THE COMPOUNDS OF A LINEAR HOMOGENEOUS GROUP. 155 



If [] 2 be the particular substitution 140), the "partial substitution" 

 141) becomes 100 



010 

 v- 



of determinant v~ l . Hence if 140) belong to 6r 4) 2, v must be a 

 square in the field. 



Inversely, if v be a square, 140) is the second compound of 

 the following substitution of determinant unity: 



v 1 /. 00 

 i/A 

 v-'/ 

 % 00 v 

 Note. The second compound contains the substitution 



12> 



rt V 



23^2 



2^="-' 



Y' i) 1 Y 



247 J1 34 V J -34 < 



In fact, the latter is the second compound of the substitution 



v 

 0100 

 0010 

 v~ 



165. Theorem. For p = 2, every substitution of 

 the relation 



satisfies 



= 1 (mod2) ; 



formed by multiplying each coefficient of the partial substitution 141) 

 by that coefficient of the matrix [a] 2 which lies symmetrical to it. 

 Gl, 2 does not contain the substitution M = ( F 12 Y 34 ). 



The left member of our relation is seen to be the expansion of 

 the expression 



'13 



C 14 



C 31 



C 41 



42 



C 34 



41 



'13 



and is therefore = 1 (mod 2), since | a,-j \ = 1. The substitution 

 does not satisfy the relation and so does not belong to the group 



2- 



