TABER. — LINEAR TRANSFORMATION. 191 



in an even power of a solution of this equation. But if ^ is a solution 

 of the second kind, for any positive integer m, we can find a linear 

 substitution satisfying this equation whose (2/?2+l)th power is equal 

 to '}>. Thus, employing the notation of pages 185, 186, let i^Q be a 

 polynomial in (/> such that 



and, as above, let 



^0 Q =: I- (^ O + Q {>), OiQ^ ^{&Q — Q&). 

 Then, if 



i// 12 i/^ = O, 

 and 



Corresponding to any linear substitution ^ of the second kind satis- 

 fying the equation 



^ i2 (^ ::= Q, 



can always be found a solution cf)^ of the first kind whose coefficients 

 are rational functions of a parameter ^, such that, by taking ^ suffi- 

 ciently small, the coefficients of ^^ may be made as nearly as we please 

 equal to the corresponding coefficients of </>. 



§3. 

 If the two sets of variables of the bilinear form 



(12 ^ a-i , a-2, . . . a:„ ^ yi , 5/2, . . . «/«)' 



of non-zero determinant, are transformed by a linear substitution whose 

 product is equal to the identical substitution ; thus, if 



(Xi, X^, . . . .T„) =r (0 ^ ^1, 4, . . . 4), 



(yi, 1/2, " ■ y,,) = (</)~^ ^ vu V2,--- vn\ 



we have 



and the necessary and sufficient condition that the transformation shall 

 be automorphic is 



^~' i2 ^ = i2. 



Any one of the class of linear substitutions which satisfy this equa- 

 tion is the mth power of a linear substitution of this class, and can be 



