ON MULTIPLE INTEGRALS. 219 



In order to simplify the expression 



v cos Saw + v sin Saw 



let us denote the s quantities Saw, &c., which are involved in it, 

 by the single symbols /9 t ... fa, and assume 



the sign of summation S extending only to that set of s out 

 of the r quantities x ... z, which corresponds to the factors in- 

 volved in the denominator 8. Of course a?, y, &c. are here, as 

 before, new variables. (In the case of s = 3, for instance, these 

 assumptions will be of the form 



It follows from this that du t ... dugd^... da. 8 will be re- 

 placed by dx ... dy . dfi^ ... d/3 a , and that the factors in S will 

 take the simpler form a 2 + /:?/, 2 + yS 2 2 , &c. ; while Saw will be- 

 come ^a? + @ 2 y + ... 



The integrations with respect to /5 extend, like those for a, 

 from o= to + oc . Let 



Then, from the obvious analogy between the forms v and v', and 

 those of the developments of cos 2/&c and sin S/ifo respectively, 

 it follows that if x, y, &c. are all positive, 



M'^ite-**-*"' (1 + 1 + ...) 



there being twice as many units within the brackets as there are 

 terms in the development of sin (f t + ,../,), or of cos (J[ + ...f t ), 

 that is to say, twice 2 8 " 1 or 2 8 . 



Moreover, if any one, as ic, of the quantities x, y, &c., is 

 negative, M' = ; and this, whether it alone is negative or any 

 others, are so too. For if x x f , let its coefficient fa be as- 

 sumed equal to fa, when the expression of M' becomes of the 

 same form as if x were positive, except that v and v are changed 



