222 GENERAL THEOREMS, &c. 



and in the second it will be the sum of a series of terms of the 

 form 



Jf 



n 8 y + ...) (ax 



Tfl 



as it is manifest that F will become j . Hence, if 



A/T (ax + ... cz) 



be such a function that its development may be substituted for it 

 in the integrations, we shall have 



f dx...f dz<t> l (m 1 x+...p 1 z) . . . < g (m& + . . . p$ ty (ax + . . . cz) 

 = 2Ff dxf dy...<l> l (m 1 x + n l y+...) ... <f> 8 (m 8 x + n s y + ...) 



^(ax + fy + ...), 



Jr-i 



where -j-^ -^r^ = tyt and all the differential coefficients of ^ of 



an order lower than the (r s) th vanish for t = 0. This is, I be- 

 lieve, in the case of 5 equal to unity, precisely equivalent to one 

 of Mr Boole's results. It might also, I imagine, be obtained 

 without having recourse to developments. 



