30 CALIFORNIA ACADEMY OF SCIENCES. [Proc 3 d Ser. 



of order fi n will be called orthogonal if it leaves invariant the 

 function 



li 2 + & + - - - + ft. 



Thus for an orthogonal substitution, 



m 



(i) S: fi— 2 o.j |j («'= i, 2, ..?«), 



we have the conditions for the coefficients (when p > 2): 



m 



(2) 2a 2 ij = i , 2a ii a ik = o 



i=l i=l 



(y, k = 1, . . . . . m ,j ^ k) . 



Generalizing the work of Jordan, Traite des Substitutions, 

 pp. 155-170, I have announced 1 the theorem: 



The group G of orthogonal substitutions of determinant 

 unity on m indices in the Galois Field of order fi n ~^ : $,ft~>'2, 

 is generated by the substitutions (each affecting only two 

 indices) : 



a, 8 (?'■=«?, + «, 



For m odd, the order of G was found to equal the order 

 co of the linear Abelian group on m-i indices, with factors 

 of composition 2 and co/2. But it is here proved that, when 

 fl n — 8 / ± 3, G has the same factors {m being odd and >3). 

 Thus are reached two triply-infinite systems of simple 

 groups of the same order, which are probably not iso- 

 morphic when wz>> 3, judging by the corresponding simple 



continuous groups of Lie. 



o 

 3. The orthogonal substitutions O , affecting only the 



indices |i and £2 , form a commutative group 1 2 of order 

 fi n — e (the number of solutions of a 2 -|- /3 2 r— 1), where 

 e= ± 1 according as — 1 is a square or a not-square. A 

 sub-group Q12 of index 2 is formed by the substitutions 



1 "Orthogonal group in a Galois Field," Bulletin of the American Mathe- 

 matical Society, Feb., 1898. The order of the group is there given, also its 

 structure for the case/ = 2. 



