1864.] 



Prof. Boole on Differential Equations. 



345 



IV. "On the Differential Equations wliicli determine the form of 

 the Roots of Algebraic Equations.-'^ By George Boole, E.R.S., 

 Professor of Mathematics in Queen^s College, Cork. Received 

 April 27, 1864. 



(Abstract.) 



Mr. Harley"^ has recently shown that any root of the equation 

 y''^ — xy+ {n~ 1) a?=0 

 satisfies the differential equation 



y D(U-l) .. (D-n+l) 2/=0.---(l) 



in which e^=x, and D = ^, provided that n be a positive integer greater 



than 2. This result, demonstrated for particular values of 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 ?2 = 2, the differential equation was found by Mr. Harley 

 to be 



T\ 3 Q 



y-^e'y^i (2) 



Solving these differential equations for the particular cases of n=2 and 

 7z = 3, Mr. Harley arrived at the actual expressions 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 w=4 

 has been published by Mr. W. H. L. Russell ; 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 diffe- 

 rential 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 (D), equations of the type 



« + 0(D)e'% = O (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 that the number of really primary forms — of forms 

 the knowledge of which, in combination with such known theorems, would 

 enable us to solve all equations of the above type that are finitely solvable — 



* Memoirs of the Literary and Philosophical Society of Mancliester. 



u 2 



