94 PROCEEDINGS OF THE AMERICAN ACADEMY, 



where k is any integer, and of Gj by ihe equations 



(I 



[e'^.'ll (j,-2 _ 1) + ^(e'X3_i)]^ 



(17 a) 



a 2 '=■ (^2 "^ '^2' 



(1) 



Nevertheless, for any value of k, the parameter group of G.y is identical 

 with the parameter group of G^, . For let S^ denote the transformation 



(21 



defined by equations (13 a), and S^ the transformation defined by 

 equations (17 a). Tiien for any system of values of ai, a^, and for any 

 value of k, we have, by properly choosing ^i, /J^, 



^a — ^^ '■> 



which symbolic equation persists for /3i, (i^i and k arbitrary, if ai, as are 

 properly chosen.* 



Let Tg^ be.an arbitrary transformation of G^, defined by the equations 



r \ r r 



X'i = X^-\- 1j aj X, Xi + — 2^- 2^ aj Ujt Xj X^Xi+ . . . 



(( = 1, 2 . . . n). 



The transformation T~ , inverse to Ta, is then defined by the equa- 

 tions which we obtain on replacing aj, ag . . . by their negatives, f 

 Compound the transformation T^ and its inverse with each transforma- 

 tion Ta of G> so as to obtain the transformation TJ1\T'^ , which is also 

 a transformation of G^ ; and let 



(18) T^.^TJaT-\X 

 Let 



(19) T„„= T,T,„T-\ 

 Then, if 



(20) T^^1\T^, 



we have 



(21) 7;,,= TTJ 



— 1 



V 



* Equations (13 a) may be regarded as defining a group with three parameters 

 a^, a,, and i, of which two, a^ and oo, vary continuously, and one, namely A-, takes 

 only integer values. But this group is not a mixed group, since we have shown 

 that h is unessential, that is, it is immaterial what value is assigned to k. 



t Transformationsgruppen, I. 52, 5.3. 



J The transformation 7",,' is said by Lie to be obtained by tlie application {Aus- 

 fuhninij) of T^ to the transformations Ta of Gr ■ Cf. Lie : Continuierliche Grup- 

 pen, 445 et sc(j. 



