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



P, Q, &c. being written for the coefficients of (y 2 + * 2 ), (/*y + v2) 

 &c. in /x ; 



•* v u 2suv yj T ; 2stw vy ; v '» 2sw 



^^ = ^ + ... + ...+ ±(^+^1.^ 

 v u miA u ?•/ 



2mm; \ m /" / 



= (m _ 1 f_ + ?_* + i.(srT.» + ?_ s )(^ + l) 



v y ?i m m m V r u u/ \ r u/ 



m _lT-2 g /m + 2_4?« + 4 3m 2 -f m 2m 2 + m 



3m + 2 m?\ 



_ Jy + csr/m + 1 m 2 2m + l\ ^+c 2 , .■"] 



« 2mn U \ r n u / u J 



„. 1 n»-l , 1 , t (m-\ 1\ 2 .. . 



Since -= + - + -( +-) , this gives 



v n u m\ r u/ 



v b b t (m-\ 1 \ 

 v u u m\ r u / 



-w.-r — A.w 2 + ;? 2 + 2B — h (m+ 1) — — 



y 2»m L ^ « « J 



:[ B .p+?-;to.4±=], 



J m — 1| 

 w 2mw 



* Whence, by putting 6=0, e=0, and diminishing y and 2 without limit, 



we have = +— 4~ — ( +— ) f° r tue focus of a small central 



v ii u m\ r u / 



direct pencil. 



