220 GENERAL THEOREMS 



by having /3/ wherever /^ occurred previously. Now none of 

 the quantities @ can occur raised to any power, and therefore 

 every term involving /3 t will change sign when /3 X is replaced by 

 8. Hence we shall have 



there being as many negative units as positive within the brackets, 

 since in the development of sin (f l + . . . f s ) or cos (f^ + . . .f s ) there 

 are 2 a ~ 2 terms independent of the sine of f^ and 2 S ~ 2 terms which 

 involve that quantity, and which therefore change sign when f t 

 does so. Hence the quantity within the bracket, and conse- 

 quently M t , is equal to zero if x be negative ; and so, of course, 

 for the other variables y ... z. 



M will, in particular cases analogous to those already noticed, 

 assume exceptional or limiting values, but of these we need not 

 take account. And thus we arrive at the following remarkable 

 theorem : 



The definite integral ofi variables x ... z 



/ O dx ... / O dz<k (m.x + ... p t z) ... < 8 (m s x+ ... 

 whose limits are given by s inequalities 



m 1 x+...p 1 z<h 1 , ...m s x+... p s z5 

 can generally be expressed as a linear function of 



- ' cz 



1.2...S 



integrals of s variables each. The form of each of these integrals 

 may be deduced from the original integral by omitting from it 

 any set of r s of the variables, and similarly the form of the 

 limiting inequalities may be got by omitting the same set of vari- 

 ables from the original inequalities (r > s). 



In certain cases, however, when the constants a, m, &c. have 

 particular values, the theorem fails because the assumption (2') 

 becomes illegitimate. This failure is indicated by certain of the 

 quantities F becoming infinite. To determine the form of F, we 

 have merely to multiply (2') by A, and then to equate to zero all 

 the s factors of which A is composed. All the quantities F, 

 except the particular one under consideration, will then dis- 

 appear, and we have s equations determining the s quantities a. 



