The Interchange of Two Differential Resolvents. 79 



On the Interchange of Two Differential Resolvents. 

 By the Rev. Robert Harley, M.A., F.R.S., Cor- 

 responding Member. 



{Received December 22nd, 1891.) 



1. The doctrine of differential resolvents rests on the 

 following theorem, discovered by Sir James Cockle, 1 viz.: — 

 That from any rational and entire algebraic equation of the 

 degree n, whereof the coefficients are functions of a single 

 parameter, we can derive a linear differential equation of 

 the degree n— 1, which is satisfied by the roots of the 

 algebraic equation. The derived equation is called with 

 eference to the algebraic equation, its "differential resol- 

 vent " ; and the two equations considered together are 

 sometimes called "co-resolvents." 2 The solution of the 

 algebraic equatibn gives the particular integrals of the 

 differential equation, and, on the other hand, the integration 

 of the differential equation gives the roots of the algebraic 

 equation. 



2. The above theorem, with its enunciation and proof 

 m was given more than thirty years ago, and, considering its 



importance in relation to both algebraic and differential 

 equations, it is curious that it has not yet found its way 

 ' into any of the ordinary text-books on these subjects. Its 

 value, however, has been recognised by some of our most 

 eminent analysts, and contributions to the general theory 



1 Cockle. "On Transcendental and Algebraic Solution." Philosophical 

 Magazine, Vol. XXI., May, 1861, pp. 379-383. 



2 Cockle. "Introductory Chapter on Co-resolvents." Quarterly Journal 

 of Mathematics, Vol. VI., 1863, pp. 9-20, 151-162, 226-229. 



