OF ARTS AND SCIENCES. 



47 



m, n, p, etc., respectively, then Q may be written asuflw \ where 

 to is a non-vacuous matrix, and 



e = r 



0, 



in which all the constituents are zero except those in the squares 

 #i> 02> @39 etc -> along the principal diagonal; these correspond re- 

 spectively to the latent roots g u g*, g 3 , etc., and are subsidiary 

 matrices each of order equal to the multiplicity of the respective 

 latent root to which it corresponds. Since O is supposed non-de- 

 generate, the characteristics of its latent roots are (m; 1, 1, ... . 1), 



(n; 1,1, 1), (p; 1,1, 1), etc., respectively. Moreover, 



since the characteristics of g 1 are (m; 1, 1, .... 1), 



in 



h = 



► ;k 



In like manner, in the subsidiary matrix 6 2 the principal diag- 

 onal is bordered on the right by unity, and the constituents in the 

 principal diagonal are all equal to g 2 , the remaining m 2 — 2 m + 1 

 constituents being zero; similarly with respect to the remaining 

 squares, 6 3 , etc. 



Let & u # 2 , etc., denote the matrices obtained from 6 by erasing 

 successively all the constituents of 6 except, first, those in the 

 square 6 1 ; second, those in the square 2 , etc. Then 



= #! + #2 + #3 + 



