DIFFERENTIAL EQUATIONS OF THE SECOND ORDER, 43 



conditions. If, when complete, the system contains N equations, then it possesses 

 23 N functionally independent solutions. Among these are to be included 



(i) The original differential equation F = ; 



(ii) The three derivatives of F = with regard to x, y, z, respectively ; 



(iii) The two distinct integrals of 



d.v _ ily <7z 



A : = |H"-~0 = HI -. 00 ' 

 say these are f, t]. 



Putting these on one side, there are thus 17 N new functionally independent 

 solutions, so that N must be not greater than 1G in order that the method may be 

 effective. If, when N = 1 G, the solution is u, then 



U = * ( 7,), 



where i// is arbitrary, is an equation of the third order that can be associated with 

 the given equation. 



The same process, with corresponding results when the appropriate conditions are 

 satisfied, is adopted for the alternative linear equation 



arising out of the reducible characteristic invariant. 



23. An example in which no equation of the first order involving only one arbitrary 

 function, or no equation of the second order involving only one arbitrary function, 

 can be associated with a given equation of the second order, is furnished by 



The general primitive is 



,, = F + G + (y + z) [F, - F, + G! - G,} 



+ TV (U + *Y (Fn ~ 2F ie + F, c + G n - 2G 13 + G 22 }, 

 where 



F = -F(x+y,z), G = G(x+z,y), 



and the subscripts 1, 2, denote derivation with respect to the first and the second of 

 the arguments in the respective cases. The associable equations are of the third 

 order at lowest ; and they are 



- 3& + 3y - S = (y + z) * (x + z, y), 



o - 3a i + 3a 2 a s (u + z ) y ( x + y> 2 ) 



where $ and "^ are arbitrary. 



In a similar way in part, and by induction in part, it may be proved that the 



integral of 



, ,. . i 21 m n 



a-h-g+f+I - y + z = 0, 



G 2 



