358 Dr. C. J. Hargreave on Differential 



the final result must be 



Qc-{xYp l + ^p x )')^c-(xYp q + ^p q )')...Qc-{xYp n + ^p n )')=0 } 



an equation of the nth degree in c, the coefficient of each power 

 of c being a symmetrical function of the n values of p. This 

 result contains only one element of arbitrariness ; whereas if we 

 had proceeded by resolution and separate integration, we should 

 have introduced n unconnected constants or arbitrary functions. 

 3. These two kinds of solution — the solution at large, and the 

 connected or systematic solution, as they may be called — may 

 be illustrated by any example which admits of being dealt with 

 both ways. Take, for example, the equation of the second 

 degree, 



p*—p- + — =0. 



r X X 



Its connected solution by a special method (Clairaut's) is 



x x 



This form, though not unique, admits of one element of varia- 

 tion only. We may write for c any function of c, and so obtain 

 an infinite series of forms such as 



vzr x J x z 



(1 l\ - m . x ■ A ^ 

 x m/ x m 



and so on. 



If we proceed by way of resolution, we obtain 



The integration of one of these will give some form of 



and that of the other will give some form of 



but the two equations and their resulting forms are entirely 

 unconnected, and the solution so obtained necessarily involves 

 two arbitrary elements. 



4. My object in this paper is to endeavour to solve generally 

 equations of the second and higher degrees by some systematic 

 process, so as to discover their connected primitive, — the prac- 



