Astro-Photographic Catalogue Plates. 



103 



a;secrf = Ac+J/ AaA(5 ^j' = — tanf^sini" 



+A^' Aa3 A^' = l{ I— I sin2 d) ( 15 )2 sin^ i" 



+A^' ^a^^6 ^3'=r|(— 5sin(?cosrf+tanc5sin2(J)(i5)2sin3i" 



+^/A«A(53 A^^ — l tan (^ sin^ 1 " 



+J5'Aa3Ad2 J5' = Jsin'^f5(i5)2sin*i" 



+^6'^«^ ^6'=Tio(i6cos*c5— i3sm2dcos2c5+sin*fy)(i5)<sin*i' 



y=Ad+Z)/Aa2 



A' 



+Z>2'Aa2Ac5 



A' 



+Z»3'A(53 



A' 



+Z)i'Aa2Adz 



A' 



+A'^'^* 



A' 



+Z>6'Aa2Ac53 



A' 



+J)/A«JAf5 



A' 



+A'A(55 



A' 



= isin2c5 (15)2 sin i" 



^2-cos2 (J(i5)2sin2 1" 



= |sin2 i" 



= — }sin2(?(i5)2sin3 i" 



^25(5 sincycos3c5 — sin3rfcos(^ (i5)*sin3i" 



^i cos 2 S (15)2 sin* i" 



= 2V ( 5 cos* f5 — 12 sin2 (5 cos2 cJ+sin* 6 ) ( 15 )* sin* i" 



^x^sin* 1" 



Many of the terms in these expansions are generally inappreci- 

 able, but it is not necessary to enter into this point here, as it is 

 better to consider it when constructing tables for the separate 

 degrees of 8. It will be noticed that x only appears multiplied 

 by sec S. This is a great advantage, for we can avoid the use of 

 X altogether, and turn our measured co-ordinates at once into 

 xsecfJ, by means of a separate scale-value table constructed for 

 each degree of d. 



It remains to show how to correct the observations for refrac- 

 tion, and to determine the scale-value and orientation from the 

 known stars on the plate. The aberration need not necessarily 

 be considered when the scale-value is thus determined from 

 known stars. 



Let us now indicate by X^^^ and Y^^^ the actual co-ordinates 

 measured on the plate. They will be expi'essed partly in divi- 

 sions of the reseau., and partly in revolutions of the micrometer 

 screw. We shall assume that the scale-value is known very 

 nearly, both for the reseau and screw. We can then prepare 

 separate scale-value tables for the reseau and screw, by means of 

 which we can at once turn X^^^ and Y^^^ into X sec 8 and Y, 

 expressed respectively in seconds of time and arc ; and these 

 tables can be made to include the effects of errors of the reseau 

 and screw. These values of Xsec<5 and ]r will then be very 

 near their true values, which we have called a; sec (J and y. The 

 corrections which must be added to them are : 

 I. Correction for refraction : 



