DIFFERENTIAL EQUATIONS OF THE SECOND ORDER, 45 



being adequate for the purpose at any stage; the relations are, in fact, satisfied 

 conditions of compatibility. This method is, therefore, ineffective. 



An effective method can, however, be obtained as follows. Restricting ourselves 

 for the moment to the equation of the second order with two compatible equations 

 also of that order'" the restriction is made only to simplify the explanations we 

 have v as expressible in terms of two arbitrary functions. Hence each of the 

 quantities /, m, n (and therefore any combination of v, I, m, n), can be expressed 

 in terms of two arbitrary functions. Now in one of the compatible equations we 

 have one arbitrary function which is to be identified with one of the arbitrary 

 functions in v ; hence it is to be expected that a proper combination of v, I, m, n, 

 is an intermediary integral of that equation involving a new arbitrary function, which 

 must be identified with the other of the arbitrary functions in v. 



Similarly for the other of the compatible equations, there is an intermediary 

 integral involving the two arbitrary functions. The conditions of coexistence of the 

 two intermediary integrals must be assigned ; it will appear that, if the conditions 

 are not satisfied identically, they provide the means of identification of the various 

 arbitrary functions. 



It is to be observed tha.t the intermediary integral or integrals thus obtained 

 cannot be regarded as intermediary integrals of the original equation in the ordinary 

 sense of the phrase, for each of them involves two arbitrary functions. But they 

 are intermediary for the respective compatible equations : each of them involves one 

 arbitrary function more than occurs in the compatible equation. The result mani- 

 festly does not imply that the original equation possesses any intermediary integral ; 

 in fact, the assumption throughout our investigations has been that no proper 

 intermediary integral exists. 



25. A methodt has been given elsewhere for constructing the intermediary integral. 

 In effect, it amounts to the use of the conditions which must be satisfied in order 

 that the derivatives 



I- nu m + gn a 4- u x = 

 hui 4- bu a 4 fit,, + ?' y = 

 gut + /, 4- < 4 w- = 



from the supposed integral 



(Z, m, n, v, .r, y, z) = 



shall cause the compatible equation 



(a, . . . , /(., /, m, n, v, x, y, 2) = 

 to be satisfied without regard to the values of the differential coefficients of v of 



* The explanations will be seen to apply, mutatis mutandis, to other cases of the second order, and 

 indeed to cases of any order, when compatible equations are known. 

 f In the memoir cited in the introductory remarks). 



