102 PROCEEDINGS OP THE AMERICAN ACADEMY 



e- — 1 



determinant A corresponding to this structure is A = , and this 



a.7 



vanislies for ao an even multiple, not zero, of tt V— !• Nevertheless 

 the group Cj is continuous, whereas the group Go is discontinuous. 



The symbolic equation e"' = e""*" '^ is equivalent to the system of 

 equations 



<^k = «'i ■ + S if c* 



(^■=l,2 . . . r), 



which define the infinitesimal transformation of the parameter group. 



But the infinitesimal transformation of the parameter group is defined by 



the equations r 



«!' = «A- + ^j^kj{a)h^*' 



(^■ = 1,2... r)* 

 Therefore 



r 



c* = ^j ^Kj (a) bj 

 {k=l,2 . . . r). 

 If A =1= 0, equations (28) give 



^A — -^J » ^3 



1 A 

 (^- :- 1, 2 . . . r). 



Therefore, if ^i . . . ^^ denote the symbols of infinitesimal transforma- 

 tion of the parameter group, we have 



1 d a^ 1 A d flj 



(i = 1, 2 . . . r).t 



To illustrate what precedes, consider the two-parameter structure 



Equation (26) gives 



bi Xi -\- b.2 Xo, — Ci Xi + Co Xo, — ^ (oi Co, — a.2 Cj) X^ — -^^ («i c, — rto Cj) Xi 



— T-f («1 ^2 — «2 Ci) Xi — ^ (Oi Cg — «2 <^l) -^'l — • • • 



whence follows 



b\ = fl ^ (^ ' — «2 — 1) ^2> 



(29) ^2 n 2 



Z>2 = Co. 



* Transformationsgruppen, I. 55, 65. 



t Engel and Schur, Transformationsgruppen, TIT. 754 el seq. and 788 et seq. 



