496 PllOCEKDINGS OF THK AMKRICAN ACADliMY. 



and the oc^ of finite transformations Tc generated by cc^ of infinitesimal 

 transformations are of the form, 



x' = e""- Xs , 



The group contains singular transformations T which are of the form, 



y'=Nx-y (N^O). 



Now T' applied to a point p on the line x= + c will transform p 

 to a point />i on the line x = — c ; and, clearly, there is no non-singular 

 transformation Tc among the transformations of the group that has the 

 same effect. If the singular transformation T is applied to a point on 

 the special invariant x = 0, p will he conveyed across the invariant 

 point ^ = 0, y = 0. But this can be done by a non-singular transfor- 

 mation whose path curve is imaginary ; for this transformation may be 

 effected by the non-singular transformation 



x' = — X, 



y' ^-y- 



Therefore, two points, ;> and />i, in the plane that lie on opitosite sides of, 

 and equidistant from, the special invariant a; = cannot always be 

 interchanged among themselves continuously by means of the tranforraa- 

 tions of the group. 



Example 2. 



Xzp., x^p., a:,/?! -f- '2x..p.2. 



Tlie -x^ of non-singular transformations T,, iiave the form, 



x'l = e''3 Xi , 



/, = -- (e^-s -e^) x, + e^^o x„ + -^ (^^'•3 -\)x,. 



and the cc" of singular transformation T have the form, 



../ 



Xi-= — Xi, 



x'. = 31 xi + X. ^Nx., , (Nzt 0) 



