134 ON THE SOLUTION OF 



If m 1, (5) and (7) become respectively 



tyx - x^rx + ^r^'x = .................. (5'). 



frx = ax-^a 2 ............................ (7'); 



in this case, (8) admits of a variety of simple solutions. Thus 

 we shall have 



&c. = &c. 

 as singular solutions of (5'). 



The preceding remarks are sufficient to indicate the exist- 

 ence of a class of functional equations, to which a considerable 

 portion of the theory of singular solutions may be applied. 

 They appear therefore to possess some interest with reference 

 to this theory, independently of the method they suggest for 

 the solution of such equations. 



In fact the theory can hardly be considered complete, unless 

 some notice is taken of the equations of which we have been 

 speaking. They have been excluded from it, because the func- 

 tion /, which they involve, is not, as in the ordinary case, a 

 known function. But this, it has been already remarked, is not 

 an essential distinction. 



On the other hand, the method by which the singular 

 solution is in the common theory deduced from the complete 

 integral, does not apply to the cases now considered. It appears 

 unnecessary to point out the reason of this difference. 



With regard to the class of differential equations, which, 

 like Clairaut's, separate into factors on differentiation, we may 

 refer to Lagrange's Legons sur le Calcul des Fonctions, 1. 16 me . 

 He there shows that if a differential equation of the first order 

 can be put into the form M=fN, where M and JVare the values 

 of a and b deduced from 





then, when differentiated, it will resolve itself into two factors, 

 one of which leads to the singular solution, and the other to 



