192 CHAPTER VII. 



196. Theorem. Tlie factors of composition of L^ p n are 



(ifp>2) 2,(^-1)^, Gp 2 -l)r, 2, 

 (if p = 2) 2 r (2 2 * - 1) 2% (2 2 - 1) 2 



e#cep w&m _p" = 2 or 3, wto the composite numbers 6 awe? 12 

 respectively are to be replaced by their prime factors. 



To determine the quaternary substitutions leaving ii^i + 2 % 

 absolutely invariant, consider the two pairs of equations 1 ), 



179) ^ + 5^ = 0, % -%=<), 



180) fe + *%=0, &-*% = (). 



The most general quaternary linear homogeneous substitution, leaving 

 invariant the pair of equations 179), for every value of % in the field, 

 is readily seen to be 



having the determinant (ad /3y) 2 . For it we have 



The group of the substitutions 181) is therefore simply isomorphic 

 with the binary group on the variables ^ + ^| 2 and ?? 2 ^%. Since 

 the transposition (2^2) transforms the pair of equations 179) into 

 the pair 180), we obtain the most general linear homogeneous sub- 

 stitution, leaving invariant the pair of equations 180), for every ^, 

 if we transform the set of substitutions 181) by (? 2 %)> giving the set 



The product of any substitution 181) by any substitution 182) gives 



i{ 



183) ill 



al> ~yD ccD yj$ 

 ft A -dC ftC dA 



1) They give the two sets of generators on the ruled surface 1^ -f | 2 ?y 2 = 0. 



