152 EVALUATION OF DEFINITE MULTIPLE INTEGRALS. 



This follows from tlie formula 



a r () *O 



x cos a xdx = V^ cos a - 



a 



m 



/ x a ~ l sin a xdx = ^ sin a ^ 



Integrating in a similar manner for y, &c. successively, we get 

 ultimately 



Now by (7), we easily see that 



f F (a+b + ...) ( f TT \ 



i v a+ fr f ---~ 1 cosa (v u) dv = +JT ^-cos -1(^+0+...) aw j- , 



' V ' 



and therefore 



Hence J a (B) = -A- .^ ' V ; '" w 1 



for all positive values of w, and = for all negative values, by 

 Fourier's theorem. Consequently the second side of (2) be- 

 comes 



1 r(a)r(fc) ... 

 tt.T(a + b+.. i 



Thus finally 



fdxjdy ... x a ~ l ^~ l ...f(mx + ny+ ... 



which is equivalent to Liouville's extension of Dirichlet's 

 theorem. 



I proceed to evaluate the definite integral 

 t dx I dy...e^ M ^'^f(mx + ny+ ...) ......... (E), 



J o Jo 



ab ... and mn . . . being all positive, and the limits being given by 



