the Spherical Aberration in a Lens-System. 83 



at a distance ?/» from the axis. I£ Si and s/ denote the 

 distances (measured from the vertex positively to the right) 

 of the intermediate point images formed by paraxial rays 



respectively before and after refraction, and A^, A* the 

 corresponding spherical aberrations for the rays OF, PI, we 

 have A0 = 5i + Ai_!, AI=.s/-t-Ai. Then C being the centre 

 of the surface AP we have as usual 



/Ltj_! sint = /Mi sinz , 



whence accurately 



/X t -! . CO __ fJLjQl 



OP " IP 



/A,_i(AO-n) 



Mi(AI-r) 



^/A0 2 -2(AO-r f )AN v/AL 2 -2(AI-r)AN- 

 Let AN = X, then 



Mt-iOt + Ai-i— n) _ fijCsi' + Ai—rj) 



A /'( 5i +A i .i) 2 -2\(^+A i _ 1 -n) v/fo' + Ai^-SXfo'+A, - r,) 



Expanding each side in powers of the A's by Taylor's 

 theorem we have 



^^{/W+^-i/'^O+^aLi/"^)-} 



G2 



