438 Mr. W. H. L. Russell on the Calculus of Symbols, 



\ Ux^^ f> dx''-^^ b dx'-^^ f 



Whence it is obvious that the general expression for the expansion of 

 f d d \^ 



(X— 4-Y — ) will depend upon principles not materially differing from 

 \ dx dy I 



those already considered. 



The symbols we have already considered are only of the first order of 



differentiation ; we shall show that there exist symbols of the second order 



which combine with certain algebraical quantities as if they were themselves 



algebraic. 



Let us take the symbols ♦ 



d^ ^ ,d d 



dx-' dxdy dy- dx dy 



and 



Kx^ + 2B^z/ + C?/^ + iMx + 2B'?/ + H . 

 Proceeding as before, we arrive at the following conditions : 



A5+Bc=0, (1) 



B6 +Cc =0, (2) 



A'6+B'c = 0, (3) 



A«' + B5' = 0, (4) 



B«'+C6'=0, (o) 



Ka +B6 =0, . (6) 



Ba +C6 =0, (7) 



A'a + B'<^ = 0, (8) 



2a A + 45B + 2cC + 4a'A' + 45'B' + e= (9) 



Whence we have, putting B= 1, 



and with the following conditions, 



ac=h\ a'c-bb'=0 



4«'A' + 46'B' + e=0; 

 the condition a'c—bb^ maybe otherwise written a'b — ab'=0, in consequence 

 of the equation ac=-P. 



It will be observed that several of the nine above equations are not inde- 

 pendent of the rest ; so that the result is much simplified. 



