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XIX. On the Differential Equations which determine the form of the Roots of Algebraic 
Equations. 
By George Boole, F.R.S . , Professor of Mathematics in Queen's College , Cork. 
Received April 27, — Read May 26, 1864. 
1. Mr. Harley* has shown that any root of the equation 
y n — ny + (n — 1 )x = 0 
satisfies the differential equation 
y- 
3n — 2 
n 
D- 
-ra+ L 
D(l)-l) . . (D— »+l) 
e (n ~ 1)9 y=0, . 
( 1 ) 
in which e e =oc, and D=~, provided that n be a positive integer greater than 2. This 
result, demonstrated for particular values of n, and raised by induction into a general 
theorem, was subsequently established rigorously by Mr. Cayley by means of Lagrange’s 
theorem. 
For the case of n— 2, the differential equation was found by Mr. Harley to be 
y-^*y=W- ( 2 ) 
Solving these differential equations for the particular cases of n— 2 and n= 3, 
Mr. Harley arrived at the actual expression of the roots of the given algebraic equation 
for these cases. That all algebraic equations up to the fifth degree can be reduced to 
the above trinomial form, is well known. 
A solution of (1) by means of definite triple integrals in the case of n=4 has been 
published by Mr. W. H. L. Bussell ; and I am informed that a general solution of the 
equation by means of a definite single integral has been obtained by the same analyst. 
While the subject seems to be more important with relation to differential than with 
reference to algebraic equations, the connexion into which the two subjects are brought 
must itself be considered as a very interesting fact. As respects the former of these 
subjects, it may be observed that it is a matter of quite fundamental importance to 
ascertain for what forms of the function <p (D), equations of the type 
u-\-<p(D)e n6 u=0 (3) 
admit of finite solution. We possess theorems which enable us to deduce from each 
known integrable form an infinite number of others. Yet there is every reason to think 
* Proceedings of the Literary and Philosophical Society of Manchester, No. 12, Session 1861-62. 
5 E 2 
