ON MULTIPLE INTEGRALS. 221 



Hence it will appear that F is equal to a fraction whose nume- 

 rator is unity, and denominator equal to the value assumed by 

 the product of the remaining r-s factors, when the values 

 already assigned for the quantities a, &c. have been substituted 

 for them ; a result which it is obvious can be immediately ex- 

 pressed in the notation of determinants. F will therefore become 

 infinite if our equating the s factors by which it is divided 

 in (2') to zero will make one or more of the remaining r s 

 factors vanish. Let it make t of these factors vanish; then 

 equating these t factors also to zero, we get in all s + 1 equations, 

 which are equivalent to s independent ones. Therefore any 

 set of s out of these s + t equations will satisfy the remaining t 

 equations. Hence 



1.2 ...I 



of the quantities F will become infinite, and therefore the second 

 side of (2') will consist of finite terms and of a finite quantity 

 expressed in the form of the sum of that number of infinite 

 terms. This indetermination of course indicates a change in the 

 form of the function, the general character of which the reader 

 will have little difficulty in perceiving. But the consideration of 

 these particular cases, some of which are interesting, must be de- 

 ferred to another occasion. 



I am inclined to believe that the process developed in this 

 paper will admit both of simplification and extension. For 

 the exponential function we may substitute with certain modifi- 

 cations any function of CKB + ...CZ, in accordance with a result 

 given by Mr Boole in his very interesting Memoir on a new 

 Method in Analysis, which is published in the Transactions of 

 the Royal Society. (This result would include the one which I 

 obtained in the last volume of the Journal, from which however 

 it might be deduced.) 



Thus, if in the theorem established in this paper we replace 

 o, 5, ... c by lea, Tcb ... kc, Jc being a wholly arbitrary quantity, 

 we may, comparing the coefficients of its powers, deduce new 

 theorems from the given one. Developing the first side of the 

 equation, the coefficient of Jc n will be 



