CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 227 

 The substitution S 2 replaces Yu, Y f j (j = 2, . . ., a x ) by respectively 



KtYn, Kt&j + Yv-d (j - 2, . . ., Oi). 



Hence, the introduction of the Y^- has the effect of setting q> = 

 in $ 2 . Proceeding similarly, we can suppose that y, #,#,... are 

 all zero but one, say ^ =j= 0. In the latter case, we set 



and find for S 2 the canonical form 



27 



In every case we reach a canonical form of the type given in 

 the theorem, for which the indices Yi, have the properties 1) and 2). 

 But the indices Zij are linear functions of the ,- with coefficients 

 which certainly involve L t and apparently 1 ) also K t . If the JKJ be 

 involved, we proceed as follows. From the canonical form actually 

 reached, S YS lf where Y is the partial substitution on the indices 

 Yij f not altering the indices Ztj 9 etc., while S does not involve the 

 indices Yij, but affects the Z ijf etc. Setting 



Y t . = y,+ y\K, + tfKf + y ( ^K?~\ 



(s = 1, . . ., a; % - 0, . . ., % - 1) 



where the 2/'s are linear functions of the ^ with coefficients in the 

 6rF[# n ], we can evidently introduce the y's as new indices in place 

 of the Y{ 99 so that Y takes the form of a substitution belonging to 

 the 6r.F[# ra ] and affecting only kcc indices. Likewise, by introducing 

 in place of the Z^ etc., an equal number of linear functions %, etc., 

 belonging to the &P[jp^j, it is possible to give to S the form of a 

 substitution in the field and affecting only m ka indices. Its 

 characteristic determinant is [Fi(KJ]P . . . Hence, by the hypothesis 

 made for the induction, S can be reduced by a linear transformation T 

 to a canonical form 



1) By the considerations in the text, we may dispense with the difficult 

 proof, analogous to that of Jordan, Traite% pp. 121 122, that the Zij do not 

 involve Ki, but the single imaginary Li. 



15* 



