140 On Homogeneous Functions, and their Index Symbol. 



where, as before, 



d , d d _ 



n=a— + 4 — +C-J- + &c. 

 da do dc 



This is obvious, since, from the supposition made relative to 

 a, b, c, &c, we can operate with the symbol n under the inte- 

 gral signs. It will be observed that the result bears a strong 

 resemblance to Liouville's well-known extension of Dirichlet's 

 integral. 



We seem to have here made a step towards the solution of 

 that which has been long a difficulty in the treatment of multiple 

 definite integrals, namely, the generalization of those in which 

 the variables enter as complicated functions in the indices of known 

 quantities. The most valuable extensions yet obtained are those 

 in which the element of the primary multiple definite integral 

 exhibits the variables under finite forms solely. 



We shall conclude by giving the two particular instances of 

 the general theorem above alluded to. It can be easily proved 

 that 



CO QO CO 



fdxfdyfdz . a-* 2 .b-v\ c-* 2 = I ttS -, — \-. -, ; 



*s o Jo *s o 8 {log a. logo. log c}* 



hence 



dxl dyf ds.Y^ + yZ + z^a-^.b-s'.c-* 2 



1 » i 



= 5«*.F(-tl) 



{loga.log6.logc}*' 

 Again, we readily see that 



CO 00 CO 



/ dxl dyf dz .a~P* .b~™ ,c~ rz .x l ~ l .y™- 1 .z n ~ l 

 _ T{l)T{m)T{n) 1 



p'.q m .r n (log«)'.(logo) m .(logc) n, 

 and hence 



GO CO 



/ dxl dyf dz.<£>(px + qy + rz)a-P*.b-w.c- r *.x l - 1 y m - l z n - i 



_ r(Q.i».r(n )g ( n , i 



p l .q m .r n v y (logfl)'.(log6) m .(logc) n ' 



38 Trinity College, Dublin, 

 December 1851. 



