360 Mr. S. D. Chalmers on the Sine Condition 



The denominator is generally nearly unity. 

 When the object is small and nearly on the axis we need 

 only consider terms of zero and first orders in x 2 , y 2 , that is, 



These three terms may be considered separately : — 



Tsf 

 (1) The lateral spherical aberration is 2pi ^~ ; 



(2) 



The comatic effects ; 



ire made 



up of 





8a; = 



wMpi*) ; 







id 



(3) fa = 



**»g? 



= 2^ s 



d/ 2 . 





fy = 





= 2# 1 # 1 #« 



>1 2 ' 



Thus 



we have 









fjL x m 1 __ lat. spherical C x , C 2 



/X 2 X 2 m 2 pi x z x \ 



where 



Ci is the # displacement due to Coma when % 12 = 0. 



C 2 is the additional x displacement due to Coma when %i 2 ^0. 



C 

 Except when x 1 is very small, the last term — becomes 



X\ 



unimportant ; if x 1 be very small the left-hand side becomes 

 indeterminate and dependent on the actual value of x 2 as 

 chosen. 



If we take %i 2 =0 we have 



m 2 = sin A, 



li x sin B 1 _ longit. spher. Coma _ &v hx 2 

 fi 2 x 2 sin A v 1 x 2 v x 2 ' 



and B and A may be obtained from the axial ray if the image 

 be small and v be the distance from the plane (1) to the point 

 at which the axial ray meets the axis. 



This result may also be demonstrated as follows : — 

 Consider the ray OPQI (fig. 2) in the plane yz passing- 

 through the system and cutting the axis in I and the theore- 

 tical image plane in J, OxP a neighbouring ray through O l 



