30 M. Falk, On the Integration of partial 



Corollary II. If F = -Fl (y) , we may put v = ti , if C^z„_^r 



be the last term, that does not vanish in the left member of (48) and 

 u a function of y only. Substituting in (48), we easily obtain 



Cr Mo,,. = FiQ/) , 

 whence 



" = i ^.A'«''^ (^»)- 



Corollary III If F= F{.t) + F,{y) , the general solution of (48) 

 will be s = rj 4- f 4" ^ , ti and v being any particular solutions of the 

 equations 



§ C, «„.,,, ^ F{a:) (51), 



g C, v„_,, = I\{y) (52) 



and Z, the general solution of (43). For putting s = rt 4- ■y -f ^, we get 

 by means of (51) and (52) 



î=:l) 



which proves our assertion. According to the corollaries I and II we may, 

 therefore, use the m and v of (49) and (50). 



Corollary IV. If some terms be missing at the beginning and the 

 end of the left member of (48), this equation is immediately integrable 

 several times and will thus be reduced to a new equation of the form (48) 

 but of a lower order. 



If, for instance, the equation (48) be 



g C, .„_,,, = F(x,y) (53), 



Avhere r and q are integers < n and n — q > r . Integrating (53) Q 

 times with respect to x , we get without difficulty 



^C,z.„,_^,=jF(x,y)dx' -f §.re— <p,Cv)== J^iGt;,3/) say; 



