Astro-Photographic Catalogue Plates. 113 



The differences in the column J — H, though small, are of a sys- 

 tematic character. They are doubtless due in part to a difference 

 between the constants of reduction emplo3^ed by M. Henry and 

 those which I have used, M. Henry having employed a predeter- 

 mined scale-value whilst I have deduced the correction of the 

 scale-value and orientation from the known stars. But in order to 

 account completely for the discordances, it would be necessary to 

 investigate the differences between M. Henry's formulae and 

 those given in the present paper. Such differences exist, especi- 

 ally in the refraction formulae, where M. Henrj- computes merely 

 the change in the magnitude of the co-ordinates x and y caused 

 b}^ refraction. But refraction not only changes the magnitude of 

 iC, but it also removes the ^ as a whole to a different position on 

 the plate. And the Aa corresponding to x depends not only on 

 the magnitude of x^ but also on its position with respect to the 

 centre of the plate. 



It may be of interest to consider the effect of errors in the 

 assumed right ascension and declination corresponding to the 

 centre of the plate. For this purpose I take up again the expan- 

 sions given in the beginning of this paper, writing them now in 

 a somewhat different form, and introducing the auxiliaries x' and y' 

 for brevity. It is sufficient for the present purpose to include 

 terms of the third order. We have, putting 



x' = ( a' — a ) COS f\ y'^ 6' — rf ( a ) 



the following : 



x = x' — tan S x' y' + l{\ — l tan2 d) x'^ } , 



y = y' -i^l tan ^ x'^ + i ( I — tan2 6) x'^ y' + ly"^\ ^' 



x' = X ~\- ta,n S xy — i (i + tan^rf) a;''-[- tan^ t5a'y2 | 



y = y — Jtanrfa;2— J?/3 — J (i + tan2d)a;2 2/ j ^^^ 



From equations (a) we have, by differentiation with respect to 

 a and d : 



dx'= — cos 6 da — tan ^ x' d ^, dy'= — d^ {d) 



Now differentiating equations (&), and substituting from equations 

 (c) and (d), we get, after slight reductions : 



dx^= — cos S da-\- tan S y . cos (^ da — x^ . cos (^ da — xy . rfd" ] 

 dy = — dS — tan ^ x . cos ^ da — xy . cos ^ da — y''- . dS J 



