102 i'i:oci:i:i>ings op the American academy 



e"- — 1 

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



do 



vanishes for o. 2 an even multiple, not zero, of ~\'— 1. Nevertheless 

 the group (•■, is continuous, whereas the group ' '< '.J is discontinuous. 



Tlie symbolic equation e ax = e a+ ' y is equivalent to the system of 

 equations 



n k = a l: -j- 6t C k 



[k= 1,2 .. . r), 



which (IctiiiL- the infinitesimal transformation of the parameter group. 



But the infinitesimal transformation of the parameter group is defined by 



the equations r 



a k = a k + S,£^(a)ft,S< 



(A- = 1, 2 . . . »).» 

 Thereforo 



r 



c* = -; £y (a) fy 

 [k =1,2... r). 

 If A 4 1 0, equations (28; give 



(fc = 1, 2 . . . r). 



Therefore, if 3L . . . 2l r denote the symbols of infinitesimal transforma- 

 tion of the parameter group, we have 



3 



A ik 9 



&j — 1 k t Jk (") — — 2 4 — r— 



i c "* i A c a 4 



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



I-, illustrate what precedes, consider the two-parameter structure 



I V, . X : ) = X 1 . 

 Equation (26 I givi 



6j A', ■; A .V c, A', + r, X. — — (a, '•., - a, e, | .V, - - * i a, e s - nt s r V Y 



a* 



- - ; (a, c, - atj c, ) A", i a, e s - Oj r,) A', — . . . 



v\ hence folli i 



/,, = c, e* — a., — 1 ) r.,, 



msformntionsgruppen, ! 

 Engi 1 and Bchnr, Transformationsgruppen, III. 764 etseg. and T. K s,/ gg^. 



