Partial Differential Operators. 53 



This theorem, it should be observed, remains true when P, 

 remaining a linear quantic in 8 M , 8y, 8 Z , . . . is any function what- 

 ever of x, y, z, . . . 



Let us agree to employ (i), (j) as umbrce, such that (i) n , (j) n 

 shall denote the factorial quantities 



i{i-i)...(i-n + l); j{j-l)...{j-n+\) 



for all values of n ; then we may express the above theorem 

 under the subjoined condensed form, which will be useful for 

 the better understanding of the sequel, 



(i)U)Y*V 



Y l * P>* = \e p 3 . F+'] * . 



Suppose now that we wish to obtain the product of three 

 factors, 



(p*)«, (P*y, (P*)*. 



Call e A i,2+ A i,3+ A 2 3 3, for the sake of brevity, E. The first term 

 in the expansion of E is unity. The second is Ai, 2 + Ai f 3 + A a> 3 , 

 which, applied to P*. P*. P*, gives 



0" (•— 1) + >"C;— 1)+ k (*— i))Pf+' + *- a . "(P*P)- 



The third term is 



*(AJ f> + AJ f , + A« 8 +2A lfl J A lf s+-2A lf ,. A 2 ,3 + 2A 1>2 A 3 , 3 >; 



the effect of the application of the first three quantities within 

 the above parentheses is to introduce terms whose sum is 



^((^-^^(/-^^(F-^^P^+^-^P^P) 2 ; 



the effect of the fourth and fifth quantities is to introduce terms 

 whose sum is 



and the effect of the sixth term is to introduce the terms 

 (t*-f)(fc«-A)P«-W+*-« (P*) 2 

 + i/l.P i+ '* + *- 3 .P*P*P; 

 giving altogether for the complete sum 



*O0 Id + «(*) + to (*)) 2 P i+ ' + *-'P 2 2 



+ y*F+>+*- 8 .P 8 + ...}*. 



And in general the effect of the term (Ai, 2 + A 2>3 + A 1(3 ) r in 

 the expansion of E will be to introduce terms containing all the 



