TABER. — LINEAR TRANSFORMATION. 187 



2 



of — ^1 fi as small as we please, and thus we may make the substi- 

 tn 



tution i// as nearly as we please equal to the identical substitution. 



Whence it follows that any linear substitution of the first kind which 



.satisfies the equation 



(that is, any linear substitution which is the second power of a solution 

 of this equation) can be generated by the repetition of a linear substi- 

 tution which is also a solution of this equation and which is infinitely 

 near the identical substitution. 



Any linear substitution </> satisfying the equation 



is of the first kind if — 1 is not a root of the characteristic equation of 

 cf) (that is, (f) is then the second power of a substitution satisfying this 

 equation). For if — 1 is not a root of the characteristic equation of 

 <t>, we may put 



n-^ Y = - 8 + 2 (S + (/))- 1 = (8 - <^) (8 + </>)- ^ 



and we then have 



Yt2-^=:n(o-^'Y)n-^ 



= n (8 - <f>) (8 -\- <^)-in-i 



= o (8 - <^) fi-i • fi (8 + </))-^ fi-i 



= (8 — n (t> n-^) {8 + n cjiQ- ^)~^ 



= (8-<^-0(s + c/>-0-^ 



= (cf> - 8) (ct> + 8)-'^ = - n-^ Y. 



From the expression for Q,~ ^ Y we also obtain 



(0-1Y + 8) (<^ + 8) =2 8; 

 and consequently, since | 0~-^ Y + 8 | 4= 0, 



<^ = -8 + 2(8 + 0-iY)-i= (8 + n-^Y)-i(S-n-iY). 

 If now & r=y (()-i Y) is a polynomial in fi~^ Y such that 

 8 + fi- 1 Y = e^ 



