigonometrical survey, was measured by a 
 Gunter's chain, the whole length of which, and the mean length 
 of the links, were obtained by means of a four-feet measure 
 (lent for the occasion by Professor Adams), which had been 
 compared with a standard of length. This distance was mea- 
 sured along the ground, which declined regularly from A to B. 
 Supposing I to be the measured length, and a the angular 
 depression, as inferred from the altitude of A observed from B, 
 the horizontal interval between A and B is I cos a. Thus there 
 were given the difference of the azimuths both of the horizontal 
 projections of AP and AB } and of the horizontal projections of 
 J5Pand BA, and the length of the projection of AB. With 
 these data the lengths of the horizontal projections of AP and 
 BP were calculated, two angles and the included side of a plane 
 triangle being given. Next, the altitude of the bisected N. or 
 S. limb of the Sun was obtained, for the given time of observa- 
 tion, by calculating, according to known rules, from the E.A., 
 P.D. and semi-diameter derived from the Nautical Almanac, and 
 the result compared with the instrumental altitude, in order 
 that the index error of the altitude-circle might be tested, and, 
 if of sensible amount, be corrected. From analogous data 
 derived from the Nautical Almanac, the Sun's azimuth at the 
 given time of the observation of azimuth was calculated, and 
 compared with the instrumental azimuth, for the purpose of 
 ascertaining the amount of correction to be applied to the latter 
 for index error. After correcting thereby the instrumental 
 azimuth of the projection of BP, the astronomical azimuth of 
 this projection became known, and, by consequence, that of the 
 projection of AP, the two projections making with each other 
 the angle of the above-mentioned plane triangle subtended by the 
 projection of AB. Thus both the length and the azimuthal bear- 
 ing of the projection of AP were determined, and consequently 
 the differences between the R.A. and P.D. of the points A and P, 
 as measured on the earth's surface, were readily inferred. Also 
 c. 23 
 
354 PRACTICAL ASTRONOMY. 
 
 the relative positions in RA. and P.D. of the points A and T 
 having been found by direct measurement (see Art. 366), the 
 differences in RA. and P.D. between the positions of the 
 pinnacle P and the Transit T were at once obtained. 
 
 368. Again, two positions C and D were chosen in the 
 South Court of the Museums Building, the former on the east 
 side and the other on the west side, and from both the top of 
 the N.E. pinnacle of the Chapel could be seen. But neither of 
 them was suitable for measurement of its distances in R.A. and 
 P.D. from the position, say K, of the clock, or for taking obser- 
 vations of the Sun ; on which accounts a position Q on the roof 
 of the S.W. angle of the building was made use of as an inter- 
 mediate station. First of all, the azimuths of P, and of the 
 mark M adjusted after placing the staff at D, were taken from 
 C, and then, the Theodolite retaining its position, the staff was 
 transferred to Q, and, the adjustment of M remaining the same, 
 its azimuth in the new position was observed from G. The 
 length CD was measured by the Gunter's chain, and, as it 
 happened, no correction was required for inclination of the 
 ground. Lastly, the Theodolite being transferred to D and the 
 staff to C, the azimuths of P, and of the mark M first at C and 
 then at Q, were taken from the position D, in the same manner 
 as the analogous azimuths were taken before from C. It will 
 thus be seen that data were obtained for calculating (as shewn 
 in Art. 367) the lengths of the projections of GP and CQ, but 
 not their directions. To obtain these the Theodolite was trans- 
 ferred to the position Q, the staff was placed at C, the heights 
 of the Telescope and the mark M were adjusted to the values 
 they had before their positions were interchanged, and the 
 azimuths of M and of the Sun at a noted time, were observed 
 from Q. The observed azimuth of the Sun was compared with 
 its value at the given time as computed from the data of the 
 Nautical Almanac, and the excess of the latter above the 
 former was applied as correction of index error of the observed 
 azimuth of the projection of QG. Hence the astronomical 
 azimuth of that projection was determined, and that of the 
 projection of CP was at once inferred from the antecedent 
 
OBSERVATIONS WITH TRANSPORTABLE INSTRUMENTS. 355 
 
 observations of the azimuths of CP and 0$ from C. Thus both 
 the lengths and the directions of the projections of CP and CQ 
 were ascertained, and consequently the differences of R.A. and 
 P.D. between C and P, and between C and Q, and, by inference, 
 between Q and P, admitted of ready calculation (see Art. 367). 
 Hence, after obtaining the relative positions in R.A. and P.D. 
 of the point Q and the place K of the clock, which could be 
 done by direct measurement, the relative positions of K and P 
 became known. The relative positions of T and P having been 
 already found (Art. 367), the differences of KA. and P.D. 
 between T and K, that is, between the Transit at the Observa- 
 tory and the Museums' clock, were immediately inferred. 
 
 369. The observations to which the foregoing statements 
 refer were taken at various times between October 27 and 
 November 5, 1866, and between October 29 and November 4, 
 1867, partly in presence of my astronomical class. The 
 measured length of the base-line AB was 6584,3 inches, and 
 that of CD 3259,4 inches. The positions A and B were west- 
 ward, and those of C and D eastward, from the point P. The 
 immediate purpose of the observations with the Theodolite and 
 the auxiliary measures was to determine the differences of R.A. 
 and P.D. between T, the place of the Observatory Transit, and 
 T' that of a small portable Transit, which was occasionally set 
 up, in the manner described in Art. 358, on the wall erected for 
 supporting out-door instruments. (See Note to Art. 306.) This 
 wall was raised a few feet above the level of the roof mentioned 
 in Art. 368, so that the Transit could be conveniently placed and 
 adjusted for approximately obtaining the local sidereal time. By 
 measurement A was 2941 inches south and 535 inches east of T. 
 By calculation from the observations at A and 5, it was found 
 that the horizontal projection of AP was 70482 inches, and, 
 taking the above measures into account, that T was north of P 
 by 42191 inches, and west of P by 59076 inches. Again, the 
 distances in R.A. and P.D. between T' and Q having first been 
 obtained by auxiliary measurement, by taking these into account 
 and calculating from the observations made with the Theodolite 
 in the South Court and on the Roof, it appeared that T was 
 
 232 
 
356 PRACTICAL ASTRONOMY. 
 
 south of P by 7457 inches and east of P by 6788 inches. 
 Hence it follows that T was north of T' by 49648 inches, and 
 west of T' by 65864 inches. By adopting these values, which 
 are measures on a surface cutting the direction of gravity at 
 right angles, and applying small corrections due to the figure of 
 the earth, it resulted that the N. latitude of the Museums' 
 Transit was less than that of the Observatory Transit by 40",8, 
 and its longitude eastward greater by 5 s , 875. Hence from the 
 known latitude and longitude of the Cambridge Transit it re- 
 sults, that the latitude of the Museums' Transit is 52. 12'. 10",8 
 north, and its longitude 28 S ,62 east of Greenwich. It may 
 be noticed that the previously known geographical place of 
 the Cambridge Transit here answers the same purpose as 
 determining by astronomical observations the geographical 
 places of certain stations in Trigonometrical Surveys. 
 
 By measures taken on March 15, 1867, it was ascertained 
 that the latitude of K, the place of the Museums' clock, was 
 greater by 132 inches than that of T', and its east longitude 
 greater than that of T' by 70 inches. Hence, finally, the 
 longitude of the clock is 28 S ,63 east of Greenwich, and its lati- 
 
 370. The Sextant. Figure 56 was taken from a drawing 
 representing one of Ramsden's sextants, which differed little in 
 construction from those afterwards made by Troughton 1 . By 
 reference to the figure, the different parts and their uses may 
 be described as follows. The rays from an object are first 
 incident on the plane reflector /, called the index-glass, and 
 after reflection are incident on a second glass, called the horizon- 
 glass, the lower half of which, i, is silvered for producing a 
 second reflection. After this reflection, the rays are received 
 by the Telescope AF and reach the eye at F, at the same time 
 that a second object is viewed directly with the Telescope by 
 rays that have passed through o, the unsilvered part of the 
 same glass. The celestial arc between the two objects subtends, 
 
 1 A sectant of Troughton's construction is figured in page 50 of Simms's 
 Treatise, and the parts, adjustments, and modes of using it are described in 
 pages 5056 of that work. 
 
OBSERVATIONS WITH TRANSPORTABLE INSTRUMENTS. 357 
 
 as is known from Optics, an angle equal to double the mutual 
 inclination of the planes of the reflectors. The arc BG is so 
 graduated that this double angle is immediately read off by 
 means of the vernier F, which is connected by a bar with the 
 index-glass 7, and turns with it about an axis passing through 
 the centre of the graduation. The vernier is detained by the 
 elastic plate s, and after being clamped to the limb by the 
 screw m, is movable by the slow-motion screw n. The gradua- 
 tion proceeds from B towards C, extending 137 from zero in 
 
 Fig. 56. 
 
 that direction, and 2 from zero in the opposite direction, and 
 each graduation-interval is 20'. As 39 graduation-intervals are 
 equal to 40 vernier- intervals, it follows that the arc can be read 
 off to 30", and by estimation to a fraction of that quantity. 
 [In Troughton's sextants, the graduation -interval is 10', which 
 
358 PRACTICAL ASTRONOMY. 
 
 by the vernier is subdivided into intervals of 10", so that the 
 graduation can be read off to an estimated fraction of 10".] 
 Two sets of dark glasses D and d are so placed that by turning 
 the former the sun-light reflected at / may be intercepted, and 
 by turning the other the sunlight entering at o may be inter- 
 cepted, whereby the light in each case can be sufficiently mode- 
 rated before it enters the eye. In certain circumstances after- 
 wards mentioned, it is preferable to place darkening glasses at 
 the eye-end F of the Telescope. 
 
 371. The adjustments of a Sextant are executed by the 
 instrument-maker, and generally so as not to require, or admit 
 of, alteration by the observer. It is right, however, that the 
 observer should know how to test their accuracy, the methods 
 of doing which will accordingly here be given. The requisite 
 adjustments are these : (1) the index-glass is required to be 
 perpendicular to the plane of the instrument ; (2) the horizontal 
 glass should be perpendicular to the same plane ; (3) after 
 setting the index of the vernier to of the arc, the horizontal- 
 glass must be parallel to the index-glass : (4) the optical axis of 
 the Telescope should be parallel to the plane of the instrument. 
 To test the first adjustment, hold the instrument horizontally 
 with the index-glass towards the eye, and look at the graduated 
 arc partly directly and partly by oblique reflection at the glass. 
 If the parts thus viewed appear at their junction to form a 
 continuous arc, the index-glass is adjusted. In fact, by this 
 means the plane of the instrument is practically defined. The 
 second adjustment is tested by sweeping with the index-bar to 
 see whether the reflected image of a star, or other object, passes 
 exactly over its position as seen directly with the Telescope. 
 If not, the error can be corrected by means of a capstan-headed 
 screw (not seen in the figure), which acts transversely on the 
 lower part of the frame that supports the horizontal glass. The 
 third adjustment is verified, after setting the vernier-index at 0, 
 by finding that the images of an object, for instance the Sun, seen 
 directly and by reflection coincide, or appear as one. If this be 
 not the case, the deviation is due to index error, the amount of 
 which, as no instrumental means are provided for correcting it, 
 
OBSERVATIONS WITH TRANSPORTABLE INSTRUMENTS. 359 
 
 is allowed for in the reduction of each observation, after being 
 ascertained by the following process. The vernier-index being 
 first set and clamped to a reading of about 30' on the side of 
 zero towards C, on looking towards the Sun two images will be 
 seen nearly in contact, or a little overlapping, and may be 
 brought to exact contact by turning the tangent-screw n. This 
 is, in fact, the process for measuring the Sun's diameter by the 
 Sextant, the images in contact being those of opposite limbs. 
 Generally it is best to hold the Sextant horizontally for this 
 purpose, the horizontal measure not being affected by refraction. 
 The measure thus obtained is to be compared with the Sun's 
 diameter as deduced for the given time from the Nautical 
 Almanac, and the algebraic excess of the latter above the 
 measured diameter is the required correction for index error. 
 This correction is also obtainable as follows by instrumental 
 observation without having recourse to the Nautical Almanac. 
 
 372. The graduation being extended, as stated in Art. 370, 
 2 on the side of zero towards B, the index may be set about 30' 
 from zero on that side, and then an exact contact of the two 
 images be effected as before. The interval between zero and 
 the place of the index for this contact may then be derived 
 from the graduation by means of the vernier. Hence if D, D' be 
 the first and second measured intervals and D+e be the interval 
 between the true place of zero and the vernier index on the 
 side towards G, D' e will be the corresponding interval on the 
 opposite side. Hence as each interval measures the Sun's 
 diameter, D + e = D' e. Consequently the correction e of D 
 is equal to ^(D r D), and is positive or negative according as 
 the right-hand interval D' is greater or less than the left-hand 
 interval D. The adopted value of e should be the mean of several 
 determinations. In taking the above-mentioned observations of 
 the Sun for determining the index error) a dark glass should be 
 placed before the eye-piece at F instead of using the shades D 
 and d, because some error might be induced by refraction through 
 the sets of glasses of which these are composed. (See Art. 370.) 
 
 373. For testing the accuracy of the fourth adjustment two 
 parallel bars are fixed in the field of view of the Telescope 
 
360 PRACTICAL ASTRONOMY. 
 
 very nearly at equal distances from its centre. These bars may 
 be placed so as to be very approximately parallel to the plane of 
 the instrument by causing the reflected image of a slow- 
 moving star, as Polaris, or of a fixed terrestrial point, to move 
 along one of the bars by combining a rotation of the Tele- 
 scope by hand with a change of inclination of the reflector / 
 by the bar. This arrangement being made, let us suppose that 
 the plane of collimation of the Telescope exactly bisects the 
 interval between the bars. Then selecting two objects, as the 
 Sun and Moon, or two stars distant 90 at least from each other, 
 bring their images into contact at the bar nearest the plane of the 
 sextant, clamp the index, and after bringing by change of position 
 of the instrument the images upon the other bar, notice whether 
 they are still in contact. Assuming the axis of the Telescope 
 to be parallel to the plane of the instrument, these two measures 
 of the distance between the objects would be taken in planes 
 equally inclined to the telescope's plane of collimation on oppo- 
 site sides, and therefore ought to be equal. If a separation of 
 the images in the second measure shews that this condition 
 is not fulfilled, the requisite adjustment of the axis is to be 
 effected tentatively by shifting two screws which fix the place 
 of an up-and-down piece, whereby the object-end of the Telescope 
 is raised or lowered. This adjustment, as originally performed 
 by the maker of the instrument, rarely requires alteration. 
 
 374. The Sextant is an instrument indispensable to the 
 navigator, being capable of effecting on ship-board several 
 of the purposes for which, as already explained, a portable 
 altazimuth is employed on land. Thus the altitude of the Sun, 
 or other heavenly body, is obtainable by taking the sea-horizon 
 for the second object, the Sextant measuring the vertical arc 
 between any object and the horizon. (When used for such 
 purposes the instrument is held with the left-hand by the 
 handle H (Fig. 56), while the vernier-bar is moved for com- 
 pleting an observation with the right-hand by turning the 
 slow-motion screw n.) The local time of day may be deduced 
 from the chronometer times of taking equal altitudes of a 
 known star, or of the Sun, before and after its meridian 
 
OBSERVATIONS WITH TRANSPORTABLE INSTRUMENTS. 3G1 
 
 passage (see Art. 355). In the case of the Sun a correction 
 has to be applied for change of its declination in the interval 
 between the observations (see Art. 356), and, whether the Sun 
 or a star is observed, a small correction is required for the ships 
 way in the same interval, the rate of sailing (exclusive of the 
 effect of currents) being ascertained by casting the log, and the 
 course being indicated by the compass 1 . The practical method 
 of finding the latitude of the place of the ship is to watch a 
 little before noon the changes of altitude of the Sun's limb, 
 as shewn by the readings of the vernier for measures of altitude 
 till the maximum height is reached, and then to verify this 
 value by seeing whether the reflected image of the limb just 
 grazes the horizon-line when the instrument is moved to and 
 fro to right and left. The Sun's altitude at the time of apparent 
 noon of the place having been thus observed, and corrections for 
 refraction and parallax applied, the Latitude of the place is 
 readily inferred from the Sun's geocentric declination at the 
 time, as calculated from the data of the Nautical Almanac. 
 This calculation, however, implies that the longitude of the 
 ship's place is at least approximately known by comparing a 
 mean time chronometer regulated to shew Greenwich mean 
 time with the chronometer used for determining the local 
 mean time. The foregoing method of finding the ship's latitude 
 is suitable for a voyage, and is much practised by seamen ; 
 but evidently when the error of a sidereal chronometer on 
 local sidereal time is found, and consequently the time at which 
 a known star crosses the meridian may be determined, the 
 latitude might be more simply deduced from an observation of 
 the meridian altitude of such star. 
 
 375. The Sextant is also employed in a very important 
 manner for measures of Lunar Distances, that is, distances in 
 celestial arc between the Moon's bright limb and the Sun's 
 limb, or the limb of a Planet, or a bright known star. In 
 taking this observation the Sextant has to be held so that 
 the eye of the observer and the two objects are in the plane 
 
 1 A formula for calculating this correction is investigated in Godfray's 
 Treatise on Astronomy, Art. 146. 
 
362 PRACTICAL ASTRONOMY. 
 
 of the instrument as defined in Art. 371. Each measure 
 requires to be accompanied by a note of time, and also by simul- 
 taneous 1 observations of the altitudes of the two points between 
 which the measure is taken, for the purposes of calculating the 
 difference of the azimuths, and clearing the observed distance 
 of refraction and parallax. The corrections of the observed alti- 
 tudes for refraction may be calculated from the usual Tables, the 
 argument being Apparent Zenith Distance. The correction (p) 
 for parallax in the case of the Moon may be calculated with suffi- 
 cient exactness by the formula smp =sinPsin^, P being the 
 horizontal parallax, and in the case of the Sun, or a Planet, by 
 the approximate formula p=P$mz. After applying these correc- 
 tions the geocentric distance is calculated from the corrected 
 zenith distances and the difference of azimuths, and from the 
 result, as will be shewn in the next Division, the Longitude of 
 the ship's place is inferred. 
 
 376. The altitude of a celestial object is also sometimes 
 observed with a Sextant on land, in which case it is necessary 
 to provide an Artificial Horizon. The one exhibited for illus- 
 tration in the Lectures consists of a small oblong iron trough 
 containing mercury for a reflecting surface, and placed under 
 a covering having two slanting sides of plate glass for protection 
 against disturbance by the wind. The plates are equally inclined 
 from a vertical plane in opposite directions, so that a ray in a 
 transverse plane, after passing through one plate, and undergoing 
 reflection at the mercury, passes through the other. The Sextant 
 measures the angle between the actual direction of the object and 
 that of its reflected image, and consequently gives the double of its 
 altitude above the horizon. In practising this method, observa- 
 tions should be taken an equal number of times with reverse 
 positions of the protecting covering, and the mean result be 
 adopted, in order to eliminate any error that may arise from 
 unequal refractive action of the two plates on the ray. For a 
 
 1 A rule is given in Simms's Treatise, p. 56, according to which the three 
 arcs can be virtually measured at the same time by a single observer, the required 
 altitudes being obtained by interpolation from altitudes taken before and after 
 the measure of distance. 
 
OBSERVATIONS WITH TRANSPORTABLE INSTRUMENTS. 368 
 
 like reason the eye-glass at F should be armed with a single 
 dark glass in place of using the shades D and d (see Arts. 370 
 and 372). 
 
 377. Troughton's Reflecting Circle. The construction and 
 uses of this instrument are for the most part the same as those 
 of the Sextant, from which it differs chiefly with respect to the 
 following particulars. It can be mounted on a tripod stand 
 having three adjusting foot-screws, and being at the same time 
 counterpoised, and capable of movements about a vertical and 
 a horizontal axis, it admits of adjustments whereby the plane 
 of collimation of the telescope is made to pass through the two 
 objects the angular distance between which has to be measured. 
 By means of a plumb line, or by the application of a small 
 spirit-level to the arms bearing the foot-screws, the position of 
 the vertical axis of motion may by the screws be approximately 
 adjusted, whence that of the horizontal axis follows by reason 
 of the mechanical construction. The Reflecting Circle thus 
 mounted is convenient for use on land in taking measures of 
 celestial arcs. Again, it is furnished with handles by which when 
 dismounted it can be so held in the hand as to allow of measures 
 being taken in reverse positions of the graduated circle. This 
 circle is graduated all round, and there are three vernier-indices 
 for reading its indications, separated by intervals of 120, the 
 minutes and seconds being read from each, and the degrees 
 being taken from that to which the tangent-screw is attached. 
 It was considered by Troughton that the mean of the three 
 readings got rid of errors that might arise from eccentricity 
 of the centre of the graduated arc relative to the axis of motion 
 of the index-glass, and that by the mean of the two sets of 
 readings in reverse positions the index error was corrected, 
 and errors due to refraction through the dark glasses were 
 counteracted. In other respects the adjustments are the same 
 as those already explained relatively to the Sextant (Art. 371), 
 and need not be stated here. By the processes above mentioned 
 the indications of the Reflecting Circle can be read with a 
 precision hardly required by the observations for which it is 
 adapted, which do not generally admit of a high degree of 
 
364 PRACTICAL ASTRONOMY. 
 
 accuracy. For seamen the Sextant is a more serviceable in- 
 strument 1 . 
 
 378. Borda's Repeating Circle. This instrument resembles 
 in some respects a Troughton's Reflecting Circle mounted on a 
 tripod stand, being like the latter counterpoised, and capable of 
 having the plane of collimation directed through two objects 
 by means of rotations about a vertical axis and a horizontal 
 axis. It differs, however, essentially from the Reflecting Circle 
 in the respect that instead of measuring celestial arcs by the 
 intervention of reflections, it employs for this purpose two 
 telescopes, which can be turned about a common axis passing 
 through the centre of a circle graduated on one face for taking 
 the measures. These telescopes and the circle (which is 
 between them) may be moved about that axis independently 
 of each other, and each telescope admits of being clamped to 
 the circle, and adjusted in position by slow-motion screws. One 
 of the telescopes has four arms radiating from it in rectangular 
 directions, and carrying four verniers by which the graduation 
 is read off. One of the arms has at its end a tangent-screw- 
 and-clamp apparatus by which the telescope is attached to the 
 graduated face of the circle and adjusted, the clamping being 
 effected by turning with the fingers the teeth of a small racked 
 circle. The other telescope is clamped to the circle and 
 adjusted by like apparatus. To this telescope a large spirit- 
 level is fixed, which has to be adjusted so that its bubble is in 
 mid-position when the axis of the telescope is pointed horizon- 
 tally (see Arts. 361 and 366). For adjusting the axis of hori- 
 zontal motion vertically, the feet of the tripod-stand are 
 adjustible by screws, which rest in three brass sockets the 
 heights of which are also regulated by screws, by which means 
 and by the adjusted spirit-level, the required vertical adjust- 
 ment may be effected and verified. This being done, the 
 adjustment of the horizontal axis of motion follows from the 
 mechanical construction. After clamping a collar and connected 
 
 1 Of Troughton's Reflecting Circle, an example of which is exhibited in the 
 Lectures, a particular description, accompanied by a Figure, is given in Simms's 
 Treatise, pp. 57 GO. 
 
OBSERVATIONS WITH TRANSPORTABLE INSTRUMENTS. 365 
 
 tongue to the vertical column supporting the instrument, the 
 whole may be turned by means of a slow-motion screw about 
 the vertical axis. By like means, when the telescopes are 
 clamped to the circle, slow motion about the axis of revolution 
 passing through the centre of the graduation is given in 
 common to the circle and the telescopes. After the plane of 
 collimation is adjusted for an observation, motion about the 
 horizontal axis is prevented both by the counterpoise and by 
 a clamp-and-screw acting upon a small fixed arc concentric 
 with that axis. The mode of using this instrument for mea- 
 suring celestial arcs is such as follows. 
 
 379. Let S l and $ 2 be the two objects in the plane of colli- 
 mation, the direction from $ t to $ 2 being that in which the 
 graduation proceeds. Let the telescope which carries the four 
 verniers, and is consequently contiguous to the graduated face 
 of the circle, be signified by A, and the telescope contiguous to 
 the opposite face of the circle by B, and let be the position 
 of the centre of the graduation. By the motions about the 
 vertical and horizontal axes, and by the apparatus for clamping 
 the telescopes to the circle, first point B towards 8 a and A 
 towards 8 V and by the slow-movements make the pointings as 
 exact as possible. Next, turn the circle and clamped telescopes 
 till B points to the object 8 V and A is consequently made to 
 point farther in the direction opposite to that of the graduation 
 by the angle $ x OS^ Hence if A be now undamped, and by 
 independent movement, with adjustment by clamp and tangent- 
 screw, be made to point exactly towards 8 V it will be moved 
 in the direction of the graduation through an angle equal to 
 twice L SflSf Hence half the excess of the second reading for 
 the pointing of A above the first gives the arc between the 
 two objects. A second measure may be obtained by proceeding 
 as follows. Turn the circle, with the telescopes clamped to it, 
 so as to make A point again towards S l9 by which operation 
 B will be brought to point farther in the direction from $ 2 
 towards S^ than its original pointing by twice the angle 
 SfiSp Then bring B by its independent motion to point to 
 >S 2 , and adjust the pointing of A to 8 l solely by the slow 
 
366 PRACTICAL ASTRONOMY. 
 
 motions of the circle about the vertical and horizontal axes 
 (because the circle-reading for the pointing of A must not be 
 altered), and that of B to $ 2 by combining the same means 
 with its proper tangent-screw adjustment relative to the circle. 
 Supposing the telescopes thus to be correctly pointed, A to S l 
 and B to $ 2 , all circumstances will be the same that they were 
 at first, and the measure may be repeated by proceeding exactly 
 in the same manner as before. Accordingly the telescope A 
 will be still further advanced in the direction of the graduation 
 by twice the angle Sf)S y If n such measures be taken, the 
 required angle will evidently be the excess of the last reading 
 for the pointing of A above the first, divided by 2n. In the 
 same manner a succession of altitudes of Polaris, or other star, 
 might be obtained, the horizontal pointing of the telescope 
 which carries the spirit-level supplying the place of a second 
 object. It was formerly thought that by the repeating principle 
 errors of reading off and of graduation might in great measure 
 be eliminated. But the methods of graduating and subdividing 
 the intervals are now brought to such perfection that there 
 appears to be no reason for having recourse to that principle, 
 of which, moreover, it is to be said that it leaves uncorrected 
 any error that is common to all the measures. 
 
 380. The foregoing explanations apply to a Borda's Repeat- 
 ing Circle (exhibited in the Lectures) which was made by 
 Troughton according to the construction that he adopted at first. 
 Subsequently he produced the instrument in the modified form 
 represented by Fig. 1 in Plate xxm. of Pearson's Practical 
 Astronomy, and described in pp. 498 509 of that work. The 
 same form is figured and described in Loomis's Introduction 
 to Practical Astronomy, pp. 103 106. The principal differ- 
 ence between this and the previous instrument is, that in the 
 new form a graduated horizontal circle surrounds, and is fixed 
 to, the supporting column at its base, and has a vernier for 
 reading the graduation, with clamping and slow-motion appa- 
 ratus. As the circle moves with the column about the vertical 
 axis, the repeating circle may by this addition be used as an 
 altazimuth instrument. 
 
MISCELLANEOUS ADDITIONAL SUBJECTS. 
 
 MISCELLANEOUS ADDITIONAL SUBJECTS. 
 
 381. Different methods of determining Terrestrial Longi- 
 tude. The methods all depend on the principle of finding the 
 mean solar, or sidereal, time at a certain reference meridian 
 (which I shall suppose to be that of Greenwich), contem- 
 poraneous with a given instant of the mean solar, or sidereal, 
 time at the meridian of the place whose longitude is required, 
 the longitude being equal to the difference, less than 12 h , 
 between the two times, and named East or West Longi- 
 tude according as the time of day at the reference meridian is 
 earlier or later than that at the other. The various means 
 that have been employed for determining differences of longi- 
 tude may be described as follows : 
 
 (1) By the transfer of chronometers. This method requires 
 arrangements to be made for setting up a transit-instrument 
 and a sidereal clock at each of the two stations whose difference 
 of longitude has to be found, for the purpose of ascertaining 
 by transit-observations of known stars the errors of the two 
 clocks on the true local sidereal times. The transit-instruments 
 should be well adjusted to the respective meridians, the 
 chronometers employed should each be transferred several 
 times alternately from one station to the other, and be compared 
 at the beginning and end of each transfer with the respective 
 clocks at the places of starting and arrival. If sidereal chrono- 
 meters be used, the comparisons are to be made by the 
 intervention of a solar chronometer (see Art. 264). Supposing 
 that the rates of the chronometers, well ascertained at the 
 same time, are allowed for, it will be seen that by these means 
 the true sidereal time at one of the places is transferred to the 
 other, and that as the difference of simultaneous local sidereal 
 times is thus known, the difference of the longitudes is deter-, 
 mined. (See what has been said on this method in Art. 262.) 
 
 By observations conducted on the principles above stated, 
 and occupying the interval from the end of June to the end of 
 
368 PRACTICAL ASTRONOMY. 
 
 September, 1844, the Astronomer Royal obtained the Longitude 
 West from Greenwich to Liverpool, from Liverpool to Kings- 
 town, from Greenwich to Kingstown directly, and from Kings- 
 town to a station in the Island of Valentia at the western 
 extreme of Ireland. For taking the transit observations of 
 stars at Greenwich and Liverpool, the transit-instruments of 
 the Observatories at those localities were used, and for ob- 
 taining like observations at Kingstown and Yalentia, transit- 
 instruments and sidereal clocks were temporarily provided, and 
 special attention was given to steadiness of mounting, and to 
 the usual exact methods of correcting transit-observations for 
 instrumental errors. According to the final results the stations 
 at Liverpool, Kingstown, and Valentia were respectively 
 ll ra .59 8 ,86, 24 m .31 s ,13, and 41 m .23 s ,12 west of Greenwich, 
 with an average probable error of O s ,04. (All particulars 
 relating to these observations are given in Vol. xvi. of the 
 Memoirs of the Royal Astronomical Society, pp. 55 276.) 
 
 (2) By Galvanic Signals. Supposing that the chrono- 
 graphic method of taking transit-observations has been arranged 
 at two localities (see Arts. 129 and 130), and a galvanic con- 
 nection by conducting wire is established between them, the 
 difference of their longitudes may be obtained as follows. The 
 transit of a given star, taken and recorded according to that 
 method by means of the transit-clock and galvanic apparatus at 
 the transmitting Station, is recorded also at the receiving 
 Station by noting the times, by the transit-clock of the 
 latter, of the jerks of a galvanic needle set in motion by the 
 contacts made at the instants of the passage of the star across 
 the transit- wires. Assuming that the errors of the clocks on 
 the true local sidereal times have been accurately determined, 
 and that the transmission of the signals between the stations is 
 instantaneous, each difference between corresponding recorded 
 times corrected for the clock-errors is equal to the difference 
 between simultaneous local sidereal times, and is therefore a 
 determination of the difference of the longitudes. The con- 
 cluded result should be the mean of all such determinations 
 derived from repeated series of observations. The result may, 
 
MISCELLANEOUS ADDITIONAL SUBJECTS. 369 
 
 however, be affected by difference of the personal equations 
 of the observers, to obviate the influence of which the series of 
 observations should be repeated after an interchange of the 
 places of the observers. Also it is desirable to transmit several 
 sets of signals in equal numbers in opposite directions, to 
 ascertain whether the transmission occupies a sensible interval 
 of time, which would be the case if the difference of longitude 
 be found to be different for the two directions of transmission. 
 (See Arts. 263 and 264.) For the convenience of recording at 
 the receiving station each set should be divided into batches of 
 about six transmissions, commencing at times previously agreed 
 upon. 
 
 A determination of the Longitude of the Cambridge Obser- 
 vatory from Greenwich was made according to the foregoing 
 method on May 17 and May 18, 1853, being, I believe, the first 
 accomplished in this country by such means. All the above- 
 mentioned precautions for ensuring accuracy were attended to ; 
 but as the galvanic connection with Greenwich did not extend 
 beyond the Cambridge Telegraph Station of the Great Eastern 
 Railway, the Cambridge times of sending and receiving signals 
 were recorded at that Station by chronometers, which before 
 leaving and after returning to the Observatory were compared 
 with the Transit-clock. The clock-errors were obtained by two 
 independent methods. In one, the errors were deduced in the 
 usual way from the apparent RA. of fundamental stars in- 
 cluded in the respective lists that were in use at the two Obser- 
 vatories, excepting that the adopted mean RA. 1853,0 of the 
 stars observed at Cambridge were all the same as their mean 
 RA. in the Greenwich list... In the other method the clock-errors 
 were inferred from transit-observations of identical stars taken 
 by the two observers on the nights of May 17 and 18, both 
 before and after recording the signal times, the observers 
 changing places after the first night. In calculating the 
 difference of the simultaneous sidereal times, the same apparent 
 R.A., derived from the Greenwich Twelve-year Catalogue, were 
 used both for the Greenwich and the Cambridge transits, in con- 
 sequence of which this process, while it increased the number of 
 c. 24 
 
370 PRACTICAL ASTRONOMY. 
 
 available stars, was not affected by small errors in their assumed 
 apparent RA. The mean results of the two methods differed by 
 only O s ,001. The final result of these calculations gave for the 
 Longitude of the Cambridge transit-instrument 23 S ,027 East. 
 Subsequently (in 1854), I applied a correction derived from 
 a computation of the effects of the forms of the pivots according 
 to the formula obtained and exemplified in Arts. 93 102, 
 whereby the Longitude East was altered to 22 S ,7'53, which is 
 less by O s ,79 than the value previously obtained by transfer of 
 chronometers 1 . 
 
 The difference of Longitude between the Greenwich and Paris 
 Observatories was determined by operations carried on from 
 May 26 to June 4, and from June 12 to June 24, 1854, which 
 closely resembled those described above as regards both the 
 galvanic signalling and the transit-observations, excepting that 
 there was direct galvanic communication between the two 
 Observatories. The observers were M. Faye and Mr Dunkin, 
 and it resulted from their observations that the Longitude of 
 the Paris Observatory is 9 m . 20 8 ,63 East from Greenwich, which 
 is about 8 ,9 less than the value which had been previously 
 accepted. (The particulars of this determination are given in 
 Vol. xxxix. of the Comptes Eendus, 1854, pp. 552 566.) 
 
 (3) By observations of Lunar Distances. This method of 
 finding the Longitude is practicable on ship-board, and is 
 specially important in long voyages. It is mainly applied in 
 determining from time to time the error of a chrono- 
 meter which should shew the navigator Greenwich mean 
 time. We have already considered in what way the Sextant 
 is employed for measuring the great-circle arc between the 
 Moon's bright Limb and either a bright star, or the Limb of the 
 Sun or a Planet, and how by applying corrections for refraction, 
 semi-diameters, and parallaxes, the geocentric, or reduced dis- 
 tance between the two objects is obtained. It remains to state 
 
 1 This account of the process employed for obtaining the Longitude of the 
 Cambridge Observatory by galvanic signals is intended to be supplementary to 
 the statements made in Arts. 263 and 264. For full details see Vol. ix., pp. 487 
 514, of the Camb. Phil. Transactions. 
 
MISCELLANEOUS ADDITIONAL SUBJECTS. 371 
 
 the process by which the longitude of the spot where the 
 observation was taken is thence inferred. This is done by 
 means of the Tables of Lunar Distances from certain bright 
 stars, and from the centres of the Sun and bright Planets, given 
 in the Nautical Almanac for every third hour of Greenwich 
 mean time each day of every month, the calculation of the 
 distances being made according to the Lunar Theory, and the 
 places of the objects selected for such calculation being well 
 known, and convenient for observation of the distances. The 
 G. M. T. corresponding to the reduced lunar distance may then be 
 inferred from the calculated value nearest that distance by a 
 simple proportion, supposing the Moon's motion to be uniform 
 during an interval of 3 h . If, however, the results obtained by 
 first differences be not sufficiently accurate, the calculations 
 may be carried to second differences by the following formula 
 adapted to this case from the known general formula of inter- 
 polation. Let D be the lunar distance reduced from the 
 observations, D lt D z , D 3 three consecutive tabular distances, of 
 which D z is the nearest to D, and let t 2 - 3 h , 2 , t z + 3 h be the 
 corresponding Greenwich Mean Times. Then if t be the re- 
 quired G. M. T. the interpolated value is given by the equation 
 
 - f , [CP-J.HJ-A) (D-DJ(D-D,)} 
 - J + \(D S - D t ) (D, - zy + (D, - DJ (D, - DJ J 
 
 as may be readily verified by substituting separately D t and Z> 3 
 for D. The longitude results from tl 
 the local mean time of the measure. 
 
 for D. The longitude results from the difference between t and 
 
 EXAMPLE. Let it be required to find the a M.T. at which the 
 reduced distance between the Moon's centre and a Aquilse was 
 39 50' 31" on March 15, 1879. 
 
 By data in the Nautical Almanac, t 2 is midnight of March 15, 
 D v = 40 25' 44", D 2 = 39 21' 6", D 3 = 38 18' 49". Hence as D = 39 
 50' 31", we get D-D l = - 2113", D- D 2 = 1765", D - D 3 = 5502", 
 D S -D^- 7615", D 3 -D 2 = - 3737", D^D^ 3878". The remainder 
 of the calculation may be conducted by five-figure logarithms as 
 follows : 
 
 242 
 
372 PEACTICAL ASTRONOMY. 
 
 L. (Z> 8 -1^) = 3,88167 Z. (J^ - 2 ) = 3,58861 
 
 L. (D 8 -J) 2 ) = 3,57252 n L. (Z^-DJ = 3,881 67 rc 
 
 X. (D. - t ) (/>, - Z> 2 ) - 7,45419 Z. (D, - D a ) (Z>. - DJ - 7,47028 n 
 
 ^.tf. =2,54581 A.C. = 2,52972^ 
 
 L. (D - Dj = 3,32490 n L. (D - D a ) = 3,24674 
 
 L. (D - D a ) = 3,24674 Z. (D - D 3 ) = 3,74052 
 
 Z. 10800 s =4,03342 Z. 10800 s = 4,03342 
 
 Sum ( = Z. - 1415 8 ,4) = 3,15087 n Sum ( = Z. - 3551 8 ,4) = 3,55040 n 
 
 .'. G.M.T. = March 15, 12 h -1415 s ,4- 3551 S ,4, 
 = March 15, 10 h 37 m 13 s . 
 
 This example is calculated in p. 509 of the Naut. Aim. for 
 1879, and the resulting G.M.T. is March 15, 10 h . 37 m . 19 8 ., 
 which, if not as accurate as that obtained by the preceding 
 calculation, has the advantage of having been computed accord- 
 ing to rules which dispense with using + and signs. 
 
 (4) By observations of occultations of stars and planets by 
 the Moon. These phenomena have been under consideration in 
 Arts. 312 314, where it is shewn that by calculations founded 
 on a noted time of the disappearance or reappearance of a star or 
 planet at the Moon's Limb, an equation of condition can be 
 formed involving, as unknown quantities, corrections of the 
 adopted K A. and P. D. of the bodies and of assumed semi- 
 diameters and parallaxes, together with corrections of the 
 assumed latitude and longitude of the place of observation. 
 For the use of such an equation in determining terrestrial 
 longitude, it is necessary to deduce subsequently, from the 
 results of Meridian Observations (as those of Greenwich), and 
 from other independent means, all the corrections except that 
 of the assumed longitude, in order that the equation may give 
 this correction and thereby the required longitude. It is, 
 however, to be remarked that the calculation of the equation of 
 condition implies that an approximate value of the longitude is 
 already ascertained, as by the observation of a lunar distance at 
 the same place, or other available means. Supposing the local 
 sidereal time of the occultation to be well ascertained, and one 
 or more observations of lunar distances to be taken about the 
 same time, the observer may proceed to calculate an approxi- 
 
MISCELLANEOUS ADDITIONAL SUBJECTS. 873 
 
 mate longitude by the foregoing method (3). The observa- 
 tions may be made either on land or ship-board ; but in the 
 latter case there must be at least one observation of lunar 
 distance not long before the occultation, and one of the same 
 star not long after, and from each the geocentric lunar distance 
 is to be calculated in the usual way, in order to deduce by 
 interpolation the geocentric lunar distance at the time of the 
 occultation. After obtaining by such means a value of the 
 longitude sufficiently approximate for calculating the equation 
 of condition, and correcting this approximation by means of 
 that equation as shewn above, the resulting longitude is likely 
 to be much more exact than one obtained directly by a 
 measured lunar distance, inasmuch as in the case of the 
 occultation the lunar distance, being the Moon's semi-diameter, 
 is accurately known. The method, as applied at sea, gives the 
 longitude of the spot at which the time of the occultation was 
 noted, independently of the change of the Moon's place and the 
 ship's way, if both motions be uniform in the interval between 
 the observations of the lunar distances. (See for details of 
 calculation the Appendix of the Naut. Aim. of 1854.) 
 
 (5) By transit-observations of the Moon and Moon-culmi- 
 nating Stars. Stars are designated as moon-culminating, when 
 they are near the Moon's parallel of Declination, and have 
 Eight Ascensions not much differing from the Moon's. From 
 transits of stars so situated, taken together with a transit of the 
 Moon's bright limb, across the meridians of two places on the 
 same day, the difference of the Longitudes of the two places 
 may be calculated on the following principles. On account of 
 the variation of the Moon's R. A., the interval between the 
 transit of a given star and the Moon's Limb will be different at 
 the two meridians, and the difference will be equal to the 
 change of R. A. of the bright limb in the interval between its 
 transits. Now that change is obtained by comparing the 
 intervals between the transits of star and limb at the two 
 places; and since the Variation in Right Ascension of the 
 Moon's bright limb in the interval between transits across two 
 meridians equidistant from Greenwich, and one hour distant 
 
374 PRACTICAL ASTRONOMY. 
 
 from each other, is given in the Naut. Aim. for the Upper and 
 Lower Culminations (being derived from data furnished by the 
 Lunar Theory), it follows that the difference of the Longitudes 
 may be inferred from a simple proportion. If, however, the 
 difference so found be large, the calculation should be repeated 
 with the value of the variation of the R. A. of the Limb in one 
 hour of Longitude corresponding to the middle of the interval 
 between the observations, which value, since the epoch would 
 be sufficiently known by the result of the first approximation, 
 might be obtained by interpolation to the second order from 
 values given in the Naut. Aim. 
 
 This method, as being differential, is independent of small 
 errors in the places of the Moon and Stars. It is also applicable 
 when the two localities are far apart. For instance, moon-cul- 
 minations observed at the Cambridge Observatory have been 
 supplied for determining by means of corresponding observations 
 the Longitudes of places in Chili. 
 
 (6) By observations of Eclipses of Jupiter's Satellites. The 
 Naut. Aim. gives, as results of theoretical calculation, the 
 configuration of the Satellites every day on which they are 
 visible from the earth, the G. M. T., to the nearest minute, of 
 Disappearance and Reappearance in occultations, of Ingress 
 and Egress in transits of Satellites and shadows of Satellites 
 across the Planet's disk, and the G. M. T., to the nearest second, 
 of the Disappearance and Reappearance of a Satellite in 
 Eclipses. Although the theoretically calculated times of Dis- 
 appearance and Reappearance, and especially the observed 
 times, cannot be supposed to be very accurate in eclipses, such 
 observations afford the navigator a ready means of at least 
 approximating to his longitude, requiring only a comparison of 
 the local mean times of the phenomena with the times 
 computed in the Naut. Aim. Observations of these eclipses 
 should therefore receive attention from voyagers and explorers. 
 As the phenomena of the eclipses, as seen from different places 
 on the earth's surface, occur sensibly at the same instants, dif- 
 ferences of longitude may be inferred from the times of the 
 same phenomenon as recorded at two or more places ; but in 
 
MISCELLANEOUS ADDITIONAL SUBJECTS. 375 
 
 that case the circumstances of the observations, as regards the 
 powers of the telescopes and atmospheric conditions, should be, 
 as nearly as may be, the same. At the Greenwich Observatory 
 the eclipses are constantly observed, and the mean times of 
 Disappearance and Reappearance strictly calculated; whence 
 by comparisons with the mean times of identical phenomena at 
 other localities, the longitudes of these from Greenwich may be 
 inferred with greater certainty than by comparison with the 
 theoretical times. The method, however, of comparison of 
 observed times is not, like the other, available on the spot, an 
 interval necessarily intervening before the comparison can be 
 made. The other phenomena above mentioned can scarcely in 
 any instance be observed with the optical means on ship-board 
 accurately enough for determinations of longitude ; on which 
 account the data in the Naut. Aim. only serve to indicate 
 approximately the times of their occurrence for observations on 
 land. At Greenwich these observations are made and reduced 
 in the same manner as those of the eclipses, because the reduced 
 times may be employed for correcting the theoretical calculation 
 of the motions of the Satellites. 
 
 (7) By Trigonometrical Surveys. The principles of this 
 method have already been exemplified by the process, explained 
 in Arts. 366 369, for determining by means of a Theodolite 
 and Gunter's chain the difference of longitude between the 
 transit-instrument of the Cambridge Observatory and the 
 Museums' Clock. For complete information respecting the 
 triangulations employed in large surveys for ascertaining 
 relative geographical positions, and the use made of the Trans- 
 portable Zenith Sector for that purpose (Arts. 352 and 353), 
 the published accounts of the Trigonometrical Surveys of Great 
 Britain and Ireland, and of the Great Trigonometrical Survey 
 of India, ought to be consulted. I shall here only add the 
 remark, that at the request of the Astronomer Royal, Col. Colby 
 ascertained, by connecting a secondary triangulation with the 
 principal one, the geodetic value of the difference of longitude 
 Between Greenwich and the Island of Valentia, for comparison 
 with the value obtained by transfer of chronometers,, and that 
 
376 PRACTICAL ASTRONOMY. 
 
 the chronometric value was found to exceed the geodetic value 
 by not more than 8 ,16, the earth's polar radius being taken to 
 be 20,853,810 feet and the equatorial radius 20,923,713 feet. 
 It was concluded by Mr Airy from the results of comparing the 
 measures by the two methods of the arcs mentioned in Art. 
 381 (1), that no improvement can be made in the assumed 
 elements of the earth's Bgure, and that consequently the length, 
 10r6499 feet, of an arc of 1" perpendicular to the meridian in 
 latitude 51. 40' deduced from those elements, may be accepted 
 as a very certain geodetical datum. (See Memoirs of the R. 
 Ast. Soc., Vol. XVL, pp. 277289.) 
 
 382. Solar Parallax. The Sun's Horizontal Parallax at 
 any place is the ratio, in seconds of arc, of the distance of 
 the place from the earth's centre to the mean distance of the 
 earth from the Sun. If the place be at the Equator, the ratio 
 is Equatorial Horizontal Parallax. Hence to calculate the 
 horizontal parallax for any given place, it is first of all necessary 
 to know the earth's figure and dimensions. These are ob- 
 tainable by such operations and calculations as those referred to 
 in Art. 381 (7), which rest on the principle of combining results 
 of geodetical and astronomical observations with information 
 derived from theoretical calculation on the hypothesis of gravi- 
 tation. But to deduce therefrom with exactness the parallactic 
 angle, and thence the value, in miles, of the earth's mean 
 distance from the Sun, (which is the dimensional unit of 
 the Solar System), is a problem of much difficulty on account 
 of the smallness of that angle, and various methods of solving 
 it have been proposed. As I have already in Arts. 308 and 309 
 treated of the determination of solar parallax by means of 
 observations of the planet Mars in opposition, I shall here 
 only take account of the methods of employing observations 
 of Transits of Venus across the Suns disk for the same 
 purpose. 
 
 383. This method, which is the one that has been chiefly 
 relied upon for calculating the Sun's parallax, is only available 
 at epochs separated by irregular intervals. According to the 
 results of calculations in Delambre's Astronomy (Vol. IL, Chap. 
 
MISCELLANEOUS ADDITIONAL SUBJECTS. 377 
 
 xxvii.), the intervals are 8 years, 121 J years, and 105 J years, 
 the longer intervals alternating with the short interval, and 
 occurring alternately. The observations at any place consist 
 essentially of recording as exactly as possible, in local times, 
 the instants of external contact of the limbs of the Sun and 
 Planet and complete immersion of the Planet, and the instants 
 of internal contact and complete emersion. The notes of time 
 are rendered somewhat uncertain by the formation of what 
 is called the black ligament, or drop, which is, apparently, a 
 diffraction phenomenon, and in greater or less degree according 
 to Telescopic and atmospheric (perhaps, also, physiological) con- 
 ditions interferes with discerning at what instants the two limbs 
 have actually a common tangent. (The phenomenon can be 
 produced by artificial representation of the disks and relative 
 motion of the Sun and Planet, by which means an approximation 
 might possibly be made to the correction which an observer 
 should apply on this account to his noted time of observation 
 in the case of the natural phenomenon 1 .) As respects the 
 principles on which the calculation of the parallax from data 
 furnished by any observations of a Transit of Yenus should 
 be conducted, and the circumstances determining the selection 
 of the most favourable places of observations in the instances of 
 the Transits of 1874 and 1882, I cannot do better than refer 
 the student to the Article by the Astronomer Royal in the 
 Monthly Notices of the R.A.S. (Vol. XVIL, No. 7, p. 208), 
 which has been already cited in a note to Art. 309 (p. 302) 
 relatively to observations of Mars for determining solar parallax. 
 I shall only add here an account of the process of obtaining 
 analytical formula of calculation appropriate to any instance of 
 an observed Transit of Venus, adopting for that purpose, and in 
 certain respects correcting, the investigation given in Wood- 
 house's Astronomy, Vol. I., Part n., Chap, xxxvin 2 . 
 
 1 See an article on this subject by Prof. Simon Newcomb in the M. N. of the 
 R. A. S., Vol. xxxvn., No. 5, p. 237. 
 
 2 Transits of Mercury occur much more frequently than those of Venus, but 
 on account of the small difference between his parallax at conjunction and that of 
 the Sun, they are comparatively useless for determining the solar parallax. (See, 
 on the Transits of both planets, Godfrey's Astronomy, Chap, xxii., Arts 349 351). 
 
378 PRACTICAL ASTRONOMY. 
 
 384. Let T be the Greenwich Mean Time of the conjunction in 
 longitude of Yenus and the Sun, as deduced from the theoretical Tables 
 of their motions, or from data contained in the Nautical Almanac, 
 and let 1 be their common geocentric longitude, and X the geocentric 
 latitude of Yenus, at that time. Suppose also the horary variations 
 of the longitudes of Yenus and the Sun to be m and m' 9 and the 
 horary variation of the latitude of Yenus to be n, as gathered for 
 the same time from the same Tables. Then if S and s be the con- 
 temporaneous semi-diameters of the Sun and Planet, and T + 1 be 
 the time of any contact of limbs as seen from the earth's centre, we 
 shall have very nearly 
 
 (mt - m't} 2 + (X + nt) 2 = (8 s) 2 , 
 
 according as the contact is external or internal. From this quadratic 
 equation, by taking account of the double sign, four values of t may 
 be obtained, two applying to the external contacts, and two to the 
 internal. If t be now taken to represent one of these values, and if 
 1 = 1 + mt, I' = 1 + m't, and X = X + nt, it follows that 
 (l-l')* + \ 2 = (Ss) 2 ...... (a) 
 
 according as t applies to an external or internal contact. The values 
 of I, I', and X, which suppose the spectator to be at the earth's centre, 
 may either be inferred from the above equalities for any of the given 
 values of t, or, as is preferable, be directly computed from the Tables 
 for any of the known epochs T+ t. To obtain an equation applicable 
 to the case of an observation made at any given place on the earth's 
 surface, the parallaxes of the Planet and Sun in longitude and 
 latitude have to be taken into account. Let a, a' be their respective 
 parallaxes in longitude, and 8, 8' the parallaxes in latitude, and 
 putting T for T+t, assume T' + t' to be the Greenwich M.T. of 
 observation. Then the apparent distance between the centres *at 
 that time is the hypothenuse of the triangle the sides of which 
 are 
 
 l-l' + ( m - m '} t'-(a-a) and X + ^'-(8-8'). 
 
 Hence, rejecting the squares and products of the small quantities 
 a a, 8 8', and t', and equating the sum of the squares to (Ss) 2 , 
 it will be found, after taking account of the equation (a), that 
 
 
 This equation may be further simplified as follows. If P and P' be 
 
MISCELLANEOUS ADDITIONAL SUBJECTS. 379 
 
 the equatorial horizontal parallaxes of Yenus and the Sun at the 
 time of the observation, and if a = aP, a! = a'P', we shall have 
 a - a' = aP - a'P = a(P-P')- (a' - a) P t which is very nearly equal 
 to a (PP'), because a and a' must be nearly equal on account of 
 the apparent proximity of the two bodies. So if 8 = IP and S' = UP, 
 8 B' = b(P P) very nearly. Consequently 
 
 (l- 
 ~ 
 
 Suppose the observation to give H for the local time of contact. 
 Then if L be the Longitude East of the place of observation, the 
 Greenwich mean time of the observation is H L. Hence 
 
 .-. f(P-P')=H-L-T'. 
 
 Such an equation as this is furnished by each of the two notes of 
 time both at ingress and at egress. Hence if^, Jff lt T^ emdf 2 , H 2 , T a ' 
 represent respectively the means of the two values at ingress and 
 egress, we shall have 
 
 f i (P-P) = H } -L-T 1 ' 
 and f 2 (P-P) = H 2 -L-T 2 '. 
 
 From either of these equations P P' may be calculated when L is 
 known. Hence as the ratio of P to P is known from the Tables, 
 the value of P, the Sun's parallax at the time of observation, may 
 be inferred, and thence, by the Solar Theory, the parallax at the 
 earth's mean distance from the Sun be deduced. This is Delisle's 
 method, which is applicable when only the ingress, or only the egress, 
 is observed at any place, but requires, as we see, an exact knowledge 
 of the longitude of the place. A good result, however, may be 
 looked for from the mean of a large number of such determinations. 
 When both the ingress and the egress are observed at the same 
 place, we get, by subtracting one equation from the other, 
 
 (/, -/ 2 ) (P- P') =*, -*.- (*V- 27), 
 
 from which L is eliminated. The deduction of the solar parallax by 
 means of this equation may be made in the manner just shewn. 
 This is Hollers method, which is preferable to the other in not 
 requiring an exact determination of the longitude of the place of 
 observation, but is generally available on the occasion of any Transit 
 in only a few localities. The formula involves in an important 
 manner H - II the duration of the Transit. 
 
380 PRACTICAL ASTRONOMY. 
 
 385. The following particulars are mentioned for the pur- 
 pose of giving information respecting the actual state of the de- 
 termination of the Sun's parallax. From the Transits of Venus 
 observed .in 1761 and 1769, especially the latter, Professor Encke 
 obtained 8", 5776 for the mean parallax, giving for the Earth's 
 mean distance from the Sun about 95 millions of miles. This 
 value was adopted in successive Nautical Almanacs, till in that 
 for 1870 it was changed to 8",95, which is the value indicated 
 by Leverrier in p. 114 of his Solar Tables published in 1858 in 
 Vol. IV. of the Annals of the Imperial Observatory of Paris. 
 This, again, is changed to 8",848 in the Nautical Almanac for 
 1882, from the result of calculations by Professor Newcomb 
 contained in the Vol. of the Washington Observations for 1865, 
 Appendix n., p. 29. A Report to Parliament of the results as 
 to the Solar Parallax derived from the Telescopic observations 
 made in the British Expedition for observing the Transit of 
 Venus on Dec. 8, 1874, was prepared by the Astronomer Royal 
 in 1877 and printed the same year. From the contents of this 
 Report, and from supplementary considerations published in the 
 Monthly Notices of the R. A. S., Vol. xxxvm., No. 1, pp. 11 16, 
 he concludes that the most complete result obtainable from 
 these observations is a mean parallax of 8",754 corresponding to 
 a mean distance from the Sun's centre equal to 93,375,000 miles. 
 In p. 199 of the same Vol., Captain Tupman, who was at the 
 head of the British Expedition, gives 8",76, subject to future 
 correction, for the general result of the comparison of similar 
 phases observed at different localities. Mr Stone, who has given 
 particular attention to the influence which the phenomenon of 
 the black-drop might have had, on the observations of 1769, 
 arrived at 8",897 0",020 as the mean of the several results 
 given by observations of Mars in 1862, the Earth's parallactic 
 inequality, the Moon's parallactic inequality, and a rediscussion 
 of the Transit of Venus observations of 1769. Finally he de- 
 duces from a comparison of Contacts in the observations of 1874 
 the result 8",88 + 0",02, which he considers to be the value en- 
 titled at present to the greatest weight. (See M. N. of the 
 R. A. S., Vol. xxxvin., pp. 279295 and pp. 341346.) From 
 
MISCELLANEOUS ADDITIONAL SUBJECTS. 
 
 381 
 
 the observations of Mars in 1862 Dr Winnecke obtains 8",96, 
 and Professor Newcomb 8",855. Hansen's deduction from the 
 Moon's parallactic inequality is 8",916. The experimental de- 
 termination of the velocity of light by M. Cornu leads to 8",86. 
 As a parallax of 8",89 corresponds to a mean distance equal to 
 91,940,000 miles, it will be seen that all the modern determina- 
 tions give a smaller mean distance than that obtained by Encke. 
 It would be premature to calculate from them a definitive value 
 because the photographic observations have not yet been fully 
 scrutinized, and it is desirable to include the information that may 
 be gained from the Transit of Venus on Dec. 6, 1882, which, it is 
 expected, will be more favourable for good results than that of 1874. 
 386. Reticles. A Telescope mounted on a stand, without 
 apparatus for moving it about a fixed axis, may be used for ob- 
 serving differences of R A. and P. D. between a known star and 
 any other object, if at its focal distance, it has either a reticle 
 formed of straight wires enclosed in a circular boundary, or a 
 Ring Micrometer. Figure 57 represents various arrangements 
 for this purpose. In every case the observations are made by 
 recording the times by a clock or chronometer of transits at 
 straight lines or circular edges. 
 
 387. Fig. 57 (1) shews the form of the reticles employed by 
 
382 PRACTICAL ASTRONOMY. 
 
 Lacaille at the Cape of Good Hope for obtaining the places of the 
 stars in his Ccdum Australe Stelliferum 1 . The reticles had the 
 shape of a rhombus, three sides of which were fine wires, and the 
 fourth was the straight edge of a brass plate. The opposite 
 angles were joined by wires, and the longer diagonal was equal 
 to the diameter of the circular boundary of the field, while the 
 shorter was of different lengths in different reticles, according to 
 the distances of the star-zones from the Pole. In the majority of 
 Lacaille's observations the shorter diagonal ab was equal to the 
 radius of the circle. By the construction the two diagonals were 
 at right angles to each other and C the point of intersection was 
 at the centre of the field. AE is a diametral wire formed by 
 the prolongation of ab both ways. The reticle could be turned 
 by hand about the axis of the Telescope, and the only adjustment 
 preliminary to the observations was to place the diameter AE in 
 coincidence with the path of a star through the field, in which 
 case the other diagonal coincides with the circle of decimation 
 through C. Suppose now repeated transits of stars to be taken 
 at the points A, a, C, b, E, for the purpose of determining exactly 
 the ratios of CA to Ca } and of GE to Cb, and let the mean of 
 the ratios so found be m. This being premised, let the obser- 
 vations give the times of transit of the known comparison-star 
 at u y t, s, and those of the compared object at u,' t', s'. Then if 
 the mean of the former times be T, and the mean of the latter 
 be jf", the algebraic excess of the R.A. of the compared object 
 above that of the star is T' T, and as the latter R. A. is known 
 the former is immediately inferred. Again, let T be the mean 
 of the intervals occupied by the passage of the comparison-star 
 along ut and ts, and T the mean of the intervals occupied by 
 the passage of the other object along ut' and t's', the intervals 
 in both cases being derivable from the transit-observations. 
 Now the ratio of Ct to Cb ut is, by similar triangles, equal to 
 the ratio of the radius of the circle to Cb, that is, the ratio 
 m. Hence Ct=m(Cb-ut). So Ct' = m ( Cb - ut'). Hence 
 tf (= Ct Ct')=m (ut' ut). But ut and ut' in arcs of a great 
 
 1 A reduction to 1750,0 of the star-places of this Catalogue, in number 9766, 
 was undertaken by a Committee of the British Association, and published in 1847. 
 
MISCELLANEOUS ADDITIONAL SUBJECTS. 383 
 
 circle, are respectively equal to 15 r cos 8 and 15r' cos 8', 8 and 8' 
 being the declinations of the two objects. Consequently tt' the 
 difference of the declinations is 
 
 15m (T cos 8' r cos 8) = 15m (r T) cos 8 t 
 
 very nearly, if 8 t be the mean between 8 and the value of 8' 
 derived from a first approximation to the value of 8 8'. It has 
 here been supposed that the two objects pass in the upper 
 half of the field, and that Ct is greater than Ct r . If they 
 pass in different halves the difference of the declinations is 
 15m (T' + r) cos 8 r We may hence derive the following general 
 rule: The difference of the P.D. is 15m(r'~T) cos 8 15 or 
 15m (T' + T) cos 8 1? according as the objects cross in the same or 
 different halves of the field, and in all cases the P.D. of the 
 compared object is equal to that of the other + the difference, 
 according as it is the higher or lower in crossing the field of an 
 inverting Telescope. 
 
 388. Fig. 57 (2) gives the form of Valtz's reticle. Two 
 parallel wires, which are chords of the circular field of view, are 
 separated by arcs of 60, so that the distance between them is 
 equal to the radius of the circle. Two contrary ends of the 
 chords are joined by a diametral wire, which, consequently, 
 makes an angle of 30 with each chord; and a second diametral 
 wire AE, passing through their middle points, cuts both at right 
 angles. The latter wire is used for adjusting the reticle, just as 
 in the case of Lacaille's reticle (Art. 387) by rotating AE so as to 
 place it in coincidence with the path of a star, in order that the 
 chords may be parallel to the circle of declination through the 
 centre of the field. The observations are then made by record- 
 ing the times of transit of the two objects across the three 
 wires ; as, for instance, the transits of the star at u t t, s, and 
 those of the comet, or other compared object, at u f , t', s f . 
 Let t v 2 , 3 , be the former recorded times, and t' v ' 2 , t' 3 , the 
 latter. Then if ^ + ^ = 2r and ' 1 + ' 3 = 2r / , the algebraic 
 excess of the transit- time of the compared object across the 
 circle of declination above the transit-time of the star is T' T. 
 Hence the' required R.A. of the object is obtained by adding 
 
384) PRACTICAL ASTRONOMY. 
 
 T T to the known R. A. of the star. Again, the difference of 
 the P.D., namely ss or uu', is readily seen to be, in arc of a 
 great circle, very nearly equal to 
 
 15 (T - T' - t z + t'J cos B t cot 30, 
 
 S l being put for the mean of the declinations of the two objects, 
 As this expression remains the same, whether the objects cross 
 the field on the same side, or on different sides, of AE, Valtz's 
 reticle has the advantage of not requiring the circumstance in 
 either case to be noted. If q be the value of that expression, which 
 is necessarily positive, the P.D. of the compared object is obtained 
 by adding to that of the star + q, according as its path across the 
 field of an inverting Telescope is higher or lower than the star's. 
 389. The method of observing differences of R. A. and P.D. 
 by means of a Ring Micrometer may be explained by referring to 
 Fig. 57 (3). A brass ring of uniform breadth is fastened to a 
 transparent glass plate, and placed at the focal distance of the 
 Telescope, so that an object, as it crosses the field, can be well 
 seen through the plate before it reaches and after it leaves the 
 ring, the space enclosed by the ring being vacant. Also 
 as the object and the ring are both seen distinctly with the eye- 
 piece, transits of a known star and a compared object (for 
 instance a comet) may be taken, as indicated by the figure, at 
 the outer and inner edges of the ring, the times of the 
 disappearances behind the ring and of the reappearances being 
 both noted. Let t t be the mean of the transit-times at v and u, 
 and 2 the mean of those at t and s, for the comparison-star, and 
 t\ } t' z the analogous times for the comet. Then if t l + t 2 2r 
 and ' 1 + ' 2 = 2r', T T will be the algebraic excess of the R. A. 
 of the comet above that of the star. Hence from the known 
 R. A. of the star, by adding T' r the required R. A. of the 
 comet, or other compared object, is obtained. Again, let r be 
 the radius (expressed in arc) of the circle which is intermediate 
 to the circular boundaries of the ring, and let 8 and B' be the 
 respective declinations of the star and comet. Then if 
 
 sin 6 = ^ (* g - j cos 8, and sin & = ($,' - */) cos 8' , 
 
MISCELLANEOUS ADDITIONAL SUBJECTS. 885 
 
 the difference of P.D. is the sum or difference of rcosO 
 and r cos #', according as the transits were taken in different or 
 the same halves of the field (as shewn by the Fig.). This 
 difference of P. D. is to be added to, or subtracted from, the P. D. 
 of the star according as the comet or star was the higher in the 
 field, and the result is the required P. D. of the comet. It is to 
 be remarked that in order to obtain trustworthy results the tran- 
 sits of the objects should be at a distance from the centre of the 
 ring, and not very near the points of greatest and least P.D. 
 This differential method clearly does not require any preliminary 
 adjustment. It is necessary, however, to determine the value 
 in arc of the radius r ; which may be done as follows. Take a 
 transit of the Sun's disk across the field, noting the times of 
 exterior and interior contacts at the circular boundaries of the 
 ring. Then if T be the interval between the two exterior 
 contacts with the circle of mean radius r, as inferred from the 
 observed times, and d be the Sun's semi-diameter, it is readily 
 seen that Twill be proportional to 2d + 2?\ Similarly it appears 
 that T', the interval between the interior contacts, is in the same 
 proportion to U - 2r, d being supposed greater than r. Hence 
 as Tis to T' as d + r to d-r, it follows that r is to d as T- T' 
 to T+ T' t from which proportion, since d the Sun's semi-diameter 
 in arc at the time of the observation is known, the value of r 
 in arc may be calculated. The adopted value should be the 
 mean of several such determinations. 
 
 390. The following method of taking differential observa- 
 tions, which, like the last, does not require preliminary adjust- 
 ment, was proposed by M. Boguslawski, the astronomer at 
 Breslau. A single wire passing through the middle of the field, 
 and movable about the Telescope-axis, is put first in one 
 arbitrary angular position, and then in a second, as represented 
 by Fig. 57 (4) (5), and transits of two known stars and an 
 unknown object (a comet) are taken at the wire in each 
 position. Let t v T, 2 be the observed times of transit of first 
 star, comet, and second star at the points /, e, a, in the first 
 position, and /, T', 2 ' the analogous times for the second 
 position, and conceive abc and ab'c to be drawn parallel to the 
 c. 2& 
 
386 
 
 PRACTICAL ASTRONOMY. 
 
 circle of declination through the centre of the field. Then if 
 a l9 2 and B lf 8 2 be respectively the RA. and P.D. of the stars, 
 and the RA. and P.D. of the comet at the time of its transit be 
 A and D, we shall have 8 2 - D to S 2 - S x in the proportion 
 of ab to ac. Also the transit of a star along fc occupies the 
 interval (a z aj ( 2 tfj, and that along eb the interval 
 T). Hence by similar triangles 
 
 a 2 A 
 
 (1) 
 
 similarly, if A + a and D + e be the R A. and P.D. of the comet 
 at the time of its second transit, 
 
 8 2 -(D + 6 ) _ *; - T - (a, - A - a) 
 
 fc-* 
 
 .(2). 
 
 The two equations (1) and (2) suffice to determine A and D, the 
 changes a and e being supposed to be known from an ephemeris 
 of the comet, or from nearly contemporaneous determinations of 
 its place by observation. 
 
 391. The Vernier, so called from the name of the inventor, 
 is used for subdividing the intervals of a graduation in cases 
 which do not require the degree of precision attainable by a 
 microscope-micrometer. As I have not hitherto explained the 
 
 FIG. 58. 
 
 principle of this contrivance and the mode of reading its indica- 
 tions, I propose to supply the omission here, referring for this 
 purpose to Fig. 58. If D be the interval between the gradua- 
 tions of the arc ee to which the vernier is applied, and D' the 
 interval of the vernier-graduations, in all cases (n + 1) D'=nD, 
 
MISCELLANEOUS ADDITIONAL SUBJECTS. 387 
 
 n being an integer. Also the vernier-intervals are reckoned in 
 the same direction as the arc-intervals. From the above 
 
 equality we have n = ^ ^ ; and if, as is often the case in 
 
 small setting circles, the degree be divided into three equal 
 parts so that D 20', and it be proposed to read to the accuracy 
 of I/, we shall also have D D' = I'. Hence D' = 19' and 
 7i = 19, and accordingly 19 arc-intervals have to be equal to 20 
 vernier-intervals, as is the case in the example shewn by Fig. 58. 
 To read the vernier-indication it is first to be noticed that the 
 zero of the* vernier is opposite to 6 and something more than 
 two intervals, or 6 40' +. The additional quantity is found by 
 observing, on looking along the vernier, that the seventh mark 
 from zero coincides with a mark of the arc-graduation. Hence 
 the required reading is 6 47'. If the reading be to the 
 accuracy of half a minute, we should have D D' = 0',5, 
 Z>' = 20'-0',5 = 19',5, and n = 39. Supposing the degree to 
 be divided into six equal parts, and the reading to be accurate 
 
 to 10", we shall have D = 10', D - D' = 10" = ^ , D' = 10' - ^ 
 
 o b 
 
 59' 
 
 = -^- , and consequently n = 59 (see Art. 370). After the fore- 
 going explanation it will be easily seen that the zero of the 
 vernier can be set to any proposed reading by means of appara- 
 tus such as that exhibited in Fig. 58, where a is a clamping- 
 screw for attaching to the arc the nut c which supports a 
 slow-motion-screw, and d is a tapped nut joined to the vernier, 
 whereby the same screw, being turned by the milled head b, 
 works so as to give to the vernier the slow-motion required for 
 the setting. 
 
 392. The Transit- reducer. For the time-reduction of 
 any transit-observation to the meridian it is requisite to calcu- 
 late the quantity 
 
 , , . N cosec S 
 
 (a -f cos z + c sin z) p; , 
 
 a, b, c being respectively the corrections of the collimation, level, 
 and azimuth errors, and /,8 the Z.D. and P.D. of the object 
 
 252 
 
388 
 
 PRACTICAL ASTRONOMY. 
 
 observed (see Art. 92). To facilitate the calculation tables are 
 formed of the particular values of the factors of a, b, c, with argu- 
 ment P.D., for the clock-stars, and of such other values as may 
 be suitable for obtaining by interpolation the factors for any given 
 P.D. (The Introductions of successive Volumes of the Cambridge 
 Observations contain tables proper for these purposes.) The 
 results of multiplying a, b, c by the respective factors are 
 obtained at Greenwich by means of sliding scales, one for each 
 product. In place of this method I began in 1849 to make 
 use of a machine, constructed according to my directions by 
 Mr Simms, whereby the total value of the foregoing formula 
 could be calculated by a single operation. The construction of 
 this machine may be gathered from the following argument. 
 In Fig. 59 the line OA QA is drawn in a fixed direction, OM 
 
 FIG. 59. 
 
 makes with this line an angle z equal to the zenith distance of 
 an object, the angle MOR is equal to the latitude of the 
 Observatory, and consequently the / AOR is the declination of 
 the object, or 90 - 8. Set off OM = b, make MP, drawn at right 
 angles to OM, equal to c, then letting fall the perpendicular PQ 
 or OA, make AQ = a, and draw AR at right angles to OA. 
 Then it follows that A = + a + b cos z + c sin z. If now OR 
 be measured on a scale the unit of which is fifteen times that 
 of the scale of the measurements of OM, MP, and AQ, we 
 
 shall have 
 
 r\ A cosecS 
 
 OR (in time) = ^- sec AOR = ( a + b cos z 4- c sin z) ^ . 
 
 For the mechanical calculation of this quantity, a space six 
 inches square, on a circular plate, was ruled all over with lines 
 
MISCELLANEOUS ADDITIONAL SUBJECTS. 389 
 
 parallel to the sides of the square, and one-twentieth of an 
 inch apart, which interval was assumed to be the measure of 
 0",1. Hence the side of the square measured 12". Accordingly 
 from the centre of the square, which coincided with the centre 
 of the circle, and corresponded to the point in the Fig., 
 distances less than 6", equal to the level and azimuth correc- 
 tions, could be set off as rectangular coordinates, the positive 
 and negative directions being indicated by + and signs along 
 the sides of the square. In this way the position of the point 
 P is marked. The position of the line OA is determined by 
 its passing through the centre of the circle, and by its being 
 parallel to a fixed fiducial edge constituting an essential part of 
 the machine. A plate, contained in a rectangular trough, and 
 held with its plane face against one side by a spring, is 
 movable along that edge, and carries with it two fine parallel 
 wires perpendicular in direction to the face of the plate. One 
 of the wires is adjustible so that it can be placed, by means of 
 a fixed scale, and an index moving with the wire, at a distance 
 from the other equal to the collimation-correction irrespective 
 of its sign, the unit of the scale being I". Two brass frames 
 carrying the wires, are marked with the signs + and , to 
 indicate the wires to be used respectively for positive and 
 negative collimation-corrections. These arrangements being 
 understood, it remains to shew how to set the circle, and to read 
 off the indication of the machine, for a particular astronomical 
 observation. 
 
 393. The circle can be turned smoothly by a handle about 
 an axle at its centre, and is graduated close to the circumference 
 from zero in an arbitrary position to 130 in the direction 
 reckoned positive, and to 60 in the opposite direction, to meet 
 the cases of P.D. above and below the Pole. In order to 
 determine the place of an index and vernier for reading the 
 graduation of the circle, the line OR, which with its graduations 
 is engraved, requires to be put, by turning the circle, into 
 coincidence with one of the parallel wires, which is, in fact, to 
 make the angle ROA equal to 90. But this angle, as we have 
 seen, is equal to the declination of the object. Hence in that 
 
390 PRACTICAL ASTRONOMY. 
 
 case the zero of the graduation is in the position corresponding 
 to of P.D. This result determines the position of an index 
 intended for setting in degrees and minutes, by means' of an 
 appropriate vernier, for the given P.D. of a proposed object. 
 The mechanical calculation then simply consists in first setting 
 the circle to the given P.D. by the index and vernier, and, after 
 bisecting the point P by the + or wire according as the 
 collimation-correction is positive or negative, reading the part 
 OR cut off by the other wire. This reading, by reason of the 
 scale of the graduation, gives the reduction to the meridian 
 in time. 
 
 394. In the instance of the machine above described the 
 diameter of the circle was 15 inches, the dimensions of the 
 scales were such that the greatest amount of correction that 
 could be read was + 1 s , and on account of the small angles made 
 by the wire with the time-scale it could not be safely used for 
 P.D. less than 30. But for the more usual P.D. it was capable 
 of calculating the correction to the nearest hundredth of a 
 second of time ; whereas by the use of the sliding-scales in the 
 same cases, requiring three multiplications and three additions, 
 the error might amount to + 8 ,02. To serve for the cases of 
 small P.D., an additional graduation, the unit of which was 1", 
 was engraved along and contiguous to the fiducial edge, and 
 reckoned in opposite directions from a zero, the position of 
 which was defined by its being bisected by either wire made 
 to cross the centre of the circle. Then, after determining the 
 position of P, and using the parallel wires, just as before, the 
 part OQ of the scale cut off by the second wire is 
 
 a + b cos z + c sin z. 
 
 The calculation of this quantity by the machine for Polaris and 
 S Ursae Minoris, or any stars within 30 from the Pole, is 
 abundantly accurate, and might, apart from other applications, 
 save much time 1 . 
 
 1 A fuller account of this instrument is given in the Monthly Notices of the 
 Astronomical Society, Vol. x., p. 182. The one made for use in the Cambridge 
 Observatory was shewn in the Great Exhibition of 1851, and received the award 
 of a Bronze Medal. 
 
MISCELLANEOUS ADDITIONAL SUBJECTS. 391 
 
 395. The Huyghenian Eye-piece. In addition to what is 
 said on this eye-piece in Art. 21, I have only to direct attention 
 to Fig. 60, in order that by a comparison with Fig. 2 in p. 23, 
 the difference between the courses of the rays in Huyghens's 
 and Ramsden's eye-pieces might be readily perceived, In the 
 former the distance between the two lenses is usually half the 
 
 FIG. 60. 
 
 sum of their focal lengths, and the image is formed at the 
 position of a diaphragm ss midway between the lenses. The 
 four-glass erecting eye-piece may be made, like Huyghens's, free 
 from injurious colour, but on account of the loss of light by 
 passage through four glasses it is not to be preferred to the 
 latter for observing the physical phenomena of celestial bodies, 
 the details of which, especially in the cases of comets and 
 nebulae, are often very faint and difficult to recognise. 
 
 396. Magnifying powers. The following process, called 
 "the method of double vision," is suitable for finding the 
 magnifying power of a small hand-telescope. A white staff is 
 set up, graduated by black transverse marks at intervals, say, of 
 one inch, and one of the intervals is blackened. An observer, 
 on looking at this interval through the Telescope, and opening 
 the unemployed eye, may perceive an enlarged image of the 
 black space thrown upon the staff. By counting the number of 
 intervals which this image covers, the magnifying power of the 
 combination of the lenses of the Telescope is obtained. 
 
 397. The power of any optical instrument composed of 
 lenses, or of lenses and mirrors, and adapted for magnifying, is 
 best ascertained by means of a small double-image micrometer, 
 called a Dynameter, invented for this purpose by Ramsden and 
 afterwards improved by Dollond. Figures 61 and 62 represent 
 
392 
 
 PRACTICAL ASTRONOMY. 
 
 the parts of a Dollond's Dynameter, the construction of which 
 is very nearly the same as that of a Dynameter made by 
 
 FIG. 61. 
 
 Fm. 62. 
 
 Troughton and Simms, which I made use of for rinding the 
 magnifying powers of the various eye-pieces pertaining to 
 the Telescopes of the Cambridge Observatory, and afterwards 
 exhibited to my astronomical class. In the two Figures/ is the 
 circular micrometer-head divided on its rim into 100 equal 
 parts, and e is a milled button for turning the micrometer. 
 The screw h (Fig. 61), which is right-handed, and is fixed to the 
 micrometer-head, acts in a tapped projection from the plate n, 
 and thereby draws the plate together with an attached 
 half-lens I. At the same time another screw i, which is 
 left-handed, turns within h, the two having axes in a common 
 direction, and as the thread-intervals are the same in both, 
 it follows that the screw i pushes the plate m and the attached 
 half-lens Tc just as much as the screw h draws the plate n and the 
 half-lens I. The relative motion of the two half-lenses is 
 therefore double the motion of each. The foregoing apparatus 
 is contained in a box acdb. The external scale g is moved 
 with the plate n. The spring o keeps the two screws in 
 bearing. It is next to be remarked that when the eye is 
 situated at some distance from the eye-glass of any optical 
 
MISCELLANEOUS ADDITIONAL SUBJECTS. 393 
 
 instrument, and looks in the same direction as when it is 
 applied to the eye-glass for observing an object, there is seen in 
 front of, and near, this glass a small bright circle, which is, in 
 fact, the image of the object-glass or -mirror, formed by pencils 
 of rays converging to this position, after emergence from 
 the eye-glass. The purpose of the Dynameter is to measure the 
 diameter of this circle, because, as is known from Optics, the 
 ratio of the diameter of the object-glass or -mirror to that of 
 the small circle is the magnifying power. (I have added in 
 an Appendix the proof of this Optical Theorem.) It is now 
 required to shew how the diameter of this circle is measured. 
 
 398. The cap lu (Fig. 62) covers a Ramsden eye-piece the 
 eye-glass of which is divided into half-lenses, and the Dynameter 
 is applied so that the bright circle is situated just beyond the 
 field glass. Hence, as the axis of the eye-piece is directed 
 towards this circle, when the half-lenses are separated by 
 turning the micrometer-head, two images of the circle will be 
 seen. (To certify that each is an image of the object-glass, it 
 suffices to affix a patch of tin-foil of any shape upon the latter.) 
 For effecting the measurement these images are brought into 
 contact, as shewn at x in Fig. 62, first on one side and then on 
 the other, and by means of a fixed index, the readings of the scale g 
 (Figs. 61 and 62) are taken, for both positions, in scale-intervals 
 and divisions of the micrometer. The intervals of the scale are 
 one-fiftieth of an inch, and hundredth parts of an interval can be 
 read by the micrometer, so that the difference of the readings 
 can be obtained with great precision in parts of an inch. This 
 difference is the measure of the diameter of the circle, because, 
 as we have seen, the two images move in opposite directions. 
 The ratio of the diameter of the object-glass or -mirror in 
 inches to the diameter measured by the Dynameter is the 
 magnifying power. 
 
 399. Interpolations. The interpolation which the astrono- 
 mer has most frequently to perform is that of finding, when 
 three values a v a 2 , a 3 of any function, separated by a common 
 interval h of time, are given, the value of the function at any 
 
394 PRACTICAL ASTRONOMY. 
 
 time t differing by less than half the interval from the epoch of the 
 middle value a 2 . Representing by X t the required value, put- 
 ting x for j , and supposing that a a -a t = d i} a 3 - a a = d z , and 
 d^ d t = e, we shall have 
 
 v d +d z e 
 
 Z * = a 2 + -^ 2 -^ + -# 2 , 
 
 the values of x and x z and their coefficients being readily 
 calculated from the data. 
 
 Occasionally it may be required to interpolate with greater 
 exactness, for instance, by using five equidistant values of the 
 function, as a v a v a v 4 , a 6 . In that case if 6 t , 6 8 , 6 8 , 6 4 be the 
 first differences 2 - a lf a 3 - a v &c., c x , c 2 , c 3 the second differences 
 \ b v &c., ^, c^ 2 the third differences c 2 -c p &c., and ^ the 
 fourth difference d 2 d v and if 
 
 X t = a + bx + Cic 2 + cfo? 3 + e^? 4 , 
 we shall have 
 
 by means of which equations the values of a, b, c, d, e are easily 
 derived from the data. In this interpolation the given time t 
 should differ by less than half h from the epoch of a & . 
 
APPENDIX. (See ART. 397.) 
 
 IT is proposed in this Appendix to demonstrate a general 
 Proposition for determining the magnifying power and bright- 
 ness of the image of an object seen by any optical combination 
 of lenses, or mirrors, or both. The demonstration is restricted 
 to the first order of approximation, and the thicknesses of the 
 lenses are not taken into account, this degree of accuracy being 
 sufficient for most practical purposes. The following is the 
 enunciation of the general Proposition. 
 
 To find the magnifying power and brightness of the image of 
 an object seen through any combination of lenses and mirrors. 
 
 The general determination of the magnifying power rests on 
 the two following Theorems applicable to any one of the lenses 
 or mirrors of the combination. The proofs of these Theorems 
 are given in the common Treatises on Optics. 
 
 (1) The linear magnitude of the object is to the linear 
 magnitude of the image, as the distance of the object from the 
 lens or mirror producing the image, is to the distance of the 
 image from the same. 
 
 (2) Each .image, whether real or virtual, may be regarded 
 as an object with respect to the next lens or mirror, taken 
 according to the course of transmission of the light. 
 
 Let n = the number of lenses and mirrors together. 
 
 D = the distance of the object from the object-glass or 
 
 object-mirror. 
 
 d = the distance of the eye-glass from the eye. 
 f it f z = the distances of the first image from the 1st and 2nd 
 glasses or mirrors. 
 
396 PRACTICAL ASTRONOMY. 
 
 fvft the distances of the second image from the 2nd and 
 3rd glasses or mirrors. 
 
 /2- 3 >/2-2 = tne distances of the (n - l) th image from the (n - l) th 
 
 and ?i th glasses or mirrors. 
 
 j^ n _! + d = the distance of the n th image from the eye. 
 Let L the linear magnitude of the object. 
 
 Then by the two Theorems above enunciated, 
 
 f 
 L i ^= the linear magnitude of the 1st image, 
 
 L -- = 2nd 
 
 r 
 
 Hence, dividing by f zn _ t + d, the distance of the n ih image 
 from the eye, 
 
 the angular magnitude of the final image 
 
 -= _ ./I V 3 ' ' 'J 2"-3 V2M-1 
 
 D'f*-fL-f^-(f^ + dy 
 
 Let c = the distance at which the object can be seen dis- 
 tinctly with the naked eye of any individual. Then the 
 
 angular magnitude of the object is , or -^. . Dividing the 
 
 c J-f c 
 
 former angular magnitude by this latter, we have for the 
 advantage gained by the optical instrument, or the magnifying 
 power, the following expression : 
 
 J\ "Js '"Jw-z 'Jzn-l 
 
 If the object be a heavenly body, necessarily -77 = 1. 
 Hence for any astronomical Telescope, 
 
 magnifying power = ^4^%H%^%^r . 
 
 /2 '/4 ' ' V2-2 ' l/in-1 T U) 
 
APPENDIX. 397 
 
 Also if the final image be seen by pencils of nearly parallel 
 rays, that is, if the observer be not short-sighted, the ratio of 
 / 2 n-i to A-i + d is nearly unity. 
 
 Another general expression for the magnifying power may 
 be deduced from the foregoing, by means of a third general 
 Theorem, which may be enunciated as follows : 
 
 (3) Supposing the pencils to fall very nearly perpendi- 
 cularly on all the lenses and mirrors, the breadth" of a given 
 pencil, where it leaves any lens or mirror, is to its breadth, 
 where it falls on the next lens or mirror, as the distance of the 
 focus, virtual or real, of the intermediate portion of the pencil 
 from the former, to its distance from the latter. 
 
 This Theorem, which depends on very simple geometrical 
 considerations, is proved in the usual Treatises on Optics. The 
 effect of the thicknesses of the lenses is not taken into account. 
 
 Let A = the breadth of a pencil falling on the object-glass 
 or object-mirror. 
 
 Then by the above Theorem, 
 
 f 
 
 A .-y = the breadth of the pencil falling on the 2nd 
 J\ 
 
 glass or mirror. 
 
 f if 
 
 A . f^f = the breadth of the pencil falling on the 3rd 
 
 /I V 3 
 
 glass or mirror. 
 
 A .y'y '"y~ 2 = the breadth of the pencil falling on the n* 
 
 /i-/3'"/ 2 n-3 , 
 
 glass or mirror. 
 Let e = the breadth of the pencil as it enters the eye. 
 
 Then e = A / ' '** * ' '**-* '' ^ 2 y + ^. 
 
 Now referring to the general expression already obtained for 
 the magnifying power, it is clear that we shall have, 
 
 . r . c A 
 
 magnifying power = -^ x . 
 
398 PRACTICAL ASTRONOMY. 
 
 This expression is of great practical utility. For a Telescope 
 we have simply, 
 
 ., . A 
 
 magmiying power = . 
 e 
 
 In the Galilean Telescope the breadth A is determined by 
 the aperture of the eye, which in this instance limits the size of 
 the pencils. In other Telescopes the size of the pencils is 
 generally limited by the object-glass or object-mirror, the dia- 
 meter of which consequently determines the value of A. 
 
 If the breadth e be greater than the diameter of the 
 pupil of the eye, part of the light of each pencil is lost, and 
 the effect is the same for the same magnifying power as if the 
 
 object-glass were of smaller diameter. If e = the diameter 
 
 
 of the pupil of the eye, then = the inferior limit of the 
 
 6 
 
 magnifying power, for any practical purpose, of a telescope 
 the diameter (A) of whose object-glass or object-mirror is 
 given. 
 
 On the other hand, if the diameter (e) of the pencil entering 
 the eye be very much smaller than the diameter (e) of the 
 pupil of the eye, the image becomes obscure and ill-defined 
 from the feebleness of the light. Consequently, to unite great 
 magnifying power with distinctness of vision it is necessary to 
 increase the size of the object-glass or object-mirror. 
 
 Suppose that the pencil of rays from a single point 
 incident on an object-glass or -mirror has the breadth e when 
 it crosses the axis of the telescope where the eye is usually 
 situated. The pencils from an unlimited number of external 
 points would all cross the axis at the same place and be of 
 the same breadth. But the aggregate of the courses of these 
 pencils between the object-glass and the position of the eye, 
 is the same as if the object-glass were an illumined object 
 from which rays are incident on the eye-piece. If the object- 
 glass be exposed to the light of clouds, or the sky, and the eye 
 be removed along the axis to some distance from the eye-piece, 
 a bright round image of the object-glass will be seen formed in 
 the front of the eye-glass. The diameter of this image is 
 
APPENDIX. 399 
 
 therefore very nearly the quantity e. Since the Dynameter, as 
 shewn in Art. 397, measures this diameter, it also measures e. 
 Hence, the diameter A of the object-glass being also measured, 
 
 A 
 
 the ratio becomes known. 
 
 6 
 
 In the case of a Microscope the magnifying power is chiefly 
 produced by the small distance D of the object from the object- 
 glass. In consequence of this arrangement the object-glass 
 must be of very short focal length and small diameter. Hence 
 
 also e must be small, in order that may not be a small 
 
 fraction. To compensate for the smallness of the breadth e of 
 the pencils the object has to be strongly illumined by artificial 
 means. 
 
 The brightness of an image produced by optical means 
 depends greatly on the loss of light by transmission through 
 lenses, or reflection at mirrors, and cannot be calculated with 
 mathematical precision. Supposing, however, that no light 
 is lost by transmission or reflection, it may be shewn as follows, 
 that the brightness of the image is the same as the brightness 
 of the object, if the pencils from the image fill the pupil of the 
 eye. 
 
 The quantity of light which enters the naked eye from a 
 given small portion of the surface of an object seen at the 
 
 e' 2 
 nearest distance of distinct vision, varies as -,-. The quantity 
 
 c i 
 
 of light which enters the eye through the optical instrument 
 from an equal portion of the image, varies as 
 
 A\, 1 
 
 D* (mag g . power) 2 ' 
 
 no light being lost by reflection or transmission. But the 
 magnifying power is equal to . -^ . Hence, by substitution it 
 
 6 JL/ 
 
 will appear that the quantity of light through the instrument 
 
 g2 
 
 varies as -^ . We have, therefore, 
 
400 PRACTICAL ASTRONOMY. 
 
 Brightness of object _ e' 2 < _ e* _ . , _ 
 Brightness of image c 2 ' c 2 e 2 
 
 which is the result it was required to obtain. 
 
 As it is known that an object seen with the naked eye 
 appears equally bright at whatever distance it is viewed, it 
 follows from the above proposition, that an object of finite 
 dimensions would appear equally bright under whatever circum- 
 stances it is seen, provided no light were lost by transmission 
 through the air or lenses, or by reflection at mirrors. Thus, if a 
 comet appears brighter in a large telescope than in a small one, 
 it is owing to the more effectual exclusion of stray light by the 
 greater length of the telescope tube. The case, however, is 
 different with respect to a star, which must be regarded as a 
 luminous point, the brightness of the image of which, as seen in 
 a telescope is, cceteris paribus, proportional to the aperture of the 
 telescope. 
 
 THE END. 
 
 CAMBRIDGE: PBINTED BY c. j. CLAY, M.A. AT THE UNIVERSITY PRESS. 
 
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