Mr. J. Bridge on the Oblique Aberration of Lenses. 345 



the tangents of the angles made with the axis of x by the pro- 

 jections on the planes of xy, xz. We must therefore write for //., 



/*(l— 2 /i2+v2 )' &C ' ; this gi 



gives 



,_fi m-l ym-ly/ if + z* ^y + vz 

 m m r Znr r \ r r 



1 /A[~/i, 2 -f-v 2 ~fi,y + vz m — 1 y^ + z 12 m — 1 



1 jiV^ + v | 2 fj,y + vz m-1 



2 ?«L ??i 2 r ?n' 2 



"V+o 





2ui 



1 7w — 1 5/ '[/j? 4- v 2 j_ /xy + VZ m— l^y 2 -}-^ 2 m— 



_fj, m — l y 

 m m r 



■ [> 2 4 v 2 f*y + vz m—1 y^ + z* m — l 1 2 ~| 

 L m r m i m J 



m—1 r ; y 2 + ^ 2 n ay + vz . ... — _ s ~l 



+^ m ir-/ A L w ~ 1 -S^ +2 ^ — (m+i).^ 2 +v 2 j 



m — 1 «r-s r w 2 4-£ 2 n ay + vz , „ ~\ 



+ "2^3 vL W " m+1 -^ 2 ^~ + WTl(^ 2 +v 2 )J- 



Next, £o yzTit? the direction after refraction at the second surface 

 of the lens whose central thickness is t. 



Let s be the radius of this surface, the curvature being sup- 

 posed in the same direction as before ; y^z 2 the point of incidence 

 on it ; fi 1 v x the direction-tangents of the emergent ray. 



The refractive index is here — : then by substitution in the 



above, we have 



fi t = /x'?n — (m — 1 ) . — 



, (m-l)m 2 r m-1 y<? + z* A 2 + v'^ > m +l ( „n , ,«! 



y a (/« — l)OT 8 r m 3 — wi-fl y 2 2 + ^ 2 2 o Aa + ^g , »» + ! / k, a\l 

 s 2 L mr s s m J 



Taking now the centre of the second surface as origin, 



1 y2 _|_ -2 



x<? + y<? + z<?=s*, ■'■ x^s-- J —j^ nearly; 



l ,,24.-2 

 (« + /--.s-0 2 4-»/ 2 + ~ 2 = ?- 2 , ••• ar + r-s-/ = »— J-^-^ ; 



