in Relation to the Coma of Optical Systems. 357 



Steinheil appears to have arrived at a correct estimate of 

 the errors in telescope systems, on the basis of a number of 

 trigonometrical calculations. 



I propose to obtain the relation between the Coma of a 

 system and the errors in the Sine Condition. 



If and A be the angles which the initial and final rays 

 make with the axis, and q the theoretical magnification of the 

 system, the Sine Condition may be stated — 



f*o 



sin 



g /n sin A 



1=0 for all values of the angle 0, 



fi and /jl being the refractive indices of the first and last 

 media. 



We shall consider centred optical systems only and take 

 the axis as the axis of :, and the object may be assumed in 

 the plane y = 0. 



The length of the theoretical image will be denoted by 

 x and the errors, in the directions x and y, for the actual 



ray, will be denoted by &e and By, and — and -* will specify 



x x 



the Coma for the ray considered. From approximate theories 

 we can see that these terms depend on the particular ray- 

 considered and may be represented as follows : — 



— = h(2 cos 2 >/r + l) sin 2 A + ; 2 (4 cos 2 ^+l) sin 4 A 



+ terms of higher order in sin A, 



— = i x (2 sint/r cos^r) sm 2 A + i 2 (-i sin-^r cos, yfr) sin 4 A 

 j 



+ terms of higher order, 



where yfr is the angle between the x z plane and the radius 



Fig. 1. 



\Mfo 



to the point of the stop through which the ray passes 



(fig. i). 



