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



giving, when BIO + A 23 is not zero, 



T 18 = A 13/0l 2 + B 13/ >3 2 + 2C 13X i3 + D 13/ >i 4 + E 13 />3 4 + 4F 13X i3 2 + 2G 13/0l V 



where 



A A Ci2 2 T> T> C 2 3" p Cio . C23 



13 = Ai2 ~ A -TS-> 13 = #23 - ^ --p- , Ois = -r -- ~f , 

 A23 + >12 A 2 3 T -t>12 A 23 + I5i2 



DM = D 12 - 4aHi2 + 2a 2 (Gi 2 + 2F 12 ) - 4a3J 12 + a 4 (E i2 + D 23 ), 

 E 13 = E 23 - 4/3J 23 + 2^ 2 (G 23 + 2F 23 ) - ^Hai + /? 4 (E 12 + D 23 ), 

 F 13 = a 2 F 2 34-^ 2 F 1 2-2a^H23-2a^ 2 J 12 + a ^ (E 12 + D 23 ), 

 GIB = * 2 G 23 + ^G 12 - 2a 2 ^H 23 - 2ajtfJi 2 + a 2 ^ ( El2 + D 23 ), 

 H 13 = /3H 12 - *P (Gis + 2F 12 ) + a'Ha, + 3a 2 /^J 12 - a?p (E 12 + D 23 ), 

 23 ) + )8J 12 + SajPHos - a^ 3 (E 12 4- D 23 ), 



.................. (II), 



in which 



a = ^12 n _ C28 



A 23 + Bi 2 ' A 23 + 1 



If, however, A 23 + B J2 is zero, (#3^3) is the Gaussian image of 

 and 



x s = x l7 - xz (hps- + ip-2 2 - 2/ X23 ) 



L C/1'> 



where 7 = - , 



an). 



Thus e = is the condition for the correction of Spherical Aberration, 

 i = ,, absence of Coma, 



/ = ,, correction of Astigmatism, 



g = is the additional condition for Flatness of Field, 

 h = is the condition for no Distortion. 



Or if we write 2F + G = 3K and 2/+ cj = 3k, then 



and we may use the conditions k = and f-g = Q together, instead 

 of / = and g = 0. 



