380 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Thus, let 



2fi = n + fi, 2 Oj = O — n. 



Then, if (f> satisfies the above equation, we have, as shown by Cayley, 



(j>n <f> = n , <t>n 1 <f> = n 1 ; 



and conversely. 



If neither ± 1 is a latent root of Cl Cl~\ then | fi | + 0, | fij | £ 0. 

 Since I Cl | + 0, ty, has a square root expressible by Sylvester's for- 

 mula in powers of Cl . For the determination of this square root the 

 solution of the algebraic equation | Cl — g | = is requisite. If Q 1 

 denotes any symmetric square root of fi , the equation $£) (£ = fi 

 may be written 



which, if i^ = fi * <£ (fi 2) _1 , becomes 



($= 1. 



Therefore, the most general expression for the matrix $ satisfying the 



above equation is 



Cl ~ 2 i/r fi l, 



in which if; is an arbitrary orthogonal matrix. 



If $ Cl (f> = Cl, we must also have <£ fix <f> = Cl v Therefore, 



ci i ^ n _ ^ Qi n ~ 2 $ ci * = fii , 



But (9) § 1 i/r is given by the product of the three matrices. 



(n-i o, n - 1 - y' )-' (q,-* Qi «<>-* + Y 'a) 



= iy (fi x - cy Y'afyr 1 (fii + n * Y '*<V) o ~** 

 (n -i fix o -i - y'^)- 1 (o -i Ox n -i + y^) 



= o i (fix - $y Y'p cy )- 1 (fix + n * Y'p ry ) o - 1, 



(n-h^n-i- y' y )-' (o^i^ry-i + y' y ) 



= ci i (Ox - $y y' v o i)- » (fix + iy y\ q i) ly 1, 



in which Y' a , Y'p, Y' v , are commutative with Cl ~* fix Cl ~h 



If we put Y a = fi * Y' a fi 2, Y 3 = iy Y'„ O i, Y v = iy Y' v O l, we 

 have for $ = fi ~2 «/r$y the expression 



(Oi-YO-MOi+Y.) (n i -Y^)- 1 Cfii + Y^)(fix-Y 7 )- 1 (fii + Y v ). 



Therefore, the group of solutions of the equation $00 = CI is a 

 sub-group of the equation cj> Cl x <j> = Cl x , generated by the totality of 

 expressions 



