CAPE ASTROGRAPHIC ZONES. VOL. VII.

CATALOGUE OF RECTANGULAR

CO-ORDINATES AND DIAMETERS

OF STAR-IMAGES

DERIVED FROM PHOTOGRAPHS TAKEN AT

THE ROYAL OBSERVATORY, CAPE OF GOOD HOPE.

COMMENCED UNDER THE DIRECTION OF Sir DAVID GILL, K.C.B., LL.D.. F.R.S., Etc..

FORMERLY H.M. ASTRONOMER AT THE CAPE.

COMPLETED AND PREPARED FOR PRESS UNDER THE SUPERVISION OF

S. S. HOUGH, M.A., F.R.S.,

H.M. ASTRONOMER AT THE CAPE.

ZONE -47° Vl ^ ^ ill:

H . H aw ,

Published by Order of the Lords Commissioners of the Admiralty in obedience to

His Majesty's Command.

LONDON: PRINTED & PUBLISHED BY HIS MAJESTY'S STATIONERY (OFFICE

To be purchased through any Bookseller or directly from H.M. STATIONERY OFFICE

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1916. [Crown Copyright Reserved.]

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CONTENTS.

Introduction to Volume VII. —

§ I. General

§ II. Instrumental Adjustments, etc. —

(a) The Photographic Telescope

(b) The Guiding Telescope . .

(c) The R&eau

(d) The Plates

(e) The Measuring Apparatus

(f) Errors of the Micrometer Screws

§ III. Measurement of the Plates . .

§ IV. Preliminary Reduction of Rectangular Co-ordinates

§ V. Corrections for Refraction and Aberration . .

§ VI. Computations of Tabular Co-ordinates of the Standard Stars

§ VII. Determin.'^tion of Plate Constants

§ VIII. iNTERCOMP.'iRISON OF PlATES

§ IX. Control Observ.^tions

§ X. Personal

§ XI. List of Pl.-vtes

§ xii. formule for conversion of co-ordinates

Explanation of the Catalogue

Rectangular Co-ordtnates, 1900-0, for .Astrographic Zone— 47°

(2273) Wt. 38308/P275/S83 500 'ill-i Harrow t: .69/192;).

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1

INTRODUCTION

TO

VOL. VIL

§ I.— GENERAL.

A general description of the work pertaining to the Cape zones of the " Carte du Ciel " Catalogue is contained in the Introduction to Vol. I. of this series.

The present volume contains the results of measures derived from plates having their centres on the declination circle —47°, which have been treated in all respects in a similar manner to those dealt with in the previous volumes.

For fuller detail, reference may be made to this general description, only such parts being repeated here as require modification for each particular zone, or which are considered essential in explanation of the contents of the volume.

§ II.— INSTRUMENTAL ADJUSTMENTS, Etc. {a) The Photographic Telescope.

A description of the photographic refractor is given at p. 120 of the History and Description of the Cape Observatory, and it will be sufficient here to state the following details : —

The equatorial mounting is of the German type, carrying two parallel telescopes securely braced together by rigid cast-iron flanges near their ends.

The main photographic telescope is of 13 inches aperture and 11 J feet focal length, giving pictures very approximately on a scale of 1 ' of arc to a miUimetre.

The breech piece of the telescope carries the plate holder mounted in such a manner that the latter can be easily removed and quickly replaced so as to return accurately to the same position. The photographic plate in its turn is held in this holder with spring pressure against six bearing points, three of which serve to define its plane, and the remaining three its position and orientation in that plane.

VI

CAPE ASTROGRAPHIC ZONES— VOL. VII.

A second plate carrier, with bearing points in exactly similar positions in relation to the plate, is used for impressing on the plate the reseau used for purposes of measurement. Thus the intersection of the central reseau lines should define a point which is " fixed " in relation to the breech end of the telescope.

By removing the plate carrier and replacing it by a small aperture coinciding with this fixed position, the object glass, as a whole, was centred on this small aperture in the usual manner.

To adjust the plate perpendicular to the optical axis, the plate was replaced in its carrier by a plane mirror on which this centre was duly marked by removal of a small circular patch of the silvering. A small glow lamp was placed in a central position outside the object glass and its reflected image in this mirror viewed through the objective. The tilt of the mirror was then regulated by the available adjusting screws until the lamp, its image, and the centre mark on the mirror were in ahgnment.

The mounting of the breech end permits of a small rotation of the plate holder, ty which the central reseau lines may be accurately adjusted respectively along, and perpendicular to, an hour circle. This adjustment can be tested by allowing a star image to trail across the sensitive plate with the driving clock stopped and measur- ing the distance of the impressed trail from an adjacent reseau line at two points symmetrically situated near opposite edges of the plate.

These adjustments, when once made, were found to possess all necessary stability, and have been occasionally verified when it has been necessary to dis- mount the object glass for cleaning, or otherwise displace any of the essential working parts.

The focal adjustment was selected so as to give the sharpest definition for stars at about 40 mm. distance from the centre of the plate, and was found to remain sufficiently permanent. The readings of the focussing scale (divided in millimetres) have, however, been slightly varied from time to time as follows : —

			mm.
1897	November 4.	Focus set at	21-5
1898	September 26.	altered to	21 -0
1899	November 7.	,, ,, ,, ... ... ... >	22-0
1902	February 14.		21-0
1905	June 23.	j» J, ,, •• • •• • • • •	21-5
1906	September 5.	set at	22-0
I9I0	February 3.	altered to	21-5

{b) TiiE Guiding Telescope.

The second or guiding telescope is of 10 inches aperture and the same focal length as the photographic telescope. Its breech end carries an eye-piece mounted •on two shdes at right angles to the axis of the telescope and to one another, so as

INTRODUCTION. Vll

to extend the field of view accessible at any one setting of the instrument. These slides are adjusted respectively parallel, and perpendicular to, an hour circle. The centre of the field of view of the eye-piece is defined by cross spider webs mounted on a frame which travels with the eye-piece, while its position in relation to the field of the objective is ascertained by means of two verniers travelling with the eye-piece and reading against fixed scales divided in millimetres.

The readings on these scales, which correspond with the centre of the field of the telescope, are found to be subject to sUght fluctuations though stationary over short intervals. They are accordingly ascertained from time to time by direct experiment. The scales are set to a reading at or near the estimated centre and the guiding telescope is then directed to a bright star, which is placed accurately at the intersection of the cross webs by means of the equatorial movements of the whole apparatus. A plate is then exposed in the photographic telescope and impressed with a reseau in the usual manner. If the scale readings have been correctly selected the image of the star should appear at the intersection of the central reseau fines. Its displacements from this position measured in millimetres in two directions at right angles, corresponding with the directions of the two eye- piece slides, determine the amounts by which the estimated readings require to be corrected.

In photographing any field, a suitable star within the field and not too far from the centre is selected, and its rectangular co-ordinates in millimetres in relation to the desired centre on an ideally adjusted plate are computed. The eye-piece is then displaced from its central position by amounts, measured on the eye-piece scales, which correspond with these rectangular co-ordinates, and previous to exposure of the plate the guiding telescope is directed so as to place the image of the guiding star in coincidence with the intersection of the cross webs.

This coincidence is maintained during exposure either by the clock movement or, if necessary, by slight hand corrections by means of a slow motion screw in declination and a differential gearing operating in the clock movement in right ascension.

(c) The R6se.\u.

The reseau consists of a sheet of silvered plate glass, of the same surface dimensions as the plates in use, on which two series of fine equi-spaced parallel lines have been engraved in a dividing engine, so as to remove the silvering along these lines. The two series of lines are respectively at right angles, and the Unes of each series are spaced at intervals of 5 mm.

A photographic copy of the reseau is impressed on each plate by placing the plate in a special carrier, so that its sensitive surface is almost m contact with tliat

Viii CAPE ASTROGRAPHIC ZONES — VOL. VII.

of the silvered surface of the reseau. The Hght from a small electric lamp, after being rendered parallel by passage through an object glass of 9 inches aperture, is then made to fall on the reseau and through the ruled apertures on to the sensitive film below.

The practice has been throughout to imprint the reseau either immediately before or immediately after exposing the plate in the main telescope.

In connection with the present programme, four different reseaux have been used, viz. : —

Gautier No. 51 from 1897 December 7 to 1899 July 11.

,, ,, ? ,, 1899 July 12 to 1901 October 30.

„ ,, 61 ,, 1901 October 31 to 1902 June 11.

., ,, 50 ,, 1902 June 12 till completion of work.

These reseaux are referred to hereafter as G51, Gx, G61, G 50, the dis- tinguishing number originally engraved on the second of them having been obliterated before use.

The division errors of the scales of these reseaux have not been subjected to direct investigation. A careful investigation of the reseau G 8*, which is similarly ruled by the same maker and probably in the same dividing engine, indicated that a very high degree of precision could be expected in the ruling, while the indirect tests furnished by the agreement of the results derived from different plates which overlap in a complex manner have confirmed this sufficiently satisfactory result.

{d) The Plates.

Except during the interval 1903 July 8 to 1904 July 20, the plates in use have been the " Ilford Special Rapid" with three exposures of durations 6m., 3m., and 20 sees., the guiding telescope being displaced by 1 mm. to the North between exposures. Between the dates above mentioned " Ilford Monarch " plates were used and the exposures reduced by one-half. A return was, however, made to the Special Rapid plates on account of their finer grain.

The developer employed was Eikonogen-Hydroquinone previous to May 1901, and subsequently Amidol.

(e) The Measuring Apparatus.

The apparatus in use for measuring the plates has been described in detail in Monthly Notices, vol. Hx., pp. 61-72. Two such instruments, distinguished here- after as Micrometer I. and Micrometer II., have been employed.

The pairs of close webs which form the outer square are mounted on fixed frames, and the two further sets of close parallel webs on sliding frames, whose * Gill and Jacoby, Acta Societatis Scientianint Fennicae, .xxiii. No. 5.

INTRODUCTION. IX

positions are governed by two micrometer screws, permitting motion of these sets •of webs bodily in a direction perpendicular to their length. The reticule admits of adjustments to ensure —

(1) that the opposite sides of the outer square shall be rendered strictly

parallel and at a distance apart corresponding to the space traversed by either set of moving webs in 10 revolutions of the micrometer screw ;

(2) that the adjacent sides of the square shall be strictly perpendicular ;

(3) that the moving webs shall be strictly parallel to the corresf)onding

sides of the square ;

(4) that the zero reading of the screws shall correspond exactly with

0 rev. or 10 rev. when the moving webs are brought into symmetrical coincidence with one or other of the sides of the fixed square.

The photographic plate is mounted on a stage beneath the microscope, which is carried on two slides at right angles to one another, so that any portion of the plate may be brought within the field of view of the microscope.

By sliding the microscope bodily in the direction of its axis, the focus may be adjusted so that the image of the plate is in the same plane as the reticule, and, by a similar motion of the objective alone, the scale of the image may be adjusted so that the distance between adjacent reseau lines corresponds with the length of the side of the outer square.

Further, the plate may be rotated in its carriage and the microscope rotated about its axis until both the reseau lines and the webs are parallel to the slides on which the plate carrier is mounted.

Thus a square of the reseau may be adjusted to coincidence with the outer square of fixed webs, while the two sets of moving webs are made to intersect symmetrically on a star image.

The readings of the micrometer screws then represent the rectangular co-ordinates of this star image expressed in screw revolutions, or tenths of a reseau interval, as unit, and referred to the sides of the fixed square which correspond with the readings 0 rev. as axes.

Except when the instruments have been dismantled for cleaning, their adjust- ments have remained remarkably steady and, with the exception of an occasional adjustment of the zero reading of the micrometer screws, when once made they could in general be left for months on end without further attention.

The change in zero reading, necessitating constant readjustment, was particularly noticeable in relation to the vertical screw of Micrometer II. about the middle of the year 1907. The shghtly rounded conical point of the screw bears

C. A. Z.— VII. 2

X CAPE ASTROGRAPHIC ZONES — VOL. VII.

with spring pressure against an agate plane. The tracing of a similar defect in another micrometer of like construction led to an examination of this end bearing in October 1907, when it was found that the jewel was split and that the sharp edges of the crack were exerting a grinding action on the point of the screw, giving rise to rapid wear. The end bearing was replaced temporarily by a hard steel disc and later, in July 1908, the jewel was renewed.

(/") Errors of the Micrometer Screws.

An investigation of the errors of the micrometer screws is contained in Vol. I. It is there shown that, though the errors due to wear are considerable, their resultant effect on the mean of readings made in opposite orientations of the plate is negligible. Accordingly no corrections on this account have been applied to the observations.

§ III.— THE MEASUREMENT OF THE PLATES.

The plates before being passed for measurement were examined to ensure —

(1) that the photographic images were sufficiently sharp and round.

Plates with markedly diffuse images, due to unfavourable atmos- pheric conditions, or with elongated images due to faulty guiding, were rejected ;

(2) that the region had been correctly identified. This was effected by a

comparison of a few of the brighter stars with positions as catalogued in the C.P.D. ;

(3) that the guiding star appeared in its correct position with an error not

exceeding 0 ' • 5 in either co-ordinate ;

(4) that the reseau had been duly impressed with suitable intensity ;

(5) that there was a sufficiency of stars shown on the plate. As a rough

test C.P.D. stars of mag. 9-0 should be shown by the third or 20 sec. exposure. In the revised series of plates the sufficiency of exposure was, however, frequently gauged by a comparison with the earlier series, taken on the same area, referred either to the same or to an adjacent centre ;

(6) that there were no other defects either of film or treatment, etc., which

called for repetition of the plate. The next operation was to mark with an ink mark on the back of the plate the stars which had been selected as standards. These stars were selected from the original series of plates and were designed to give an average of about ten stars

INTRODUCTION. XI

on each plate, distributed as uniformly as possible over the area of the plate, and of such magnitude as to give well-defined sharp images on the plates, while at the same time they should be not too faint for accurate observation with the meridian circle. The selected stars are those contained in the Cape Catalogue of 8560 Astrographic Standard Stars for the Equinox 1900.

The plate was then placed in the measuring machine, first with the edge marked " A " on the reseau at the top, or in a position corresponding with the " black scales.'" This implies that the numbering of the lines of the reseau square brought into the field of view could be read off by means of the black figures en- graved on the indicating scales. The small indices reading against these scales admit of slight adjustment to adapt them to the particular setting of the plate, and the only further adjustment usually necessary is that for the orientation of the plate, in order to render the reseau fines strictly parallel to the sUdes on which the plate carrier moves, and to the fixed spider webs in the field of the microscope.

Two observers now take part in the work, one handUng the instrument and the other acting as recorder, the observers alternately reUeving one another.

The observer at the instrument starts with the row of squares contained between the lines numbered 55, 54 and sweeps these under the microscope in order, proceeding in the direction from 2 to 26 in the horizontal scale. When a square is reached containing images of one or more stars, this square is brought into exact coincidence or symmetry with the square of " fixed " webs contained in the field of view of the measuring microscope by means of the coarse motions governing the motion of the plate, and subsequently by means of the fine motions operating on the micrometer box. As a rule only the images resulting from the longest exposure (6"") were measured. Exception was made in cases where this image was disfigured by a flaw on the plate or otherwise ; in this case the second image was measured, and a reduction derived from comparison with neighbouring stars applied to reduce to the first-image system.

After adjusting the square the observer at the instrument reads off the numbers of the reseau lines of the square pointed on, e.g.,

54-55. 7-8,

and these are duly entered on a form by the recorder.

The observer then points the movable webs in turn on the image of each star contained within the square, proceeding in order from left to right in the field of view.

After each setting, the two micrometers are read by the obser\'er, and an estimate of the diameter of the star-image is made. For the brighter stars this estimate is expressed in terms of the space between the close sets of parallel web>

(2273)

Xii CAPE ASTROGRAPHIC ZONES — VOL. VII.

contained in the micrometer as unit, and this forms a convenient means ol dis- criminating the stars' magnitudes, but the method is not equally satisfactory for the fainter stars. Below a certain degree of brightness which yields a diameter corre- sponding with about -6 of the unit involved, all stars give images of sensibly equal dimensions, but showing marked contrast in the degree of intensity. For such stars an estimate of the intensity of the image is recorded instead of a micasurement of the diameter. Five different degrees of density have been recognised, and these have bsen indicated by the num.bers — 1, — 2, — 3, — 4, — 5. — 1 indicates that the star image only just fails to attain the full density of the images ol the brighter stars on the plate, and — 5 refers to the faintest stars whose second images are distinctly visible.

The readings of the micrometers and the diameter are read out by the observer and duly entered by the recorder. Whenever a " standard " star is reached the observer and recorder temporarily change places, and measurements are made by both of them.

On completion of the row 54-55, the observer proceeds in like manner to sweep along the row 53-54 in similar order, viz., from 2 to 26 on the horizontal scale.

The same series of operations is continued until the whole plate has been covered, the two observers, however, changing places occasionally, but always, except temporarily for the standard stars, on the completion of a row.

The plate is then reversed in its carrier through 180° in orientation, i.e. set up in a convenient position for " red scale " readings. This implies that the numberings attached to the reseau lines now correspond with the red instead of the black figures on the indicating scales, and a new adjustm.ent of the pointers and of the orientation of the plate is necessary.

The whole process of measurement is then repeated in the same order as before, but, of course, with reversed directions of motion, each observer measuring the same star images as before in the reversed position. The results of this second measure- ment are entered in convenient spaces in the recording form adjacent to those derived from the previous measurement.

It is evident that, since the sides of the squares are set to correspond with 0 rev. and 10 rev. of the micrometer screws, the sum of the readings taken in the two positions of the plate should, apart from errors of measurement, amount to exactly 10 revs. \^'henever this sum in the first instance differed from 10 revs, by more than 0'-020 (=0"-6) the measures have been subjected to further examination, and, if no fault was immediately traceable, they have been discarded and the measures both in the " black scale " and the " red scale " positions repeated.

INTRODUCTION. XIU

§ IV.— PRELIMINARY REDUCTION OF RECTANGULAR

CO-ORDINATES.

The rectangular co-ordinates of each star image, referred to the central reseau lines as axes, may now be derived by a combination of the reseau readings with the micrometer equivalents, each being expressed in similar units. The unit at first adopted is the millimetre on the plate.

Denoting by R^, R'j, the micrometer readings, expressed in screw revolutions,

for the two co-ordinates of a star at its first measurement and by R2, R 'g similar

readings at the second measurement in the opposite orientation, we have

Ri4-R2= lo--"- e R'i-|-R'g= 10"" - e'

where 6, 0' are small quantities not exceeding (y-020. If we refer the second

measurements to the same origin as the first by subtracting each from 10 rev.,

we obtain as the equivalents of the readings R^, R\ resulting from the second set

of measures,

lo-^— Rj=Ri-t- d and io'^-R'2=R'i-|- 0'

whence, from the mean of the two measures, we derive as the rectangular co- ordinates of the star-image referred to the comer of the reseau square as origin and the revolution of the screws as unit

-(R,-Hi0), R\+id' the sign of the former being reversed to correspond with the direction of increasing R.A.

We may convert these measures into millimetres by dividing by 2 since one revolution of the screw corresponds with half a millimetre on the plate. Further, adding the reseau equivalents, I, I', i.e. the rectangular co-ordinates in milHmetres of the origin in the reseau square in relation to the origin at the intersection of the central reseau lines, we obtain the co-ordinates of the star images in millimetres

by the formulae

i-i(Ri+ie). r-hj(R\+ie')

or the equivalent formulae, which have been used for control, where I, I' are given by the following table of reseau equivalents : —

	Reseau Equivalents for	Vertical Scale (I).
Lilies.	Equivalent.	Lines.	Equivalent.	Lines.	Equivalent.	Lines.	Equivalent.
31-32	4-60	37-38	+ 30	43-44	0	49-50	-30
32-33	+ 55	38-39	+ 25	44-45	- 5	50-51	-35
33-34	+ 50	39-40	-f-20	45-46	— 10	51-52	-40
34-35	+45	40-41	+ '5	46-47	-«5	52-53	-45
35-36	+40	41-42	+ 10	47-48	—20	53-54	-50
36-37	+ 35	42-4^	+ <;	48-49	-25	54-55	-55

XIV

CAPE ASTROGRAPHIC ZONES — VOL. VII.

Reseau Equivalents for Horizontal Scale (I ') .

Lines.	Equivalent.	Lines.	Equivalent.	Lines.	Equivalent.	Lines.	Equivalent.
2-3	+ 55	8-9	+25	14-15	- 5	20-21	-35
3-4-	+ 50	9-10	+20	15-16	— 10	21-22	—40
4-5	+45	lO-II	+15	16-17	-15	22-23	-45
5-6	+ 40	II-I2	+10	17-18	— 20	23-24	-50
^7	+ 35	12-13	+ 5	18-19	-25	24-25	-55
7-8	+ 30	13-14	0	19-20	-30	25-26	-60

Finally, it is necessary to express the co-ordinates in units which correspond more accurately with a minute of arc at the centre of the plate. A preliminary determination of the scale equivalent of the plates, depending on the focal length of the telescope, showed that one millimetre on the plate is less than a minute of arc by one part in 300. The rectangular co-ordinates derived as above have therefore all been diminished by Trwoth part of themselves, and the final formulae for the rectangular co-ordinates derived from the measurement are

x=I-i(R,+i 9)-^5[I-i(Ri+i 0)]

A specimen of the computation form used is appended.

Measures of Co-ordinates — Catalogue Plates. Plate 7566. Date, 1903 September i. Reseau G. 50. Measurers, NM, C.

o„ -47°.	Vertical.	Horizontal.	:i„ Oh .im.
No. Notes. 1	1 a 2	Black. 10— Sum. Red. 3	Line. Equivalent. i Mean. 4	Sum. Sum -=-300. X. 5	riate Corrections. X. 6	Black. 10— Sum. Red. 7	Line. Equivalent. i Mean. 8	Sum. Sum -=-300. y- 9	Plate Corrections. y. 10	CP.D. No. C.P.D. Mag. Mean. Diam. 11
6 *	i-i i-i i-i I- 1	2-590 — 10 7-420	54. 55 -55" - 1-293	-56-293 -188 -56-105	- -152 — - 2 -56-259	9-360 — 2 0-642 2-276 — 12 7-736	II, 12 + !C- + 4-680	+ 14-680 -049 + 14-631	- - 45 + • 10 + 14-596	46°io62i 9-8 I -10
153 *	5-710 — 0 4-290	31,32 +60- - 2-855	+ 57-145 -190 + 56-955	- • 72 + • 5 + 56-888	23.24 -50- + I-I35	-48-865 -163 —48 - 702	- • 37 - ■ 34 -48-773	47°i6 10-2 I -10

INTRODUCTION. XV

In the first column are entered any notes made by the observer, usuallv by means of convenient symbols, and space is left for the entry of a rotation number following the order of the x co-ordinate throughout the plate.

Column 2 contains the measures of the diameters or estimates of density of images as the case may be.

The micrometer readings R^, Rg, are entered in the third column under the precepts " Black," " Red," and the quantity 0 is inserted between them.

In the fourth column are entered the numbers of the reseau lines corres{X)nding with the vertical or right ascension co-ordinate of the star image, and beneath them the equivalent of these readings in millimetres. Beneath this again is entered the quantity — |(Ri-|-|6) derived from the two first entries in column 3.

The fifth column contains the sum of the last two entries in the preceding column taken algebraically, viz. the quantity I — ^^1+^ 9)- ^"d beneath it the quotient obtained by dividing it by 300. The numerical difference of these two quantities gives the co-ordinate x which is then inserted beneath them.

Columns 7, 8, 9, for the computation of the co-ordinate y, corresf>ond exactly with columns 3, 4, 5. Columns 6 and 10 serve for the application of " plate constant " corrections derived in a manner to be hereafter explained.

The final column gives the number and magnitude of the star as identified in the C.P.D. and the mean diameter derived from the entries in column 2.

§ v.— CORRECTIONS FOR REFRACTION AND ABERRATION.

The corrections for differential refraction, as affecting the rectangular co-ordinates of star images on photographic plates, have been very fully developed by Kapteyn {Bulletin du Comite Permanent, t. iii. p. 71). The significant terms may be expressed as follows : —

Ax=ax+by, Ay—Ax+By,

where, with sufficient accuracy,

a=k' sec* q,

b=k' tan q (tan/>— tan d),

A=k' tan q (tan p+tan 6),

B=k'sec^^p,

p, q being defined by the equations

tan^=tan C cos to, tan 9= tan f sin to,

where ^, oi, S denote respectively the zenith distance, parallactic angle and declination

of the centre of the plate and k' is the constant of refraction, dependent on the

readings of the barometer and thermometer.

XVI

CAPE ASTROGRAPHIC ZONES — VOL. VII.

It is convenient to divide each of these expressions for A.r, Ay into three parts as follows : —

Ax=[Bx]+[by]+[(a-B)x] Ay^[By]~[bx]+[{A+b)x]

Now the terms [Bx], [By] in Ax, Ay are evidently equivalent in effect to a variation of the scale equivalent of the reseau, while the second terms [by], [bx] are equivalent to a small change in the orientation of the axes of reference.

Hence we may ignore these terms provided we derive the instantaneous scale -equivalent of the reseau and its orientation from the internal evidence afforded by the measures of star images independently for each plate.

The corrections expressed by the remaining terms are, however, of a different character and indicate a distortion of the field of view as represented on the plate.

Hence if we put

oi.=a—B=k' (sec^ q—sec^ p)= —k' tan" C cos2fo ji=\-\-h=2k' tan p tan q=k' tan" C sin2w

the distortional parts of the refraction, which need alone concern us, are given by

Ax—xx, J3'= fix.

Since a, ^ both contain tan'^ J^ as a factor, and all the plates have been taken at small zenith distances, these corrections will always be small. They have been computed with a mean value of k' corresponding to average atmospheric conditions, as functions of the declination and hour angle of the plate centre at the middle of exposure. Within the limits of accuracy sought their variations, even for extreme atmospheric conditions, are quite insensible. The values used for plates with centres on the declination circle — 47° are contained in the following table.

Refraction Table. (S = — 47°).

Hour .Angle of Plate Centre.	(I	/i
)i tn O O		000016		■ 000000
O 20		000013	+	•OOOOIO
o 40	—	000008	+	■0000 I g
I 0	+	000002	+	•000028
I 20	+	000014	+	• 00003 5
I 40	+	00003 I	-r	■ 000040
2 0	+	000053	+	• 000042

The signs quoted are those appropriate for western hour-angles. For eastern hour-angles the sign of [3 must be reversed.

The expressions given by Kapteyn for the differential effects of aberration show that the principal parts of the aberration are equivalent in effect to a small variation

INTRODUCTION. XVU

in the scale and orientation of the plate. The outstanding effects are of the second order in relation to the co-ordinates, and will be quite insensible for plates of the dimensions used. Thus the corrections due to aberration may be regarded as absorbed in those due to the scale and orientation of the plate.

§ VI.— COMPUTATION OF TABULAR CO-ORDINATES OF THE STANDARD STARS.

The tabular positions of the comparison stars used for the standardisation of the plates are contained in the Cape Catalogue of 8560 A strographic Standard Stars, derived from meridian observations made with the transit circle during the years 1896-1899 inclusive.

For convenience in computation the star places were all referred to the epoch 1900-0 by means of the catalogue proper motions, zero values, however, being assigned to these proper motions where no determination was included in the catalogue.

The rectangular co-ordinates, together with their annual variations on account of proper motion, referred to ideal plate centres, were then computed by means of the formulae given by Jacoby,* retaining terms as far as the third order. To facilitate this computation, auxiliary tables were prepared for each zone of plate centres. The tables used for zone — 47° are given on p. x\iii.

The tables have been derived from a slight modification of Jacoby 's formulae, involving the declination of the star (S) instead of that of the plate centre.

In Jacoby's notation the quantities contained in the tables represent the terms

Table A ...	(UoJ d*sin»i'-FA'jJa») cos d.
Table B ...	D'j.da*.
Table C ...	D'2zla*.d a+D'jJ «>* ;

whence, Aa, AS, which denote the differences of the R.A. and declination of the star from those of the plate centre, being expressed in minutes of arc

x=/ia COS <5-f Table A, y=A (5-Table B+Table C.

Table A takes the sign of Aa ; Table C that of AS.

The arguments involved in these tables will be apparent with the exception of the vertical argument in Table C. The argument here used and denoted by " B correction " is the quantity derived in the previous entry from Table B. The term Aa cos 8 in x was computed with five figure logarithms.

* BuUetin du ComiU International Permanent, t. iii, p. 3. C. A. Z.-VII. 3

XVIU

CAPE ASTROGRAPHIC ZONES — VOL. VII.

Table A. Unit O'-OOl.

\^^ Ad.								Ad. y
^---.^	0'	10'	20'	30'	40'	50'	60'	y^
Zlacosi?"^\								^A a cos 6.
0	o	o	o	o	0	0	o	0
10	0	o	o	0	I	I	2	10
20	o	o	o	I	2	2	3	20
30	o	0	0-	I	3	4	5	30
40	I	I	I	2	4	5	7	40
50	2	2	2	3'	5	7	9	50
60	2-	2-	3-	5	7	9	1 1 •	60

Table B.

A(x.	0'	1'	2'	3'	4'	5'	6'	7'	8'	9'	Ja.
0	•ooo	•000	000	001	•001	002	•003	004	•005	•006	0'
10	•007	•009	010	012	•014	016	•019	021	•023	•026	10
20	•029	• 032	035	038	•042	045	• 049	053	•057	•061	20
30	•061;	•070	074	079	•084	089	•094	099	•105	•no	30
40	• 116	•122	128	134	•140	147	•153	160	•167	■174	40
50	■ 181	•189	196	204	•211	219	■227	236	•244	•252	50
60	•261	•270	279	288	■297	.306	•316	326	•335	•345	60
70	•355	•365	376	386	•397	•408	•419	430	•441	•452	70
80	•464	■476	487	500	•512	•524	•536	549	■561	•574	80

Table C. Unit O'-OOl.

\^^ Ad.								A d. y"'
^^^^^	0'	10'	20'	30'	40	50'	60	y
B. CoiT>\								yB.CoTT.
00	0	0	0	I	2	3^	6	00
10	0	0	0	I		3	6	10
20	0	0	0	o-	I-	3	5-	20
•30	0	0	0	0-	I	3	5-	30
40	0	0	0	0		3	5	40
50	0	0	0	0		z-	5	•50
60	0	0	0	0		2-	A-	60

With all necessary precision for zones so remote from the pole, the annual variations of x and y are expressed in minutes of arc as follows —

/*x=ii"„cos d, iJ.j,=Tki^g,

where \i^, [ig are the annual proper motions as contained in the catalogue, expressed in seconds of time and arc respectively.

INTRODUCTION. XIX

§ VII.— DETERMINATION OF PLATE-CONSTANTS.

After applying to the co-ordinates, as measured on the plates, corrections on

account of the distortional effects of differential refraction, it is necessary further to

correct the measures on account of the imperfections in centring and orientation of

the reseau and variations in the scale. For this purpose, it is necessary to introduce

for each plate four additional constants, a, b, S, P, the corrections being expressible

by means of the formulae —

Ax=a-\-Sx—Py, /iy=b+Px+Sy.

The quantities a, b here denote the corrections for errors in centring,