Astro-Photographic Catalogue Plates. 115 



from equations (e), replacing x and y hy X andF, which does not 

 cause an appreciable loss of accuracy in these small terms. This 

 gives : 



(X—z)+pX+(r—l&n&. cos 5 da) F+ (x + cosS da) + X2.cos8da + XFdS = 

 (^Y—y)+p r — (r— tana. co&& do.) X+{^+cU) +,XY . cosB da+ Y^d& = Q 



}w 



Equations {h) are complete equations for determining the con- 

 stants of reduction of a plate. 



These constants are : 



j9, r, the constants of scale value and orientation, 

 y^, 4', the errors of centreing the plate, 

 cos dda, dSj the errors of pointing the telescope. 



The corrections required by the observed co-ordinates of a star, 

 beyond those already given in my reduction of M. Henry's plate, 

 are evidently 



X2 . cos 6 da-Jr XYdS for X 



XY. co&6da-{- Y^dS " Y 



When X and Y are each equal to 1°, none of these terms could 

 amount to 0."01 until cos d da or d8 become as large as 33'''. 



Equations {h) bring out several very interesting points. If 

 we neglect the very small terms in Z^, XZand T'^ we see that 

 the equations are of the same form as those used in the reduction 

 of M. Henry's plate. But in that case it is evident that we did 

 not obtain the true orientation constant r, but the quantity 



r — tan 6 . cos S da 



It follows that for plates taken at considerable declinations we 

 must not expect the orientation constant to come out the same 

 for each plate, as in each individual case the value obtained will 

 depend on the accidental error of pointing the telescope. Equa- 

 tions (Ti), moreover, show that it is impossible to determine this 

 accidental error without retaining the terms in -X^, etc., which 

 are so small that they would not determine cos d da and d8 with 

 accuracy. Similarly it is impossible to separate cos 8 da and d8, 

 which depend on the pointing of the telescope, from y and (p, 

 which depend on the adjustment of the plate. 



