186 On Differential Equations 



in the fii'st two of which y is to be taken to represent {po), 

 are co-resolvents. The first is the algebraical resolvent, the 

 second is the difterential resolvent, and the third is a func- 

 tional resolvent. If the variable x be not greater than 

 positive unity, or less than negative unity, the relation 



^{x) = 2 sm. I ^ - - (iv) 



is a solution, and the only common solution, of the above 

 system of co-resolvents, while, on the other hand, all the 

 roots of (i) are solutions of (ii). The theory of co-resolvents 

 not only throws light on those of algebraical and of differ- 

 ential equations, but it enables us to make the solution of 

 whole classes of functional equations depend upon that of 

 alo^ebraical or of differential equations. The well-known 

 application of the theory of algebraical equations to the 

 solution of linear differential equations with constant coeffi- 

 cients, the analogies between the theories of algebraical and 

 differential equations pointed out by Libri and Lioiiville 

 and the communication between the two theories established 

 by Abel, when he explored the track upon which Euler had 

 entered, may give interest to the new communication be- 

 tween those theories opened by the method of co-resolvents. 



22. Before proceeding to that development of the theory 

 which it is the object of this section to explain, I ought to 

 say that its present advanced state is, in no small measure, 

 attributable to Mr. Harley. That eminent mathematician, by 

 his earlier inquiries into the forms of the differential resolv- 

 ents of certain trinomial algebraical equations, obtained 

 results which not only excited attention at the time, but 

 which have also, to a great extent, determined the cuiTcnt 

 of subsequent research. Traces of Mi\ Harley 's investiga- 

 tions appear in almost every paper that has since been 

 published. His approach to the following theorem was 

 simultaneous with my own. 



23. If u represent the ?7ith power of any root of the 

 algebraical equation 



yn .j,yr_l^^() _ _ - (v) 



then It, considered as a function of x, satisfies the linear 

 differential equation 



\_Dfu-Y 



\7Yi + rD ' 



^ ^— + n-v a;« u = (vi) 



in J ^ ' 



