104 



PROCEEDINGS OF THE AMEDICAN ACADEMY 



a. a.y 



Cl = h — ^02 + if 63 = ^j Cy 



r 



{a)bj, 



(32) 



C2 = ^2 

 C, = ia 



1 



Therefore, the infinitesimal transformations of the parameter group are 



5 



^1 = 



9 Ui 



^2 = - 



^3= ' 



ao 



2 dcH 



9 tto^ 



2 5 «! 9 a^ 



By means of the methods exphiined above, I have examined the deter- 

 minant A, and the adjoined group, corresponding to all possible types of 

 structure of two-, three-, and four-parameter complex groups,* and the 

 results are summarized in the table on pages 591-597, Vol. XXXV. 

 of These Proceedings. For all types considered, the several elements 

 (a, 7), (a, (a, y)), etc., of (26) were calculated, and the A determined 

 by the actual summation of the resulting series. Since making these 

 calculations, Professor Taber has discovered a method of obtaining A 

 immediately from (a, y) ; f namely, we have 



(a^y) 



•s: 



I 



flo, Co 



^llj + 



a 



n "-1 



a 



3) 



Ci;y + 



• • )x,. 



Let, now, ^ denote the matrix 



f — ^ttjCjii, 





2 aj Cj 21 



— 1jajCj,2 



<. -^ Ctj Cj 1,- , -^ (tj Cj 2r 



_ ■<? 



.jO^Cj,, J 



. e* — 1 

 Then A is the determinant of the matrix ; that is, if pi . . . p^ are 



"^ '• fe''' — 1\ 

 the roots of the characteristic equation of ^, A = 11. 1 ). The 



1 \ Pi J 



* These structures are enumerated by Lie on pp. 565, 571, 574-589, Continuier- 

 liche Gruppen ; anrl also on pp. 713, 716, 723-730, Transformationsgruppen, III. 



t This metliod had previously been given by Professor Engel. See Transfor- 

 mationsgruppen, III. 788. 



