DIFFERENTIAL EQUATIONS OF THE n"' ORDER. 29 



Treating this identity in a similar way, we get 



S ^'^ ^i-~d^ ' = ^- ^)" \^-^f^ - 2Wr-y' (»0 + 4r^'/"(»^)] ■ • ■ (47). 



This process we may repeat as many times, as we please, and we 

 see, that the exponent of that «, which multiplies the function ^ in the left 

 member, determines the order of the highest derived function of /(m) , 

 that occurrs. 



Now, as the root m is repeated A times in f {m) = , we have 

 f(m)=f'(m) = f" (m) ... =/<*-^'(m) = 0, and, therefore, the right 

 members in (45) and the equations, that are derived from it by the (A — 1) 

 first differentiations with respect to m, will all vanish. These equations, 

 therefore, shew that the equation (43) will be identically satisfied by every 

 one of the expressions, that stand in the enunciation of the theorem. 



Corollary. As we have obtained A distinct solutions belonging to a 

 root repeated A times in (44), Ave can, also here, get n distinct solutions 

 and , therefore , by theorem II the general solution of (43) , even when (44) 

 has equal roots. That general solution Avill be c = a sum of n terras, 

 presenting for every simple root m,. a term (p,. {y — m,,v) and for every 

 root »n, repeated A times, a sum of terms of the form 



■^{y — »1er) -\- x-\y {y — m -v) -\- x-'^.;, (y — nix) -j- ... -\- x'' ^ 4a-i (y — mœ) . 



These theorems are , of course , valid , also when (44) has imaginary 

 roots, but then the sohrtions will involve functions of imaginary functions. 



We now proceed to set forth some corollaries to theorem I, as far 

 as it applies to the equation 



i=:n 



SC:.^W.= V= F(.v,y) (48), 



shewing how to obtain a particular solution of tliis equation. 



Corollary I. If T' = F (x) , u may be considered as a function 

 of X only; then the equation (48) becomes, when z =^ ti ^ 



C,v„,, = F(x), 



whence it follows, that we may put 



1 f 



u =^ p I F{x) dx" 



/n 

 denoting n successive integrations with respect to x. 



