220 Mr. R. Moon on the Integration of the General Linear 



may admit of being derived from a single partial differential 

 equation of the first order, together with the mode of ascertain- 

 ing such equation in the cases in which it exists. 



I now propose to show the circumstances under which (1) is 

 derivable from a single integral equation involving one or more 

 arbitrary functions, as well as the form of the integral equation 

 when any such obtains. 



Let the integral equations involve the arbitrary functions (f>, ty, 

 &c. which appear in it under the forms <£, </>', (j>" . . . , -ty, ty',^ 1 . . . ; 

 and, solving with respect to <£, let it stand 



<$>(u) = f{zyz), (2) 



where u is a definite function of xyz , and where F may involve 

 cf>', cf>", . . . , ty, yjr f , yfr" . . . , but does not involve <£. 

 Differentiating (2), we get 



o=4Ku) | -F( ?/ ) + {y (u) J-n*)}?. 



(3) 



I shall first assume that the number of derivatives of ^> con- 

 tained in (1) is finite, and that ^ m ~ l) {u) is the highest derivative ; 

 in which case each of the equations (3) will contain ft m) {n) or 

 U* (unless, indeed, u contains x or y only, a case which I pass 

 by for the present), so that (3) may be replaced by 



(4) 



q=f{xyzV),f 

 whence we have 



|=F W+ F(,) + EHU).U 1 .(g + J P ) 



=FW + F(,)F + F(U).U 1 . g + Jp) ■ 



and putting F, for Ffc) + F(«) . F, and ^ for ^ + ~ F, 



and similarly with respect to H? y f x f y -M, Uj standing for 



ay 



(j>( >n +V(u), we shall have 



* If (2) contain no derivative of $, each of the equations (3) will contain 

 <$>' ; and putting U for <£' the ensuing reasoning will remain unaltered. 



