ON MULTIPLE INTEGRALS. 215 



[a- V(- 1) (am+ a,m,)} ... {c- V(- 1) (a 



Now, assume that* 



1 



(a + am + a,w,) ... (c + a^> + a^J 



...... (2); 



where -F a& , F w , &c. are independent of a and a, . This assump- 



r r _ ^ 

 tion is justifiable because it introduces -- - disposable quan- 



tities F t viz. as many as there are combinations two and two of 

 the r quantities a, b . . . c, and it will be easily seen that there are 

 the same number of conditions to be satisfied. 



Consequently as 



e (u+aV)V(-i) = cos ( aw+ a ' tt ') + V(~ 1) sin (a.u + aV), 

 we shall have 



H= F& [db (am + arn^} (an + ,w,)} cos (au + a,u t ) 



+ {a (an + can) + b (am + a,w y )} sin (an + aft) \ 



divided by 



{a 2 + (am + am')*} {tf + (an + an') 2 } 



Let us next assume u = mx + w?/, w y = w/c + w^, a; and y being 

 here two new variables ; also a' = am + a^, , and /3' = aw + a y n, : 

 then the coefficient of F& in the expression of H will become 



(aE - a'ff) cos (q y a? + ^y) + (aff + bof) sin (g'x 



Moreover dudu 1 dada l will be replaced by dxdyda'dfi ; and 

 therefore, as we have 



1 >A r^ r +CD /-+ 00 



/ = -2 1 <f>udu I $ l u t du l I 6?a I da / Hj we shall have 



I = j^2 2^ a& M0 (wia; 4- w?/) ^ (m t x + ny] dxdy M, 



