116 Reduction of Stellar Photographs. 



But if we assume that the adjustment of the plate is perfect, or 

 in other words, that a perpendicular let fall upon the plate from 

 the optical centre of the object gla&s^cuts the plate at the inter- 

 section of the co-ordinate axes used in its measurement, then 

 X and 4' ^re 0, and wo have for our former constants k and c : 



h = cos S da^ e = dS 



IJpon this assumption, the corrections which have to be added 

 to the measured X and Y on the plate are : 



i)X+(r— fctanrf) Y+k-\-k X^^e XY f or X 



p F— (r — fctanfJ) X+c + A;XF+e F2 " F 



which are of exactly the same form as the expressions (3) except 

 for the negligible terms in X^^ XY, and Y\ 



One other matter deserves brief notice in the present paper. 

 It will sometimes happen that certain plates do not contain 

 enough well-determined stars to furnish satisfactory values of p 

 and r. In that case we can use the known stars upon the neigh- 

 boring overlapping plates, as was first suggested b}^ M. Loewy. 

 "We may suppose that the values of p, r, k and^ are very small ; 

 or, if not, that their values are very approximately known. Us- 

 ing these approximate values we now determine for all the known 

 stars, and for a number of unknown stars well distributed in 

 the overlapping portion of the two plates, the values of ^d and 

 Aa. This is done exactly in the manner already described ; or, 

 in other words, we reduce some of the stars preliminarily with- 

 out the help of the overlapping plate. 



Now let us indicate by p', r'^k' and c', the corrections re- 

 quired by the assumed approximate values of ^, r, k and c ; and by 

 the subscripts 1 and 2, any corresponding quantities belonging 

 to the two plates. We shall then have from each known and un- 

 known star the following four equations : 



/ 6' —6^ = \6 — r^. 15 Xi + j^/ Y^ + c/ 



^ M a' — a^ =: Aj a -|- r/ . X^ sec 6^ -|- p^' . yij, Fj sec ^-^ + tV ^i' sec 6^ 

 \ a' —a^ = ^.^a-\-r.J . X, sec 6.^ + p.J . j\ F^ sec 6.^ -\- ^^ fc/ sec rf^ 



From these equations we obtain b}^ subtraction 



