584 PROCEEDINGS OF TUE AMERICAN ACADEMY. 



sufficiently small* On the other hand if A„ = 0, and the b's 'arc prop- 

 erly chosen, the transformation T^^ T„ may be essentially singular however \ 

 small t may be taken. Such a transformation T^ 1 term non- essentially , 

 singular. i 



If Ta is non-essentially singular, that is, if T'^j T^ is essentially singu- 

 lar however small t may be, a system of values b\, b'^, . . . 5',., of the 

 parameters can be found such that, however small t may be, T^ Tt^, is , 

 essentially singular ; and conversely. 



§3. 



Let «!, a^, . . . a^, and ^i, b^, . . . b^, be any two systems of finite 

 arbitrarily chosen values of the parameters of G, and let the transforma- 

 tion Ta be defined by the symbolic equation 



(14) T„ = n Ta, 

 where i is a variable quantity independent of the x's. We then have 



(15) aj = (pj(ai, . . . a„tbi, . . . tb,) 



= 1,2, ... r). 



The differential equations satisfied by the a's are 



See Bulletin of the American Mathematical Society for February, 1900, 

 p. 202. 



Two groups G and (?'^' are of the same structure (Zusammensetzung) 

 if the structural constants [Zusammensetzungconstanten) Cj^i and cfli are 

 identical. For two groups of the same structure, the system of differ- 

 ential equations satisfied by the r dependent variables Oj ==: (pj (a, t b) 

 are the same. But the equations of the group may restrict the number 

 of systems of the functions aj, which differ in the initial values of the 

 a/s, in certain cases so that there shall be but one system of functions (p. 

 Consequently, in the case of two groups of the same structure, one 

 may contain essentially singular transformations and the other may 

 contain no essentially singular transformation. Two such groups cannot 

 properly be said to be isomorphic, since one is continuous and the other 

 discontinuous. 



* If An i= 0, Ttb Ta may be essentially singular for an infinite number of values 

 of t. But this assemblage of values of t has no derived assemblage. 



