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

 ' dx 1 



2nabck 



J? 



c 2 +x ^(a* + x) (b*+x) (c 2 + «*•)' 

 We have therefore in the general case 



dV rrr(g—x)dxdydz n p r dx 



~ (a 2 +.r)A 



^v „ , r dx 



% a £ Y h J - 



^v £ i r dx 



d & ' */ (<r + X) A 



If we multiply the first of these by dg, the second by dh, and 

 the third by dk, and add them, we may certainly make the 

 integral of the result 



(6.) 



V = 



7ra 



V L « 2 + ^ 6 2 + .r c 2 + *J A' 



whether g, h, and & be implicitly contained in (6.) or not, if 

 we suppose A a function of g, h, k. Let us take the partial 

 differentials of it relative to g, h and k, and equal them to 

 their values given in (6.), we shall have, leaving out quanti- 

 ties which destroy one another, 



d pxdx _ rd jY g 2 h* F \1~)^ 2 



rfgv/ A ~Vdc 2 {\ a 2 + * 6 2 + # c*+x/Ajdg 

 rd f( g 2 £ 2 £ 2 \l\rfc a 



T/d* lA a 2 + .r i 2 +^ c 2 +.r/AJrf£ 



. l {£ A 2 F \^c 2 _ 

 ~~ ^ W + & 8 + c 2 V rfg ~ 

 In like manner, 



d rxdx _ d r xdx — n 



dhJ ~A~ ~ °' dA./ "A~ - °* 



— -r— is a quantity independent of g, h, k, and 



also of c. It must be remembered in this process that c 2 =c 2 

 + a 2 — y 2 , 6 2 ■= c 2 + £ 2 — y 2 . But we cannot determine A by 

 making g, h, k and c infinite, for then both sides of the above 

 equation will vanish independently of any particular value of 

 this quantity. Nor is it easy to see how we can determine it 

 from hence. But we see immediately from (6.) that 



p p[ a* I? c 2 \ dx 



***V Xa^Tx + WTx + c^TxJ "A 



dx 



3 



