150 



CHAPTER VI. 



I: 



where AIJ is the minor of ^ | complementary to a,^, the products 

 jD x /S and B rjSy iS will induce upon the Ft the substitutions called for 

 by the theorem. First, the product D S will induce upon the F t 

 the substitution 





The matrices of the two products D S and /S^X are respectively 



-4,1 



21 



22 



Da,. 



DA m2 ...DA 



mm ) 



Here the second matrix is derived from the first by the law expressed 

 by our theorem. 



Next, the product S r>s ^S induces upon the F { the substitution 



! = A ik F k -f - 



are 



The matrices of the two products 



. CC mm 



and 

 A ir 



-2m 



ml 



The second matrix is seen to be derived from the first according to 

 the law expressed in the theorem. 



Corollary. The second compound of any linear homogeneous 

 group G- m gives rise to a linear group on the m Pfaffians F lf . . ., F m 

 which is identical with the m I st compound of Gr m . 



158. We can establish in an analogous manner the theorem: The 

 linear substitution [a] 2 of the second compound of any m-ary linear 

 homogeneous group G m , which corresponds to the substitution (,-y) of G m , 



effects upon the C m2 Pfaffians . . 



/,,%,...,4, 



Rh --] L- /,- / 



\<V<... 



2 



a linear homogeneous substitution identical with the substitution [] m _s 

 of the (m 2} nd compound of G m . 



The group induced by the second compound of G m upon these 

 Pfaffians is therefore the (m 2) nd compound of G m . 



