8o KANSAS UNIVICRSri'V (^)UARTKKI,>'. 



The invariant points of either transformation are given by the 



equation 



cx--p(d — a)x — b=ro. 



be . . . 



Putting d=: in this it breaks up into 



{^-t){-+-})=°- 



thus showing that and are the invariant points of the 



c a 



transformations. 



Pfopcrlx J. — The i^^roup G, dnifains a pair of pscitdo-traiisponiia- 

 tions, each of lohich transfonns atl points of the line 1 into one op tfie 

 invariant points of tfie i^ronp: tiie cliaraeteristic anftaiinonie ratios of 

 tfiese pseui/o-transf()r mat ions are o a)id qd respect ivetx. 



As stated above, tlie identical transformation of tlie group G^ is 

 produced by the degenerate conic Ooo parallel to A A.' O is the 

 point of contact of this conic with 1. The conic of the system S 

 touching 1 at ^/O, a point inhnitesimally near to O, produces an 

 //-(/^"////ovV//^?/ transformation; i. e. a transformation which shifts every 

 point of the line 1 an infinitesimal distance. The point dO may be 

 either to the right or to the left of O. In the one case the infinit- 

 esimal transformation is positive, in the other negative. The two 

 ma}' be expressed b}' the same analytical formula; and hence Lie's 

 theorem asserts that Ci j contains one and only one infinitesimal 

 transformation. Let i liic/ represent its characteristic anharmonic 

 ratio. If the infinitesimal transformation be repeated an infinite 

 number of times, the result will be ecpiivalent to a transformation 

 of the same group, whose characteristic anharmonic ratio is (i ±:^/)". 

 This is a finite quantity different from unity. Hence we infer that 

 the whole group may be generated from this infinitesimal trans- 

 formation. 



Propertx 6. — llie ,i;'ronp G, contai/is one and o)il\ one inpnitesiniat 

 transfo)inati(>n : tfie lofiole i^^i'oup max l>e i:;enej-ated prom tfiis infnitesi- 

 mal transformation. 



Let us next consider transformations of the type T|^ , which leave 

 one and onl}' one point A invariant. It is evident that c»i hyper- 

 bolas can be drawn touching 1 and 1' and having AA' for an 

 asymptote; (see Fig. 3). Call this system S.' Every point on the 

 line 1 is the point of contact of some hyperbola of the system S ' ; 

 and hence we infer that the 00' transformations determined by S' 

 form a continuous system. 



Let T;^ denote a transformation of this system which transforms 



