0.2V A MULTIPLE DEFINITE INTEGRAL. 171 



and two planes parallel to one another and perpendicular to (xy). 

 The equations of these planes are respectively 



mx +ny = h, and mx + ny h', 



and the introduction of u and v is equivalent to changing the 

 axes of co-ordinates, so that one of the new axes, that of u 9 

 is perpendicular to these planes, while the other is parallel to 

 them. 



In order to find the value of 

 dv 



/+OO 

 _oo [a* + (mu + nv)' 2 } {b 2 + (nu mv) 

 I 



27 , assume 



{a 2 + (mu + nvY\ {& + ( nu - V Y] 



_ A (mu -f nv) + Bna C (nu mv) + Dmb 

 a 2 + (mu + nv)* ~b 2 + (nu-mvf ' 



It is evident that the terms in A and C will disappear on 

 integration between infinite limits : those in B and D become 

 respectively nrB and TrD, and the integral in question is 

 therefore 



Now it may be shown that 



D n 

 Jo = 



a {w 2 + (ma + nb)*} {u* + (ma - nb} 



m 



" 



b {u*+(ma + nbY} [u*+(ma-nb)*} ' 



Consequently B+D = = r , -. 7^2 ; 



ao u + (ma + no) 



and thus the integral sought is seen to be equal to 



ma + nb f h ' fudu, 



av ] h w 2 + (ma + nb)*' 



which was to be proved. 



Similar considerations apply in the case of more variables, 

 and doubtless by induction our general result might be established. 

 But the method we have followed, besides being more analytical, 

 is also very much simpler. 



