in Relation to the Coma of Optical Systems. 359 



are the coordinates of the point in which the ray meets any 

 plane at right angles to the axis. 



Considering a complete system we have 



fLoX Q m — ^2(^2^2—^2)5 °r 

 fi x m = fi 2 (x 2 + Sx 2 ) m 2 —fi 2 fyj2i 

 if the ideal values of the coordinates on the image plane be x 2 . 

 Thus 



/W)?»o , __ (m 2 8z 2 —l 2 8y 2 ) _ S.v _ J2_ fy 

 [i 2 x 2 m 2 /j, 2 x 2 m 2 x 2 m 2 x 2 



It remains to obtain the value of 

 Bx _ l^_ fy 



•''■2 W?2 X 2 



in suitable form. 



The aberrations of any centred system may be expressed as 



^ = | / ,{i i '(^p/.x. 2 )}' 



where 



Xii/i represents the point at which the ray meets the 



stop plane, 



x 2 y 2 the point in which the ray should meet the theo- 

 retical image plane, 



Pi 2 =^L 2 +.vr\ P2 2 ='?v +,'/•/, Xi2= + (#i#2+#iy8)i 



and F is a function of the second and higher orders in 



pf, p 2 2 , and xi2- 



Then 



, 9 3F 3F 



6# 2 = ^i5T~2+^2 5 — , 



m 2 Bd' 2 — i 2 B>/., = 2(m 2 ^i— %i)< , + (n).,r.,~l,?/ )^ — ; 



op 1- OX12 



r 5 / * \ .1 3F dF 



.*. p. 2 {m 2 bx 2 --l 2 by 2 ) = 2fM x m ^—z +^ 2W?2 # 2 - — ; 



opr " 0x12 



. W^^ _ 2 f l o !C o m o 5F __ dF 

 /u 2 # 2 m 2 P2<c 2 m 2 'dp^ 0X\^ 



2 BF 9F 



^2#2™2 " 1 oM 



"Bpi 2 



