TABER. — LINEAR TRANSFORMATION. 181 



IX. 



NOTE ON THE AUTOMORPHIC LINEAR TRANS- 

 FORMATION OF A BILINEAR FORM. 



By Henry Taber. 



Presented April 10, 1895. 

 § 1. 



In the BnUetin of the New York Mathematical Society for July, 

 1894, I have shown that not every proper orthogonal substitution can 

 be generated by the repetition of an infinitesimal orthogonal substitu- 

 tion. That is to say, if we designate a substitution of the orthogonal 

 group as of the first or second kind according as it is or is not the 

 second power of a substitution of the group, there are then proper 

 orthogonal substitutions of the second kind ; and whereas any substi- 

 tution of the first kind can be generated by the repetition of an infini- 

 tesimal substitution of the group, no substitution of the second kind 

 can be generated thus. Nevertheless, by the repetition of an infini- 

 tesimal substitution of the orthogonal group, we can obtain a substitu- 

 tion of the first kind which shall be as nearly as we please equal to 

 any proper substitution whatever of the second kind. 



I also pointed out in this paper that for the orthogonal group every 

 substitution of the first kind was the mth. power of a substitution of the 

 group, for any positive integer m ; and that every substitution of the 

 second kind was the (2 m -j- l)th power of a substitution of the group. 



It follows of course at once that an exactly similar theory exists for 

 the group of linear substitutions which transform automorphically a 

 symmetric bilinear form with cogrediant variables. An exactly similar 

 theory exists also for the group of linear substitutions which transform 

 automorphically an alternate bilinear form with cogrediant variables, 

 and for the group of linear substitutions which transform automorphi- 

 cally a general bilinear form (neither symmetric nor alternate) with 

 cogrediant variables, as remarked in a note at the conclusion of the 

 above mentioned article. 



On the other hand, any linear substitution of the group of linear 

 substitutions which transform automorphically a bilinear form with 



