Partial Differential Equation of the Second Order. 221 



|=F, + F<(U).U,^ 



dq 



IT 



d(u) 

 dy y 



d(u) 



(5) 



S -/..+/'(D).U 1 . ir 



=/, + /'(U).U 1 



^M 



All the independent relations that can exist between the par- 

 tial differential coefficients -(-> -—> -£* — , and the variables x, ?/, 



dx dy dx dy ,J ' 



z } p, q, are embodied in the seven equations (2), (4), and (5) ; 

 and the whole of such relations which are free from <f) itself are 

 comprised in equations (4) and (5). Moreover, since p and q 

 have been eliminated from (5) by means of (4), the equations (5) 

 embody all the relations free from </> which can exist between 

 the same partial differential coefficients and the variables x 3 y } z. 

 But if we put Rj, S„ T v V 2 for the values assumed by R, S, 

 T, V when we put Y,fiorp, q respectively, we shall have, in the 



equations dp d d . 



dx dy 



(5a) 



two relations between the same partial differential coefficients 

 and x } y, z which do not contain <f>. 



Hence, if we substitute in these last the values of the partial 

 differential coefficients given by (5), we shall have in each case a 

 result which must hold identically, viz. 



= R 1 F/, + S 1 F y + T 1 F, + V - 



+ 



| Ki< g +Sirf |)_ F(U)+Ti |) / ,. (U) ^ IJi 



0=/-F, + {/'(U) 



d{u) 



dx 



F'(U) 



d(u) 

 dy 



}u,; 



whence, since U, does not enter into F or/, we have 

 = R I F I + S 1 F J + T 1 F.. + V, 



F'(U)+T^|V(U), 



dx dy 



0=/,-F„ 



0^(U)f-F' ( U)M 



(6) 



