78 PROCEEDINGS OP THE AMERICAN ACADEMY. 



I shall therefore assume that w > 4, that the roots of the character- 

 istic equaiiou of jf are not all of the same sigu ; and I shall show that 

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

 infinitesimal transformation of group G if it is an even power of a 

 transformation of this group. 



If the matrix or linear homogeneous transformation ^ transforms Jf 

 automorphically, it satisfies the matrical equation 



^ n «^ = O, (1) 



in whch ^ denotes the transverse or conjugate of 0. Conversely, if 

 this equation is satisfied, ^ transforms jp automorphically. The deter- 

 minant of any transformation satisfying this equation is equal to either 

 -f 1 or —1. By definition the totality of real proper solutions of equa- 

 tion (1) constitutes group G. 



If </) is any real solution of equation (1), we may put 



cji = (f)o (f)i = cfii cf)o, 



where <^o, <^i, are polynomials in ^, are both real, and are both solutions 

 of equation (1), that is 



Moreover — 1 is not a root of the characteristic equation of ^i, therefore 

 the determinant of <^i is equal to +1 ;* and consequently ^i is a trans- 

 formation of group G. Finally, 



and therefore 



</>' = 4>' i>i' = <^i'. (2) 



That is, the second power of any real solution of equation (1) is the 

 second power of a transformation of group G.f 



transformation of this group can be generated by the repetition of a real infini- 

 tesimal orthogonal transformation. 



* The roots, other than ±1, of the characteristic equation of any solution of 

 equation (1) occur in pairs, the product of two of the same pair being unity. 

 The determinant of a linear transformation is equal to the product of the roots 

 of its characteristic equation. 



t If — 1 is a root of multiplicity vi of the characteristic equation of (p, and if 

 the roots of this equation other than —1 are gi, r?.., etc. of multiplicity, respectively. 

 p-,, pn, etc., then 



where 



[(./) + !)'»- ( .71 + 1)'"?' [{<i> + 1)"* - {fj.2 + 1)'"]P' 



[- (.'/I + 1)"'P' ' I- (.9. + 1)'"]^' " "• 



