170 ON A MULTIPLE DEFINITE INTEGRAL. 



integration, as the limits extend from GO to + co . Consequently 

 the last written expression becomes 



1 [* s J r j [ +c cosamx 7 f +00 cosawy 7 

 / fudul cos au da = y-dx I f dv... 



n-Jh JQ J- a'-t-or J- M I +y 



AT [ +M COSOimX , 7T _ wma p 



^low g- 5- du = e &c. = &c., 



J_ w a +x a 



and thus the integral becomes 



_."-! rh' r < 



-? A <&j cos a^~ (Wia+n6+ - - )a da, 

 06. ..J*" J n 



or, Tr"- 1 



which was to be proved. 



It may be well to verify this result in a particular case. 

 Let v = 2 ; then we have to prove that 



/* f(mx + ny) ma + nb [ h> fudu 



j y (a* + x*)(V + y*)~ ~ab~~h 



for simplicity, we will suppose that 



Let mx + ny =u, and therefore x = mu -f wv, 

 nx w^ = v, y nu mv, 



u and v being two new variables. As u* + v 2 = a? 2 + ^ 2 , efo c?y is 

 to be replaced by du dv, and thus 



jdxjdy 



! +y) 

 ^ 



^comes 



w J {a 2 + (mw + H 2 } {^ 2 + (nu - <mv)*} ' 



The limits are easily seen to be + co co for v ; h' and h for u. 

 For the integral expresses the volume of that portion of a solid 

 bounded by the surface, whose equation is 



- 



which is included between the plane of (#?/), the bounding surface, 



