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

 and 



v'(/ir 2 — y.\x + [i,y + vz)=fJ(vr* — z.\x + fj,y + vz)+A(fiZ — vy) ; 

 .' . (Jbr* — y(\x + /j,y + vz) ) 2 = /// 2 (r 4 — ?' 2 . \x + /xy-{- vz) 2 ) 

 + 2/jJ . A(\x + fiy + vz] 2 — r*)y 



+ A 2 (y 2 + ^V— 2fMy .Xx + py + vz), 

 whence 



, y 1 



^ — mr* * /wA ' 2 — ? ' 2 + (^ + W + vz f ± —^{p^—y-^ + W + vz ) • 

 As the process is thus far symmetrical, similar expressions will 

 hold for \' and v'. 



The equations above are satisfied by two lines in the plane 

 passing through the normal and the incident ray, one on each 

 side of the normal ; the upper sign belongs to the refracted ray. 



Expansion of fi' and v', 

 x= * / r 2 -y 2 -s 3 =r(l-2 ?L J £ -) \= ^l-^ 2 -v 2 =l-|(|a 2 + v 2 ). 

 By substitution we have 



v ~\~ z r 

 \x + fiy + vz=r + fiy + vz— ^- v — [y?+v 



r *-(Xx + w + vzf = r*(£^-2 fJ ?^ + ^ + v^ 



TO V 2 -r 2 +(^ + ^ + v^) 2 =r 2 [m 2 -(^^-2^±^ + ^ + v 2 )] 

 Vmrr*— r 2 + (A^ + /u,y + v^) 2 



.... /=yfi- i (^_2<*±» +l ,. + ^)] + e._r 



^ rL 2»</V r J r r J A m r 



Lift 2in\ r z r /J 



_//, nz-1 y m-1 y( tf + z* n fiy + vz , 2 , ^ 



m m r am* r \ r* r J 



As the axis of x is no longer to be symmetrical with the other 

 two, it will be more convenient that fi, v, fi', v' should represent 



