482 PROCEEDINGS OP THE AMERICAN ACADEMY. 



which are of the same degree, have the same roots (obviously distinct, 

 siuce no two particles could coincide in a position of equilibrium), so 

 that they differ only by a constant factor. Accordingly the polynomial 

 (fi, whose roots determine a position of equilibrium, satisfies a differential 

 equation of the form : 



dx' ax 



Substituting for y the polynomial ib, we find at once by comparing the 

 coefficients of a?* on both sides that (7— — 2ck. Hence: 



llie positions of equilibrium of a group of k unit particles repelling 

 one another inversely as the distance, and situated in the field of force 

 K ( — ex), are the roots of the polynomials of the ^th degree which satisfy 

 the differential equation 



(4) %y.^-2cx'i^ + 2cky = 0. 



The mechanical problem shows that this equation must have at least 

 one polynomial solution of the ^'th degree, a fiict which may be readily 

 verified by substituting in (4) a polynomial with undetermined coefficients. 

 This substitution shows that there is only one polynomial solution. 



The polynomials thus obtained are known as Herraite's Polynomials, 

 having been first discussed by this mathematician * from the following 

 point of view. Consider the function e~'^'^'. If we differentiate this 

 function k times with regard to x we obtain the original function multi- 

 plied by a certain polynomial in x. Among the properties of this func- 

 tion which Hermite develops is the fact that it satisfies the differential 

 equation (4). This shows its identity with our polynomial ^. The fact 

 that all the roots of ^ are real and distinct, which followed from our 

 mechanical problem, follows also at once from Hermite's defiuition 



If we make the transformation 



2 



2 = e~^ • y, 



the differential equation (4) takes the form 

 d'^z 



(O) — ,:=[C^X--C(2X-+ 1)]Z, 



* Comptes Rendus, 18G4, pp. 93 and 26G. 



