584 PROCEEDINGS OP THE AMERICAN ACADEMY. 



sufficiently small.* On the other hand if A„ Oj and the b's are prop- 

 erly chosen, the transformation T tt T a may be essentially singular however 

 small \ mini be taken. Such a transformation T„ I term non-essentially 

 singular. 



It '/', is non-essentially Bingular, that is, if /',. '/'„ is essentially singu- 

 lar however small / may be, a system of values //',, //._,. . . . b',, of the 

 parameters can be found such that, however small t may he. 'J 1 ., 7', b , is 

 utially singular; ami conversely. 



§ 8. 



Let a u a. 2 , . . . tt r , and b\, />.,, . . . b r , be any two systems of finite 

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

 tion T„ be defined by the symbolic equation 



(II) T a = T lf ,T ri . 



where / is a variable quantity independent of the j's. We then have 



(15) aj = q> j (a u . . . a r , tb u . . . tb r ) 



= 1,2, . . . r). 



The differential equations satisfied by the a's are 



nr , f da i da * d«r\ <t>a n , , , 



(1G) \di* dT> - - • di) = j7ir I {Kh ' ■ ■ • b - y 



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

 p. 202. 



Two groups G and G {1) are of the same structure (Zusammensetzung) 

 if the structural constants (Zusammensetzungconstanten) <- iU and c% t are 

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

 ential equations satisfied by the r dependent variables a, = <$. (a, t b) 

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

 of systems of the functions a r which differ in the initial values of the 

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

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

 may contain essentially singular transformations ami the other may 

 contain no essentially singular transformation. Two such groups cannot 

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

 discontinuous. 



* If a„ -\. 0, 7', i, '/'„ may be essentially Bingular for an infinite number of values 



of /. L5ut this assemblage of values of/ has no derived assemblage. 



