THE COMPOUNDS OF A LINEAR HOMOGENEOUS GROUP. 149 



Consider the Pfaffian Fj, j being a fixed integer <; m. By the 

 last section, it is unchanged by the substitutions 



B r , s ,i (r, s = 1, 2, . . ., j - 1, j + 1, . . ., m). 



Furthermore, _B^ ^ alters no element of Fj and hence leaves Fj 

 unchanged. Finally, we prove that B ft j^ replaces Fj by 



j^_l_ (_ iy- 

 Indeed, B^j^ replaces Fj by 



= [1 2 3...J-1 j + l...m] + i[j 2 3...J-1 j + !...] 



Interchanging 1 with r, we see that B r j^ replaces 



[r 2 3...J-1 j + l...r-l 1 r + 1 . . . w] = - 



Hence B rt j,i induces upon the Pfaffians FI the substitution 



By inspection, Z>^ gives rise to the substitution 



S } : Fj-Fj, Fi-DFt (- 1, . . ., j - 1, j + 1, - . ., m). 



Our theorem is therefore true for the particular substitutions .#/,, a > -Z)i 

 which generate the group G m . 



To complete the proof of the theorem, we show that, if S==(utj) 

 induces upon the F t the substitution 



