Investigating Aberrations of Symmetrical Optical System. 507 



incidence of the ray on the plane of the first surface of the 

 system. 



With an infinitely-distant object in the plane zOx, the 

 image, if one exist, is also to be found in this plane. We 

 shall therefore study the position of the point in which the 

 emergent ray strikes this plane. 



Denoting by \_V , m\ n' ; a', u f , v'~\ the coordinates of the 

 emergent ray, we suppose it to cut the plane zOx in the 



point (r, o, ro- 



Then from equations (8) 



/ If'/ ' I vl 



U = — 771 £ J » =-TO £'. 



Putting m=0, w' = — m'U', our fundamental formulae 

 become 



m'/u = Q + i{Bn 2 -2I)nv + 'F(u 2 + v 2 )\ 1 



-m'glu = $ + j [ \Kn 2 -2Jnv-t-L{u 2 + v 2 )}. 



(i.) Incident beam parallel to the axis. 

 Here n = 0, so that m'/u = Q ) + ^'F(u 2 + v 2 ), 

 -m'%u = & + ±L{u 2 + v 2 ). 



Thus ? = -{$ + iL(u 2 + v 2 )}l{Q + iF(u'+v 2 )}, 

 = - S/Q + (FS - QL) (u 2 + v 2 )I2Q 2 . 



if f is to be the same for every ray, i. e. is to be inde- 

 pendent of u, v, we must have 



FS-QL = 0. 



This is the condition for no spherical aberration. 



If h is the semi-aperture, then h 2 = u 2 + v' 2 , and the amount 

 of spherical aberration when the above condition is not 

 satisfied is seen to be 



A>(FS-QL)/2Q 2 . 



It is worth while to remember that — Q is the power of 

 the system. 



(ii.) Point-to-point correction. 



Returning to the case in which w=£0, we have 



_ S + j{Hn 2 -2J^ + L(u 2 + r 2 )} 

 * ~ Q + i{Bn 2 -2T)nv + F(u 2 + v 2 )} 



= -S/Q + {(BS---QH)n 2 -2(DS-JQ)m> 



+ (FS-LQ)(u 2 + v 2 )}/2Q 2 

 to the second order. 



