Aberrations of a Symmetrical Optical Instrument. 271 



Assuming a plane object perpendicular to the axis, let the 

 successive Gauss image-planes for the several surfaces be 

 constructed, and let s i} s/ be the distances from 0; of those 

 connected with the z'th surface before and and after refraction 



Fig. 1. 



respectively. Let K„ K i+1 be the positions on these planes 

 of the images of a point on the object supposing there were 

 no aberration. Then CiKiK i+1 is a straight line. Let a ray 

 from the same point on the object incident on the surfaee at 

 Qi(& Vi, £) cut this line at F l (x l -h Sa?,-, ft+fy, Si+As-j, and 

 let the refracted ray Q;P, +1 cut it at 



Pi+iOt\+i + o\r t+1 , y t+1 +6> <+1 , sZ+Asi'), 



(*»>&>«*), and (x i+ i, y i+v sj) being the coordinates of k\ 

 and K i+1 . 



Then we have 



fc' + tf+^fc, 



or 



^ - 2 \ ?•<» J ' ne S lectlrl S 4tl1 order terms . (1 ) 

 Also it readily follows from the law of refraction that 



Q.Pi Q,P, +1 

 ^_,(J,P, ^C,P, +1 ( 2 ) 



Q l IV = (.r.4&,-f,) = +G',.-tfy,-^ + (, l + A, l -g2 

 = (.<-, + o.r,) 2 + (y ( -f S^y+{s i + A^-2^ i U, + B.v t ) 



