324 



also some remarks as to the mode of calculation of the new tables, 

 and also as to a peculiar symmetry of the numbers in the tables of 

 each set, a symmetry which, so far as I am aware, has not hitherto 

 been observed, and the existence of which appears to constitute an 

 important theorem in the subject. The theorem in question might, I 

 think, be deduced from a very elegant formula of M. Borchardt 

 (referred to in the sequel), which gives the generating function of any 

 symmetric function of the roots, and contains potentially a method 

 for the calculation of the tables (5), but which, from the example I 

 have given, would not appear to be a very convenient one for actual 

 calculation. 



VI. "Memoir on the Conditions for the Existence of given 

 Systems of Equalities among the Roots of an Equation/' 

 By ARTHUR CATLEY, Esq., F.R.S. Received December 18, 



1856. 



(Abstract.) 



It is well known that there is a symmetric function of the roots of 

 an equation, viz. the product of the squares of the differences of the 

 roots, which vanishes when any two roots are put equal to each other, 

 and that consequently such function expressed in terms of the coeffi- 

 cients and equated to zero, gives the condition for the existence of a 

 pair of equal roots. And it was remarked long ago by Professor 

 Sylvester, in some of his earlier papers in the f Philosophical Maga- 

 zine,' that the like method could be applied to finding the conditions 

 for the existence of other systems of equalities among the roots, viz. 

 that it was possible to form symmetric functions, each of them a sum 

 of terms containing the product of a certain number of the differences 

 of the roots, and such that the entire function might vanish for the 

 particular system of equalities in question ; and that such functions 

 expressed in terms of the coefficients and equated to zero would give 

 the required conditions. The object of the present memoir is to 

 extend this theory, and render it exhaustive by showing how to form 

 a series of types of all the different functions which vanish for one 

 or more systems* of equalities among the roots ; and in particular 

 to obtain by the method distinctive conditions for all the different 



