300 THIRD REPORT — 1833. 



combinations may be either the sums of every two of the quanti- 

 ties, X, x^,... T„_i, such as X + x^, x + x^^, &c., or their products, 

 such as X Xi, or other rational linear functions of those quanti- 

 ties, involving two of them only, such as, x + x^^ + x x^, x + x^ 

 •^ 2 X x^, ov X + x-^ + k X x■^, where k may be any given num- 

 ber whatsoever. If we take any one of these sets of combina- 

 tions, we can form rational expressions for their sum, for the 

 sum of their products, two and two, three and three, and so on, 

 in terms of the coefficients p^, p.^, . . . pn, of the original equa- 

 tion (1.), by means of the common theory of symmetrical func- 

 tions *, and consequently, we can form the corresponding equa- 

 tions of m {2 m — 1) dimensions which will have rational and 

 known coefficients. Such equations being of odd dimensions 

 must have at least one real root ; or, in other words, there must 

 exist at least one real value of one of the sums of two roots, 

 such as a; + x^, of one of the products, such as x x^, of one 

 of the functions, x + x^ + x x^, or a: + a;] + k x Xy If the 

 symbols which form the real sum x -\- x^ are the same with those 

 which form the real value of the product x x^ then, under such 

 circumstances, x and x^ are expressible by real magnitudes af- 

 fected with the ordinary signs of algebra f . We shall now pro- 

 ceed to show that this must be the case. 



If we form the equations successively whose roots are x -\- x^ 

 + k X x-^, corresponding to different values of k, we shall have 

 one real root at least in each of them. If we form more than 

 m (2 m — 1), such equations for different values of k, we must 

 at least have amongst them the same combination of x and x^^ 

 forming the real root, in as much as there are only m {2 m — \) 

 such combinations which are different from each other. Let k 

 and X'l be the values of k which give such combinations, and 

 let «' and (6' be the values of the real roots corresponding ; then 

 we must have 



X + x^ -\- k X x^ ^= u.' 

 X + .r, + k^ X x^ = /3' 



* The formation of symmetrical combinations of any number oi symbolical 

 quantities x, x^, . . . a-n-i, and the determination of their symbolical values 

 in terms of their sums {px), their products two and two (pq), three and three 

 (JO3), and so on, involves no principle which is not contained in the direct 

 processes of algebra, and is altogether independent of the theory of equations. 

 The theorems for this purpose may be found in the first chapter of Waring's 

 Meditationes Algebraicce, in Lagrange's Traite sur la Resolution des Equations 

 Ni/meriques, chap. i. and notes 3 and 10, and with more or less detail in 

 nearly all treatises on Algebra. 



t If J + •'^1 = a and xx^ = /i . , where « and /3 are real magnitudes, then 



j; = -^- + s'' s " — /3 ? the values of which are either real or of the form 

 (cos (I + \^ — 1 sin 6) ^^/3, where the modulus \//3 is real. 



