On an Impromptu Demonstration of Mr. Cay ley. 199 



has been said above regarding circular oscillations may be ap- 

 plied to elliptical ones, and the resistance of the air may be 

 neglected ; but in practice it might be necessary to take both 

 this and the amplitude of the vibrations into account. 



After the foregoing remarks, it is to be hoped that the ability 

 possessed by certain substances, particularly those operated on 

 by electro-magnetism, to cause the plane of polarization to rotate, 

 and to transmit the right- and left-handed aether vibrations with 

 unequal velocities, will find a more complete explanation. 



XXXII. On Mr. Cayley^s Impromptu Demonstration of the Rule 

 for determining at sight the Degree of any symmetrical Func- 

 tion of the Roots of an Equation expressed in terms of the 

 Coefficients. By J. J. Sylvester, F.R.S.^ 



FOR a considerable time past, among the few cultivators of 

 the higher algebra, a proposition relative to the theory of 

 the symmetrical functions of the roots of an equation has been 

 in private circulation, which, to say nothing of the important 

 applications of which it has been found susceptible to the calculus 

 of forms, merits (by reason of its extreme simplicity), although, 

 strange to say, it has, I believe, not yet obtained, a place in 

 elementary treatises on algebra. The proposition alluded to I 

 have reason to think first came to be observed in connexion with 

 my well-known formulae for Sturm^s auxiliary functions in terms 

 of the roots given in this Magazine. The theorem is briefly as 

 follows. If a, b, c, &c. be the roots of an equation 



x^'+p^x''-^ +p^ . x""-^ -j- &c. = 0, 

 any symmetric function such as 2a" .h^ ,0^ . . . , where a, /5, 7 . . . 

 are positive integers arranged according to the order of their 

 magnitudes in a descending (or, to speak more strictly, non- 

 ascending) order, when expressed as a function of the coefficients, 

 will be made up of terms of the form p^^-p^^p^^ • • -i? , such 

 that ^1 + ^2 + ^3+ ••• +^A will be equal to (a) for some terms, 

 but will for no term exceed (a) ; a being, as above described, 

 that one of the indices a, y5, 7 . . . which is not less than any of 

 the others, 



I had prepared, and indeed despatched, a somewhat elaborate 

 proof of this theorem for the Cambridge and Dubhn Mathema- 

 tical Journal ; but on proceeding to explain my method to Mr. 

 Cayley, elicited from that sagacious analyst the following excel- 

 lent impromptu, which I think too valuable to be lost ; and as 

 it is now a twelvemonth or two since our conversation on the 



* Communicated by the Author. 



