Mr. G. Boole on a Method for Differential Equations, 7 



Now (1.) gives Tr,„pv = pT:,n-iV, 



whatever v may be; let p v = n, then v = p~^ii, and we have 



■7r„ji = pn„,.ip-^u; 

 so that the symbol 7r,„ is equivalent in operation to the com- 

 pound symbol p7r,„_if)~'; writing then 7r,„_,=^7r„j_2p~', we 



have T,„?< = p2 7r„j_2p-^; 



and finally, 7r„ji=p"''7rQp-'"'ii. 



Hence the equation 7r,„?i=0 gives 



p"'TrQp-'"u=0; 

 .-. M = p"'7r-ip-'«0 (6.) 



This equation is unquestionably true, whatever may be the 

 interpretation of the symbols 7r„j and p, provided that they 

 satisfy the combination law (1,). 



Now taking Mr. Bronwin's first equation, viz. 



X ( D^ -f /. -) u + 2 m D ?/ = 0, 



and making 7r,„ =x (D^ + k-) + 2mD, p = D^ + F, 



we have * 7rQ=x(D^ + k^), 



whence (6.) gives 



But {,r(D2 + F)}-i = (D2 + F)-i^->, 



therefore u= (DHO'""'-^'"' (D2 + F)-'«0. 



Without entering into any special examination of the above 

 result, and merely resting on the analogy of many similar 

 cases, I should at once assert that, when ?« is greater than 0, 

 we ought to retain two of the constants which arise from the 

 performance of the inverse operation {D--i-k'^)~"'0, and that 

 it is not necessary that we should retain more than two. Thus 

 one form of solution is unquestionably 



11= (D- W- Z:-)"' - ' a;~ ' {c cos k x + c' sin kx ) ; 



and there are, I believe, niany equivalent forms. The nature 

 of Mr. Bronwin's error consists in his virtually rejecting the 

 factors A'~' and cos k x + c' sin k x, and his result is accordingly 

 nugatory whenever m >■ 0. Similar remarks apply to the 

 otiier ecjuations in his paper, of which however 1 have only 

 examined a small number in detail. 



Perhaps, in connexion with this subject, it may not be quite 

 irrelevant to mention, that about two years since 1 obtained 

 the solution of the purely symbolical equation 



