40 Prof. Maxwell, On Ajjjyroximate Multiple [Mar. 12, 



to be integrated is a function of a linear function of the two 

 independent variables. 



- Thus if 



/''^'2 /"2/2 



1=1 I udxdy (1), 



where u is a function of r, and 



r= a + hx + ci/ (2), 



let a^ = 2^^2 + ^i)+2^('^2-^i) • i^)> 



2/ = 2 (2/2 + 2/1) + 2^ (2/2-2/1) (4), 



^ = 4(^2-^1) (2/2-2/1)] j ^'^p(^l -(5). 



If we write 



ro = a + 2^{x^ + ^,) + 2^iy2 + V,) (6), 



^-2^i^,-^i) (7), 



7 = 2 'y (2^2 - 2/1) (8), 



? = /5p + 7^ (9), 



we may consider w as a function of ^ of the form 



u = A„ + A^^+A^^ + &c (10), 



Ifjudpdq = A, + Ia^ (/3^ + 7^) +A^(j^^^ + l /3V + 1 7^) 



+ &c (11). 



Now let Uq be the value of u corresponding to ^ = 0, 



u^ and u\ ?= ± ^1, 



w^andw'g ^=±^2, 



and if we assume 



I={X^- X^ (2/2 - 2/1) l^o'^fo + ^1 (^^1 + '^^'l) + -^2 (^2 + ^^'2) 



+ &C.} (12), 



^= (^2 - ^1) (2/2 - 2/1) {(^0 + 2^1 + 2i?2 + &c.) ^, 



+ (2i^,r,^ + 27?,^,=' + &c.)A + &c.] (IS), 



