232 THE REV. ROBERT HARLEY ON A CERTAIN CLASS 



involving a!\ 36", 30"^ d)\ is equal to 



A+ . «" 3^" 3c^' ^" 



. a' 36' 3c' </' 



. a 36 3c G? 



I . a lb c 



X a lb c . 



XVI. — On a Certain Class of Linear Differential Equations. 

 By the Rev. Robert Harley, F.R.A.S., Corresponding 

 Member of the Literary and Philosophical Society of 

 Manchester. 



Eead November 4, 1862. 



In the Philosophical Magazine for May of last year Mr. 

 Cockle showed, in a paper entitled " On Transcendental 

 and Algebraic Solution/^ that from any algebraic equation 

 of the degree n, whereof the coefficients are functions of a 

 variable, there may be derived a linear differential equa- 

 tion of the order n—i, which will be satisfied by any 

 one of the roots of the given algebraic equation. The 

 connexion of this theorem with a certain general process 

 for the solution of algebraic equations, led me to consider 

 its application to the form 



y^—ny+ (/j— i)<^=o. 



(I) 



to which it is known that any equation of the nth degree, 



when n is not greater than 5, can, by the aid of equations 



of inferior degrees, be reduced. 



In the course of my investigations I was conducted to 



the conclusion that for all integral values of n between the 



limits 



?^=2, ?i = 5. 



