436 Mr. W. H. L. Russell on the Calculus of Symbols 



+ ... 



+ ... 



We are consequently able to see that the general term of 



IS 



where 



p + q-i-...=r, (1) 



p^ + q'+...=s, (2) 



and 



r + s —m. . , (3) 



Hence the general term of ~^'^~dx) "^'^ 



fjn—m, 



n a r q' p' r^q p x x ^ya+/3+ . . . -r . . . -s 



Tinder the conditions (1), (2), (3), and also 



a + ... +a + /3+ . . . =7^. 



Calling the expanded form of * easily seen that we 



can resolve all linear partial differential equations of the form 



The same property is possessed by a great number of other symbols. 

 Let us examine the condition that 



{aco + + c) £ - {dx +Vy + c') 



and 



A^^ + 2 B^y + + 2Ea? + 2 Fy + H 

 may combine according to the algebraical law. The condition is easily 

 seen to be 



(Ax + By + E) (ax + 5y + c) — (Cy + + F) (a'x -\-b'y+c')= 0, 

 from whence we have 



Aa -Ba'=0, B6-C6'=0, 

 Ab +Ba =Ca^+B5', 

 Be +Eb =Cc' + F5', 

 Ea +Ac =F«' + Bc', 



and 



Ec =Fc'. 



