CHAPTER VI. THE COMPOUNDS OF A LINEAR etc. 145 



CHAPTER VI, 



THE COMPOUNDS OF A LINEAR HOMOGENEOUS GROUP. 1 ) 

 153. It was shown in 98 that the linear substitutions 



A: IE 



combine according to the law 



A' f = A'A: I! 



where 



.1=1 



(* = 1, . . ., m) 

 (, j 1, . . ., m). 



In Sylvester's umbral notation, the general # th minor of the 

 determinant jay! is as follows: 





The formula expressing the # th minors of a/J in terms of the 

 q ih minors of ay| and of | ay | is the following 2 ): 



I, . . . 



133) 



Ji 



the summation extending over the C m)q combinations I 

 the m integers 1, 2, . . ., m taken q at a time. 



Consider the linear homogeneous substitutions on 



I/, /..,. 7 



Z 2? . . ., ? 2 of 



variables 



' * 2 



where the sets 



and (? 1? Z 2 , . . ., 2 2 ) take independently 



1) This chapter gives a new exposition of results published by the author 

 in the following journals: Bulletin of the Amer. Math. Soc., vol. 6 (1898), 

 pp. 120135; Proceed. Lond. Math. Soc., vol. 30 (1898), pp. 70 98; Trans- 

 actions of the Amer. Math. Soc.', vol. 1 (1900), pp. 91 96. 



2) Scott, Theory of determinants, p. 53. 



DlCKSON, Linear Groups. 10 



