Partial Differential Equation of the Second Order, 231 



Hence, when the equation 



= Rr + S* + T* + R(«p + £gr + y*) . . . (24) 



admits of being derived from a single integral involvingone or more 

 arbitrary functions, and in which the number of the derivatives 

 of one of those functions is not infinite, then, except in certain 

 particular cases presently to be adverted to, in which the foregoing 

 method requires modification, (24) will either be derivable from 

 a single partial differential equation of the first order, in order 

 to which at least one equation of condition must be satisfied by 

 R>, S, &c, or else p and q must be capable of assuming the form 



p = m 1 f+f a) q=f 

 where 



/ a =^ + ^ + ^</>" + &c. + « OT _ 1 ^- 1 ); 



a, a v # 2 . . ., a m ^i, C,fp being functions of xy only, m x being one 

 of the roots of 



= Rm 2 +Sm-fT, 



and the u of $>{u) is determined by 



_ du du 



= -j m, -j-i 



ax ay 



— these results being dependent, however, upon the fact of the coeffi- 

 cients R, S, fyc. satisfying as many equations of condition as there 

 are derivatives of <£ occurring in f and f a . 



We have/p determined by the equation (14a) ; and the re- 

 maining indeterminate quantities C, a v a 2 , &c. are given by the 

 foregoing theory, although practically, their values, as well as the 

 forms of the equations of condition, will probably be most satis- 

 factorily obtained by assuming, as it is evident we may do, 

 Az=${u) + A^'W +A 2 (/>» +kc. + A m _ 1 ^ m - 1 \u), 



in which case it immediately follows, from comparison of the 

 value of p obtained from this equation with that above given, 

 that we shall have 



C - _ I — 

 A dy 



du . dAi 



du . dk q 



dy 



a du dk 3 



2 dy dy 

 &c. &c. 



