104 



PROCEEDINGS OF Till-: AMERICAN ACADEMY 





ci = h — 2*2 + y h = 2 j iu (a) bj, 



r , 



J a 



= *,**(«)**. 



c 3 = b 8 — 2, £ 8j (a) hj . 



Therefore, the infinitesimal transformations of the parameter group are 



".: 



<3 = _ — 



- 9 "i ' 9 a% 





2 5 Oi <T ".. 



By means of the methods explained 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 591-5D7, Vol. XXXV 

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

 (a, y), (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, 



1 it, now, <£ denote the matrix 



C — IttjCjn, 





+ 



X,. 



2 Oj c }2l 



- ~m ff. (' ■ J -J . — OjCd 



— i a . 



— %OjC Jlr , — 2ajCj ir 



■ - *i«iCjrr J 



,,,. are 

 be 



- - 1 

 I hen A is the determinant <»f the matrix ; that is. if pj . 



* fe°> - 1 \ 



the roota "f the characteristic equation of 0, A = IF, ( - ). Tl 



» \ Pi J 



' These structures are enumerated by Lie on pp. €66, 671 9, Continuier 



liche Gruppen ; and also on pp. 713, 7!''., 728 780, Transformationsgruppen, III. 

 \ This method had previous!} been given by Professor EngeL See Transfor- 

 ruppen, III. 



