148 CHAPTER VI. 



The square of [1 2 . . . m] is the skew -symmetric determinant 







Jtl 







where Yy = Y j{ . 



By 100, 6r = GLH(m, p n ) is generated by the substitutions 

 Br,s,?i and D m . The corresponding substitutions of the second com- 

 pound 6r m) 2 will therefore generate the latter group. To ^1,2, i and D 

 there correspond the respective substitutions of 6r m>2 : 



2,.. -.,1113 ^ 



But A is unaltered by an interchange of any two subscripts as 1 

 with 3; for, the resulting determinant may be derived from A by 

 interchanging the first and third rows and the first and third columns. 

 It therefore suffices to prove that A remains invariant, up to a 

 multiplicative constant, upon applying the substitutions "B\,^\ and D 1 . 

 By inspection, D multiplies A by D 2 . Also Bi^a transforms A 

 into the determinant 















v_!_ivvv v n 



- t ml"T Ajt -m2 J -m2 -*-m3 -*-TO4 ...v 



This reduces at once to A since F 12 + I r 21 = 0. 



157. Theorem. For m odd, the substitution [a] 2 of 

 compound gives rise to the substitution 



second 



upon the Pfaffians F$ E^ [1 2 . . . j 1 j -j- 1 . . . ni], if 

 minor 1 ) complementary to a^ in the determinant 



(* 1, . . ., m) 



denotes the 



1) Or the adjoint of cc i . without its prefixed sign. 



