80 



CHAPTER I. 



Corollary II. The group of Unary linear homogeneous substitutions 

 of determinant unity is generated by the substitutions -B^g,* and 



T: Si- -I,, 6 = 6,- 



Indeed, T transforms .#1,2, a into I?2,i,;.. 



101. Transformation of indices. - We can introduce in place 

 of . . . ro the m new indices 



67) ni - 



provided the determinant 0;* | =(= 0. In fact, the substitution 



A: |[- = "^ ccjj^j (i = 1, 2, . . ., m) 



1 . . .m 



will replace ^- byfe^fe? which, by solving 67), can be put into 

 /,* 



m 



the form ^%/ ; -%. The substitution A becomes 



> =1 

 B-*AB: , = ( - 1, 2, - ., m) 



where 5 denotes the substitution 67) replacing the fj f by the ^. 

 In fact 



B _! AJ > _ 



The determinant of the transformed substitution equals that of A } 



This result is, however, a special case (p = 0) of the next theorem. 



102. Theorem. I he characteristic determinant (with parameter Q) 

 of a linear homogeneous substitution A, 



23 



^'s unchanged under every linear transformation of indices. 



It is only necessary to prove the theorem for the following types 

 of transformations of indices, since by 100 every linear trans- 

 formation can be derived from them: 



