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

 The second and fourth of these, (6 U ) and (6 iv ), give us 



1 ax 



Q d{u) d{u) T d\u) 



doc dy l dy 



therefore if m l9 m 2 be the roots of the equation 



we shall have 



ax x dy 



0=V(V)-mf'(V). 

 Integrating the last, we get, since m x does not contain U, 



~F(xyzV)=mJ(xyzV) +f a (xyz), 



where f a does not contain U or higher derivatives of <fi, but may 

 contain lower derivatives of 0. 

 Hence we have 



^x = mJx J r-^T-f J rU, 



V p , ^( m l) ^ , Jf 



Vy :=m J»+ -^ -f+fav, 

 where 



/-=/.' w +/» •/=/» + K/+/J -/.'W. 



Substituting these values, (6 1 ) and (6 i,:i ) become 



«£/' 



+ T./.+T 



1' 



o=«y,-/.+4+-^-/. 



(7) 



«- T, 



Dividing by R p putting — ?n l + m 2 for ^, and m^g for =r±, 



the first of these becomes, taking account of the value of , niify—f x 

 given by the second, 



v 



Suppose, now, that ^ is of the form ap -r f3q -\- yz + S, and 

 R, S, T, V, «, /3, 7, 8 are functions of a? and y only ; and the last 



