DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 35 



with 



= Ap + (fcH - 0) r/ - (iG - ty). 



This system of four equations must be rendered complete by constructing the 

 additional equations that arise out of the JACOBi-PoissoN conditions. If, when 

 complete, the system contains n equations, then it possesses 13 n functionally 

 independent solutions. Among these must be included (i) the original differential 

 equation 



F (a, b, c,f, g, k, I, m, n, v, x, y, z) = ; 



(ii) the two distinct integrals of 



dx^ _ dy dz 



~A ' ~ -iH -6 ~~ i-G - 6ty ' 

 say these are f, yj. 



Putting these on one side, there are thus 10 n new functionally independent 

 solutions. A not uncommon case is n = 9, when there is one new solution, say u. 

 Then we have 



u = i/, ( 17), 



where $ is an arbitrary functional form ; and this equation coexists with the original 

 equation 



F 0. 



16. Thus far we have considered only one of the two equations into A = is 

 resoluble. When we consider the other equation, viz.. 



Ap + (|H + 0) q - (1Q + 0$ = 0, 



the sole difference in the general analysis is manifestly a change in the sign of ; and 

 we therefore obtain the corresponding system of linear homogeneous partial differ- 

 ential equations, determining an integral combination (if any), by changing the sign 

 of 9 in the preceding system. The method of integration is the same as before. 



It may happen that neither of these two systems possesses a solution distinct from 

 the differential equation. If, however, either (or both) should possess such a solution, 

 then n must be less than 10, and certain conditions viz., those in order that the 

 system when complete should contain not more than nine equations must be satisfied. 

 These are the conditions in order that one equation or two equations, if the result 

 hold for both systems of the second order involving an arbitrary function of two 

 arguments should be associable with the given equation. 



And it should be noted that the characteristic invariant of an equation associable 

 with the given equation is satisfied by that linear equation in the characteristic 

 invariant of the given equation which is used to derive the new equation. The 

 result is general, and the proof of the general result is immediate. 



F 2 



