336 Mr MURPHY'S THIRD MEMOIR ON THE A 



The term by which b is here multiplied, is the coefficient of m in 

 Qn-q-i,„^^), which is easily found from the expansions in Art. 13; hence, 



n{n-\) (« + ])(w + 2) r 1 1 I 1 1 Ux,. 



The general solution also fails when m is an integer, for then some 

 of the terms in the expansion of Q„ or g-.^+u will become infinite, and 

 the principle of vanishing fractions will simply enough in this case 

 also be applicable in determining the complete solution ; but if we put 

 for Q„, q„ their differential forms, the solution will never fail, for the 

 failures arise from the entrance of logarithms into the result, and these 

 will actually enter in the latter forms; changing our constants, the 

 complete solution for all cases is 



it is therefore necessary to shew that the functions by which the ar- 

 bitrary constants are multiplied, are particular solutions. 



Putting v-itty-"^, then -t- =(» + »») (1-2^) («')"+""-', 

 and -^ =(m + »?) {n + m - 1) (1 - 2tf («')"+'"-^- 2 (?< +«?) {tt'f *''-'. 



Hence tt' .-^ -{n + m—\){l-2f) -r.-\-2{n-\-m) .v-0, 

 and by successive differentiations the following equations arise: 

 («■').^-(« + '«-2)(l-20.^+2(2« + 2«^-l)^=0, 



(«')f^-(» + ^«-3)(l-20.^+2(3« + 3»e-3)g=0. 



