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THE EXPEESSION OF CEETAIN SYMMETEIC FUNCTIONS 

 AS AN AGGEEGATE OF FEACTIONS. 



By Thomas Mum, LL.D. 



(Eead May 30, 1906.) 



1. In the Noav. Annates de Math., xv., pp. 86-91, the following 

 theorem is enunciated by E. Prouhet, namely, Si a, b, c, ..., f, g, ..., 



1 sont n quantites inegales et racines de V equation 0(x) = O, la somme 

 des produits de ces racines prises m d m sera egale a 



I _ t u-^ Jafo - /g)°"" + '- («- m*-cY -(/-g) ' 

 ( } f(a).f'{b)...f'{g) 



Though not so stated, it is evidently intended that the elements 

 a, b, c, ...,/, g are m in number. Further, in the original there is 

 a misprint of an I for a b in the numerator of the fraction, but it is 

 otherwise clear that by (a - l) 2 (a- c) 2 ... (f-g) 2 is meant the product- 

 which Sylvester used to denote by £(a, b, c, ...,/, g). 



It is also important to note that <p'(a) . <f>'(b) ... <ji'(g) is exactly 

 divisible by £(a, b, c, ...,/, </), the quotient being the product of 

 m(n—m) binomial factors, namely, the binomials got by subtracting 

 from each of the elements a, b, c, ..., g each of the n-m elements 

 h, ..., I. Thus, when n — 4: and m = S, the identity is 



, , , a 2 b 2 c 2 , a 2 b 2 d 2 



abc-\-abd-\-acd J r ocd= -, jr-yr — ^-, 7,+ 



(a — d) (b -d) (c- d) (a -c)(b- c) (d - c) 

 a 2 c 2 d 2 b 2 c 2 d 2 



+ (a-b) (c - b) (3 - b) + (b-a)(c-a)(d-a) ' 



The object of the present note is to point out that the property is 

 not confined to the simple symmetric functions 2<2, Sa6, 'Zabc, ... 

 and especially that, as a consequence of this, when the number of 

 elements is even we obtain a generalisation of Jacobi's theorem 

 regarding the difference- product. 



