The Interchange of Two Differential Resolvents. 87 



constants, it should be observed that the sum of all the 

 roots of (a) is a constant, viz., o or 2 ; let us therefore write 

 for the moment, c in place of 2y. Using the last found 

 differential equation (a' ), and summing for the n roots of 

 (a), we have 



?i n -\J)f- 1 c - (n - 1)"" 1 ( 7 ^ " VZt) C *"~ 1 = ^^ + C;iC " '' 

 or, by reduction, 



-c(n~l ) n ~H — -= J = c x nx + c 2 n, 



a relation which must hold universally. Hence 4 = o, and 



c n 



n % -3n + \\ n - x ' 



c t 





When n is greater than 2, S^ = o, and therefore c = o, <r a = 0, 

 and the differential equation becomes 



„-.[D]-V - (» - D-{^i - ^T J Vl " - °' 



which agrees, as it ought to do, with (a). 



In the particular case n = 2, we have Sj = 2, and there- 

 fore <:=2 and Ci—l. Hence 



2(D)y - (2D - 3)ay = ff, 



the differential resolvent of 



y 1 - 2y + x = 0. 



II. Interchanges similar to the foregoing may some- 

 times be effected without any change of the variables x and 

 y. I give an example. 



If u represent the m* power of any root of the algebraic 

 equation 



ay" + by r + ex — . . . (A) 



where the co-efficients a> b, c, are independent of x, then 



