particular Differential Equatiori. 259 



or 



./ i'l\ 

 p = ai\m — n — I. 



As the two values of j) now found are the same as those 

 found in the two first solutions, the two equations solved are 

 the same ; and the two sets of solutions are only particular 

 solutions, for the solutions themselves are evidently different. 



We now proceed to solve the equations (2.). Putting D 



for -j-^ and supposing (p, 9, and A functions o^ x, we make 



t: =(P D + S , 77 =(S D + 9 ; 

 then, if we use the accent for different coefficient, 



m n Tm~n ' \TrtVY n ' Tm'Jn ' Tn m' ' rm « ' "m n 



n m TmTn ' VrnT »« ' Tn rn ' Tm ni '^ Tn m n m 



Make 



"■,„ "■„ = "■„ ff + f(m)\, 



and we find, by substituting the above values. 



Integrating the first of these, we have (p^^=(p^ , for an arbitrary 

 constant would not add to the generality. The second thus 

 becomes 



From 7r,„X=Xw,„+i by substitution we find 



A^,„=0, A9,„=<p,„-. 



A 



Suppose that A does not contain m or w, and by integrating 

 these we have <p,„ =<p„ = a constant, or rather a function of a?, 

 without m or ?/, which we shall denote by <p ; and 



whcre/(^) is any function of :r whatever, provided it does not 

 contain 7)i or 7i. An(i/{m)k = (p(6i^^ — d' ) gives 



f{m)\= {?i-m)<pDy<p^y, 



a\ = fD[f-\ (3.) 



S2 



