Rev. B. Bronwin on certain Definite Multiple Integrals. 577 

 J Lp + q p—q. 



'■ — n\ sin u du 



p + q p-qj V{p + q)(p-q) 

 b c 4 g 2 sin 2 u + a 4 k 2 cos 2 u 



a c 2 sin 2 u + a 2 cos 2 u 



. a c 4 ^ 2 sin 2 u + b 4 k 2 cos 2 «"l sin « rfw 



"r 



os 2 «1 sin w du 

 b c 2 sin 2 w + W- cos 2 # J A ' 



where A = (c 2 sin 2 m + a 2 cos 2 w)* (c 2 sin 2 « + b 2 cos 2 «)*• 



If we transform this by making sin 2 u = -g , and to 



abridge A = ^/(a^ + x) (£ 2 + #)(c" 2 + .r), we find 



tt / /'/g 2 , ^ 2 , 1 a 2 k 2 -c 2 g 2 



J \_a l b l c 2 a z -\-x 



1 Z» 2 F-c 2 ^ 2 \</^ 



+ c 2 6 2 + a; J A ' 



the integral to be taken from x — to x = oo . In this value 

 of U, v has been taken in the plane of x and y. But if we 

 make a and c, g and £, and then b and c, A and k change places, 

 we shall have two other values of this quantity. Adding the 

 three values together, dividing the sum by 3, and multiplying 



fit O *y 



by — ~. we find in virtue of (d.), 

 * abc v " 



Jo \ a *-£ l b *~ h * ~c*-k 2 \ 



As this expression is complicated, we will find V by an- 

 other method. From the formulae already given, we easily 

 perceive that 



rrr{k—z)dxdydz rrTi^ . , , 



III „ 3 — = I I I dtxsmu cos u du dv 



= / / Rsmucosududv= 2 1 1 — sin u cos u du dv 



_4>k S* /*sinu cos 2 ududv _^itk /* sin u cos 2 u du 

 c 2 JJ p + qcosll) c 2 J Vp 2 — q 2 



. 7 7 /* 2 cos 2 u sin u du 



=4w ab k I 

 *S o 



.(5.) 



V (c 2 sin 2 w + a 2 cos 2 w)(c 2 sin 2 ?^ + 6 2 cos 2 w) „ 



And if we transform this by making sin 2 w = — , it will 



J * Vc 2 + x 



become 



Phil. Mag. S. 3. Vol. 28. No. 188. May 1846. 2 D 



