TABER, — LINEAR TRANSFORMATION. 183 



where e^ denotes the convergent infinite series 



Since ^ is commutative with fi, ;( is also commutative with Q ; con- 



1 l^ 



sequently for any positive integer w, — ;^ and therefore e" are com- 

 mutative with fi. Whence it follows that if 



1 

 ^ = e'" , 

 we have 



and 



^m _ gX _ ^_ 



That is, any linear substitution which transforms automorphically the 

 bilinear form 



with contragrediant variables is the ?nth power of a linear substitution 

 which also transforms this form automorphically. By taking m suffi- 

 ciently great, the coefficients of the linear substitution ;^ can be made 

 as nearly as we please equal to zero, and thus the linear substitution 



ij/ = e"' may be made as nearly as we please equal to the identical 

 substitution. But however great m may be, we have, nevertheless, 



* The infinite series 



eX = S + X + 2X' + 3!X^ + • • • + ^ X-" + • • • 



is convergent for any linear substitution x ; and we liave 



(eX)-i = e- X. 



and if m is any positive integer, 



If X and x' are commutative, we also have 



gX gX ^ gX + X . 



For any linear substitution <j) whose determinant is not zero a polynomial 

 X = f{^) in <p can be found such that 



