DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 47 



In a similar manner, by writing 



it can be shown that 



3 = X 



is an intermediary of the third of the equations. When the conditions of coexistence 

 of these two are assigned, they determine the arbitrary functions t/> "nd x m tlie 

 forms 



so that we have 



Z - TO = ( ; y + 2) (</ n - 2</ 13 



I - n = (y + z) (/ u - 2/ ia +/) + /; -/ 2 - . 9l + c,, 



and the primitive can be obtained by the customary process, leading to the form 

 v = 2 



where ./'=./' (sc + y, s) and </ = r/ (, + ~', Z/) are arbitrary functions. 



26. If one or both of the equations compatible with the original equation were of 

 the third order, we should then seek an equation of the second order involving 

 one arbitrary function more than that equation of the third order ; and \ve should 

 proceed in a manner similar to that of the preceding plan, the conditions of coexist- 

 ence of the different equations furnishing the means of identification or comparison 

 of the arbitrary functions that occur. 



If there be no equation of the third order, we should similarly proceed to obtain 

 possible equations of the fourth order, if any; and so on with the orders in succession. 

 The method is one of general application if equations of any order compatible with 

 the original equation exist. 



SECTION III. 

 Equations having an irresoluUe characteristic invariant. 



'27. The investigations contained in the preceding sections of this paper have 

 referred for the most part to those equations 



F (a, b, c,f, a, h, I, w, n, v, x, y, z) = 0, 

 whose characteristic invariant 



o aF , SF , ,?F 3F 8F 3F 



r-& +MV; + (r % -**j - q y + a = 



is resoluble into equations that are linear in p and q. Those contained in the present 

 section refer to equations whose characteristic invariant is irresoluble. 



