Homogeneous Quadratic Polynomials. 



139 



is, I believe, new ; and being very simple, and reposing upon a 

 theorem of interest in itself, and capable no doubt of many other 

 applications, will, I think be interesting to the mathematical 

 readers of the Magazine. 



Let , 



/w= 



(1, 1)+X(1, 2) ... (1,«) 



(2,1) (2,2) + \ ... (2,«) 

 (3,1) (3,2) (3,3)+X...(3,n) 



{n,l) (re, 2) ... («, n)+\ 



it is easily proved that/(\) x/( — \) 



= [1, 11-A.2 [1,2] ... [l,n] 

 [2,1] [2,2]-\^..;[2,n] 



[n, 1] [re, 2] ... \n,n-\-\\ 



[t, e] = (t,l)x(l,e) + (t,2)x(2,e)+ ... +{c,n) x{n, e). 



If, now, for all values of r and s (r, s) = (s, r), i. e. if/(0) be- 

 comes the complete determinant to a symmetrical matrix, then 

 eveiy term [r, s] in the derived mati'ix becomes a sum of squares, 

 and is essentially positive, and ( — 1)^./(A) x/( — X) assumes the 

 form 



(xy-F(\2)"-' + G(X2)"-'+ ... ±L, 

 where F, G, . . . L will evidently be all positive ; for it may be 

 shown that F will be the sum of the squares of the separate terms, 

 i. e. of the last minor determinants of the given matrix, G the 

 sum of the squares of the last but one minors, and so on, L being 

 the square of the complete determinant. For instance, if 

 f{X) = 



where 



-/W x/(-\)=\«-P\'» + G\2_H, 



F=«2 + 62 + c2 + 2«2 + 2/32 + 272 

 G=(«A-72)2 + (6e-a2)2+(/3^-«c)2 



+ 2(a«-/37)2 + 2(A/9-7«)2 + 2(c7-a/3)2 

 H = 



