270 Mr. S. D. Chalmers. [June 18, 





 Jn the complete system 



= - a 

 i2 i2 + BOI) Ci2 



and 



Ci --* 

 Ai 2 1 - a ' 



= {2E 12 - 4/?J 12 + 6/?2K 12 - 



Ui2 

 2 /Hoa^T "I 2 / ~ H 02 , IT \ 



7T-^-T + J 23? =TS i r^+Hu r 

 C 23 I y 3 J CM I a 3 



oi - ^ (D 12 + E i)} 



2 / J 01_uTT \ 2 / - J 02,T I 



DC if" J23 i ;= c.l-T- Jl2 j 



= Zl {aJ 12 - 3a^K 12 + 3a 2 /?H 12 + j8 3 J i - a/? 3 (D 12 + E i) - aJ 12 } 

 = 3K 12 - 



2 

 r? 



C 23 L 7 



,. 



We have A = provided Di 2 + Ai 2 3 = ; but this is the condition that 

 the single lens should be spherically corrected with regard to the 

 centre of the stop.* 



It is further evident that it is impossible to exactly satisfy A=0, 

 k = 0, and i 0, and it is easy to obtain the minimum residual error 

 for a given stop and field, t 



* This result is given by Professor Lummer. 



f This was pointed out by H. Bruns, " Das Eikonal," ' Abh. der 2. Klasse K. 

 Sachs. Akad.,' Leipzig, 1895. In both cases the methods employed are essentially 

 different from the above. 



