138 Mr. J. J. Sylvester on 



oxygen, although the former combination is one of the weakest, 

 and the latter one of the strongest with which we are acquainted*.'' 

 If Faraday's electro-chemical equivalent numbers represent 

 neutralizing quantities of electricity, as atomic numbers repre- 

 sent saturating ratios of combinable bodies, it should happen 

 that combination of different kinds of matter, in atomic ratios, 

 should take place without development of free electricity, the 

 two states having been in all cases exactly sufficient to neutralize 

 each other. But the contrary condition of chemical combination 

 is notorious ; for there are few instances of chemical action which 

 are not accompanied by the evolution of electricity, and such 

 evolution is easily recognizable, provided the bodies concerned 

 are conductors. Instances in abundance are furnished by the 

 experiments of Lavoisier, Laplace, Becquerel, Pouillet and others. 

 [To be continued.] 



XIX. A Demonstration of the Theorem that every Homogeneous 

 Quadratic Polynomial is reducible by real orthogonal substitu- 

 tions to the form of a sum of Positive and Negative Squares. 

 By J. J. Sylvester, Barrister-at-Lawf. 



TT is well known that the reduction of any quadratic polynomial 

 -■- (1, l)a;2 + 2(l,2)^?/+(2,2)?/2-t-&c. ... +{n,n).t'' 



to the form « j . ^^ + Wg , t;^ + ... + a^^ . 6'^, where ^, rj, . . . 6 are 

 linear functions of x, y, . . . t, such that x^ + 7/^+ . . , -|- 1'^ re- 

 mains identical with ^ + nf+ .. .6^ (which identity is the cha- 

 racteristic test of orthogonal transformation), depends upon the 

 solution of the equatioii 



(1,1) +X (1,2) ... (l,n) 



(2,2) (2, 2)+\...(2, 7i) =0. 



{n, 1) (n, 2) ... (w,n)4-X 



The roots of this equation give a^, a^^, . . . a^', and if they are 

 real, it is easily shown that the connexions between x,y, . . . t ; 

 ^, rj, . . .$, are also real. M. Cauchy has somewhere given a 

 proof of the theorem J, that the roots of \ in the above equation 

 must necessarily always be I'eal ; but the annexed demonstration 



* Traits de Chimie. Pai-is edition, 1845, p. 100. 



t Communicated by the Author. 



X Jacobi and M. Borchardt have also given demonstrations ; that of the 

 latter consists in showing that Sturm's functions for ascertaining the total 

 number of real roots expressed by my formulic (many years ago given in 

 this Magazine) are all, in the case of /(X), representable as the sums of 

 squares, and are therefore essentially positive. 



