particular Differential Eguatiofi. 4)17 



and independent, and the remaining coefficients formed from 

 these according to a general law of successive derivation. It 

 is thus seen that the performance of the direct operations 

 T;h+ iT,,, +2 • •"■„,+,. has caused the virtual disappearance of the 

 superfluous arbitrary constants introduced by the performance 

 of the operations which are inverse. 



Let us now consider the solution given in the Mathematical 

 Journal, viz. 



11 =7r , , 7r , „ . . TT , 7r~ ' 7r~ ' 0. 



tn + l m + 2 m + r n m + r 



Effecting, as before, the requisite reductions, we successively get 



where w is the complete integral of the differential equation 



(at^ ^ \dw , ,, ,, 



— + ht + c\-jy — {m + r—7i){at-\-b)w = CQ. 



Integrating this equation in a series, we have 



i«=Ao + Ai^ + A./ + &c., . . . . (11.) 

 where Aq is a new arbitrary constant, and the remaining co- 

 efficients are determined by the law (8.), provided that we 

 regard C/j_, as vanishing ibr every value of jo except p=l. 

 Here, then, the coefficients Aq A, must be considered arbi- 

 trary, and the remaining coefficients as formed from these in 

 .subjection to the law (9.). Hence 



((I .o ,, \~"'r* ^(r+l^ . r(?-+2y 



= {-t~ + U,-c) |a,. -i^) +A,...-^j.^-- 



+ A,+, r(3) •••/ 



The two first coefficients of the bracketed series being 

 distinct functions of the arbitrary and independent constants 

 Aq a,, are themselves arbitrary and independent, and the 

 remaining coefficients are derived from these according to the 

 same continuous law as are the coefficients of the former solu- 

 tion. The two solutions therefore agree; and the rejection of 

 the inverse factors tt"^ j._, 7r~^^._2 . .tt"' in the symbolical so- 

 lution (4.) does not at all aflect the final issue. 



After tliis it is scarcely necessary to observe, that the solu- 

 tion (3.) is also general and not particular ; and that while the 

 arbitrary constants introduced by the inverse factors 7r,7''^,^+,. 



Phil. Man. S. 3. Vol. 32. No. 21 7. June 1818. 2 E 



\ (12.) 



