374 PROCEEDINGS OF THE AMERICAN ACADEMY. 



4. Conversely, if is any matrix for which 6 to = 0to6 and 

 e (e-0)n __ ^ t b en <£ _ . e eo j s a solution of the equation £to<f> = to. 

 In particular, if 6 is symmetric, e 9n is a solution of this equation, and 

 so also is e» , in which n is any positive integer. But (e« en )" = e en . 

 Therefore, every expression e 0a in which 6 is symmetric is a solution 

 of the equation <£Q</> = to, and can be generated by the product into 

 itself an infinite number of times of a matrix which is a solution of 

 this equation, and differs only infinitesimally from the matrix unity. 



Let $ = e eil in which is symmetric. If the positive integer n is 



sufficiently great, no odd multiple of it \/—l is a latent root of - $ to ; 



n 



and therefore —1 is not a latent root of e* Bil . But <$> = (e» en ) n . 

 Therefore, <£ is the nth power of a matrix given by Cayley's ex- 

 pression. 



5. Every solution of the equation <£fi<£ = to, given by Cayley's 

 expression, can be put in the form e 9n in which 6 is symmetric. Thus, 

 let 



<f> = (to — Y)- 1 (O + Y) 



= to- 1 . (i - yd- 1 )- 1 (i + Yto- 1 ) . to, 



in which Y is symmetric. If now &' =f(—Yto~ 1 ) is a polynomial in 

 — YI2 -1 , satisfying the equation 



e»' = 1 — YO- 1 , 



then &" =f(Yto~ 1 ) will satisfy the equation 



ed" = 1 + YO" 1 .* 



* Since | <p | 4= 0, and therefore | 1 - T fl- 1 1 £ 0, 1 1 + Tfl- 1 1 ^ 0, such 



polynomials in —Tfl- 1 and TH -1 can always be formed; and if e-^ -Yn ' 

 = 1 — Tfl- 1 , equating the transverse of either member, we have 



e /(n-'Y) _ 1 + n _i T 



That is, 



Therefore, 



fl-»«/( Yfl_l )n = gO-'/frn-^o 

 - /(n-'.YQ-'.c) 



= l + n- J T 



= n-Mi + rn- 1 )n. 



Jlva- 1 ) 



= 1 + rn- 1 . 



