186 PROCEEDINGS OF THE AMERICAN ACADEMY. 



In particular, it & CI — — CI &, (f) = e^ satisfies the preceding 

 equation. 

 Let now 



then 



n e^ — ^ n {& n + n &) n-^ 



= i (o/(</.) o-i + "/(0-') n-') 



= i (/(fi «/) 0-1) +/(fi <jf>-i o-i)) 



Again, let 

 then 



Since 60 and 0i fi are polynomials in ^, they are commutative. 

 Therefore 



J 2 __ g2,>fi _. g2 9„n+2e,n ^ e2 9„n g2 9jn _ - g2 e, n 



g2 9on ^ g,>n + n,? -- g_ 

 2 



smce 

 If now 



and 



Consequently, any linear substitution, as ^^ which is the second 

 power of a linear substitution satisfying the equation 



(p ii (p ^= i2j 



is the wth power for any positive integer m of a solution of this 

 equation. By taking m sufl[iciently great, we can make the coefficients 



* See last note, page 185. 



