DIFFERENTIAL EQUATIONS OF THE n"" ORDER. 25 



whence by integration 



i,-ßY ^ Aœ [f{y)-\-y]. 



Another complete solution will be obtained by connecting- directly 



Xi = a with the given equation, whence ^19 = ^,^01= • ' '--^ ; and, sub- 



a, 



stituting these values, as before, in the equation dz = z-^^ dx -\- z^ -y dy , 



we get the new complete solution 



z = ax -^ -^ f(y) + ^ . 



We conclude this paragraphe by resuming the formulje to be used 

 in the integration of linear partial ditferential equations of the n'* order. 



Notation: .„. „ = ^;^„ . 



m . n 



The differential equation being 



8^^--.,.= ^. (12), 



form the equation 



g(— iy^m"-'=0 (20) 



1=0 



a7id solve it. For every distinct root m of this equation, calcidate the 

 corresponding system of W's from the equations 



W,= U,, W,+ W,_, m=U, (i = l, 2, ... , ^n-1)) . . . (22) 



and substitute these W's together with the same root m in the equations 



dy — mdx =: (21), 



i=n-l 



S n\dz„_,_,^ = Vdx (23). 



1=0 



Noiv, regarding (21) and (23) as simultaneous, seek their system of 

 solution in the form <p = a, , -^ = ß ; then the corresponding first inte- 

 Nova Acta Eeg. Soc. Sc. Ups. Ser. III. 4 



