EVALUATION OF DEFINITE MULTIPLE INTEGRALS. 153 



By the general formula 

 E= - j h fudu ("da 1 dx \ dy...e- (WMX+nby+ --- } cosa(mx+ny+...-u) 



77V hi J o / o ^ o 



&)). 



Let F= [ *dx {^ dy ... e - (mtta ***+-> cos a (mx 4 ny + ... - u) ; 



'o J o 



/oo / 



then .F= cos aw cfce eft/ ... e -< 1Baa * n w 1 --) C os a (ra# -f w# + ...) 



J Jo 



-- } sin a (mx 



7x| Jy .. 



J o 



which we may put equal to G cos au + H sin aw. 

 First to find the value of Gr 



= [ dxi dy ...e- (max+nby+ - } cos a (mx + ny +...). 



J o ^o 



Develope the cosine ; the result is composed of terms, each 

 containing sines or cosines of all the variables. 



Also by the formulae 



a 



/; 



e~ max sin amxdx = 



we see that every factor whether sine or cosine introduces on 



integration a factor of the form ^ ^ . Moreover a sine 



ffi \o> ~r a ) 



factor introduces a in the numerator, a cosine factor a or b or, &c. 

 Let P represent the continued product of a, b, c, &c. and 



D that of w (a 2 + a 2 ) n (5 2 + a 2 ), &c. Then G = ^ S(7^> , 



where //. is a positive integer less than the whole number of 

 variables in E, and equal to the number of factors in the 



denominator of -j , and C is some coefficient. 

 ab ... 



A little consideration shows, that if we develope 



