Partial Differential Equation of the Second Order. 229 



derived from an integral equation such as to give rise to the 

 following values of p and q, viz. : — 



p = rn i f r V-f^.V_ l + (m i C + H . + S f p )z, 

 5= /,.U + G», 



where /„, f p , and C are determined as above indicated. 



The foregoing result depends upon the assumption that f p 

 and A do not contain U_ 2 . Let us now assume, on the contrary, 

 that these functions contain U_ 2 * 



If we had 



we should have 



/' ft (y)=A(y)+A(U_ 2 )u_ 1 |; 



and the terms in these derivatives of /g which involve U_ 1 

 would give rise to a term in (13 1 ) of form 



-AGU.OU^g-g), 



which must vanish, — thus implying either that/' 6 (U_ 2 ) =0, i. e. 

 that/, does not contain U_ 2 , or else that 



du du 

 dy dx 



%x~ -jz.—^i 



which last is impossible unless m, 2 = m X) a particular case of 

 which we may at present omit the consideration. 

 But, if A involves U_ 2 , assume 



A=/8(^U_ 2 ), 

 whence we get 



AW =/» +i 0* + !/•„>- {A, W-A(u_ 2 ) ^j . u_ v 



A(y)=AW+|(^+^>-{A.(2')-A(u_ 2 )J}.u_ i , 

 A(*)= p+ s Aj 



and substituting these in (13% we shall get, equating to zero the 

 coefficient of U_ 1 , in the result, 



o=-(^J-^)A(u_ 2 )+ ma /V2/)-A 1 W+(«+/-+4)4, 



