PROCEEDINGS OF THE AMERICAN ACADEMY. 



XVIII. 



ON THE GROUP OF REAL LINEAR TRANSFORMA- 

 TIONS WHOSE INVARIANT IS AN ALTERNATE 

 BILINEAR FORM. 



By Henry Taber. 



Presented February 12, 1896. 



Let G denote the group of linear automorphic transformations of 

 the alternate bilinear form 



2n 2n 

 1 1 



with cogrediant variables and of non-zero determinant. On page 575 

 et seq., Volume XLVI. of the Mathematische Annalen, I have shown 

 that a transformation of group G can be generated by the repetition 

 of an infinitesimal transformation of group G if, and only if, it is the 

 second power of a transformation of group G. I now find, if J is 

 real, that the same theorem holds for the sub-group of real trans- 

 formations of group G. That is, if J is real, a real transformation 

 of group G can be generated by the repetition of a real infinitesimal 

 transformation of group G if, and only if, it is the second power of a 

 real transformation of tliis group. Furthermore, if JF is real, the sec- 

 ond power of a real transformation of group G is the (2 ???)th power of 

 a real transformation of this group for any even exponent 2 m* 

 If the transformation T'is defined by the system of equations 



x\ = a^i Xi + «,-2 a:2 + . . . . -f «r, 2n ^2n (^ = 1 9 2, . . . . 2 n — 1,2 n), 

 let 7a denote the transformation defined by the equations 



X'r= (Orl^l + ^r2^2 + • • • • + f'^r. 2n^2n) Xx^ (f = 1 , 2, . . . 2n — 1, 2n), 



A being a root of multiplicity m of the characteristic equation of T. 



* For an odd exponent 2 m + 1, any real transformation of group G is the 

 (2 m -f l)th power of a real transformation of this group. 



