Mr. J. Bridge on the Oblique Aberration of Lenses. 353 

 In the small terms y v g^ 5l, |l may be considered as equal 



i. be 



t0 yu z» —, — . 



u u 



The conditions that all rays proceeding from any one point in 

 a plane perpendicular to the'axis may meet in one point on the 

 screen placed to receive the image, are 



U t + U 2 + ... =0 (and a similar one for f) 



for all values of y, z, b, c, involving among other conditions, that 



n^-1 %-l 1 i 



*, + — + --= ' -JT+X+--* 



fvfv &c. being the principal focal lengths of the lenses, the 

 thicknesses being neglected. This equation is impossible so 

 long as the lenses remain in contact. 



For two lenses not in contact, but whose interior surfaces are 

 separated by an interval d, the aberration is found as follows. 



Let v l — d=u i , and let y v g % be the point of incidence on the 

 second lens; C 2 = m 2 -|- 1, 2m = 2(C 2 -l). Then 



and by substitution we obtain 



2* = 1l + Hi . A . h_( "hzi} . J_\ 



»' 2 w 2 u 2 u ™ 9 \ r 2 u^J 



U V u <2 «2 W 2 2 // 



It may sometimes be convenient to use the polar forms of 

 Rome of the expressions, as follows. 



K z=p since, y=pcos«, A = <7cos& c^asm/3, |^- =t an^, 



Phil, May. S. 4. Vol. 9. No. 00. May 1855. 2° A 



