Equations of the First Order. 357 



(where/j, / 2 , . ./» are all arbitrary and unconnected), without inva- 

 lidating the argument of that section. There cannot, therefore, 

 be any definite connexion (of equality or otherwise) between the 

 different arbitrary functions or constants thus introduced. 



2. In what manner, then, must the constants be introduced so 

 as to admit of their being definitely connected? A partial 

 answer to this question may be found in every case in which an 

 equation of the nth degree is integrable without resolution. I 

 have already observed that equations of the nth. degree are not 

 in practice integrated by resolution and separate integration, but 

 by special methods of a totally different character, by some pro- 

 cess, in short, which connects the separate integrals in a manner 

 which can be neither mistaken nor avoided. This circumstance 

 is shown in a very clear and remarkable manner in the special 

 mode of solution which is employed for the integration of equa- 

 tions of the nth degree of the form 



y = xfp + cj>p, 



which includes Clairaut's form,' and also includes, in effect, 

 homogeneous equations. It would be idle to attempt to solve 

 such an equation as this by resolution with regard to p, and 

 then n separate searches after a complete primitive, the results 

 to be afterwards connected together in a fortuitous and unsys- 

 tematic manner. The solution in fact is effected quite other- 

 wise — by a process, that is, which enables us to find an equation 

 between p } x and an arbitrary constant. At this point, the 

 necessary connexion between the n factors of the result presents 

 itself without the possibility of mistake ; for since p has n values, 

 PvP& "Pru which are indistinguishable inter sc, it is manifest 

 that, whatever may be the form or character of the result in other 

 respects, it must be a symmetrical function of these n values of 

 p. The process which leads to this symmetrical result is simply 

 the differentiation of the original equation, which gives 



p-fp=(xfp + <l> ! p)-£, 



or 



dx _ xfp + <fip 

 dp~ p-fp ' 



a linear differential equation between x and p of the first order, 

 and therefore integrable in the form 



c = x¥p + <£>p. 



And since p has n values, which are the n roots of 



