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

 and then 



D 2_ (0*— /*o)(£—&) — (y—ro) fa- *7o)) a 



(^-/g 2 +("-v ) 2 



Now 



^~^o='l nearly, 



where M and N are symmetrical about the axis, 

 . by + cz 



x= + 



M(*/ 2 + * 2 )+N' 



•M-^-s&S) 



y 2 + * 2 \ u(f + z*) 



D=-^.N to -»- 



M */y 2 + * 2 



Y-Y =-t>.N. f*7V * Z-Z =+r;.N fV^. -y 

 «(y 2 + ,sr) w^ + ^j * 



Z-Z _ _y 



Y-y " *■ 



The line joining the two points in the intersection is therefore 

 perpendicular to the line joining the two points of incidence. 



From these formulae may be deduced the expressions for the 

 primary and secondary foci of an oblique centrical pencil. 



If the focus of incident rays be supposed to be a point in the 

 plane of zx, b = 0. 



Then for a ray incident in the axis of z, y = 0, and it is clear 

 that D = 0, or it meets the central ray. 



V c z 



Xj = ^ — - {3m + l)-^ } if — be supposed very small as com- 

 pared with — . 

 u 



For a ray incident in the axis of y, z = 0, N = ; .'.0 = 0, and 



X2= 2^/ (m+1 ^- 



If for brevity we write the expressions for — in the form 



7} b 



— = — h T + U, we have for the coordinate u, of the point where 

 v u '* r 



a ray after refraction at two lenses in contact meets the plane 



perpendicular to their common axis through the focus for central 



direct rays, 



^ 2 =^- + T 2 + U 2 =- + T 1 +T 2 + U, + U 2 . 



