Kansas University Quarterly 



Vol. V. OCTOBER, 1896. No. 



Continuous Groups of Projective Transfor- 

 mations Treated Synthetically. 



BY H. B. NEWSON. 

 *Pai't ir Continued. 



i^2 Groups in the Plane. 



I have defined a projective transformation in a plane in the sense 

 in which the the term will be used in this paper, and have given a 

 simple method of constructing it. Having given four points A,B, 

 C,D, no three of which are in the same straight line, we may 

 choose as their corresponding points A',B',C',D'; thereby a pro- 

 jective transformation T of the plane is completely determined such 

 that any point P is transformed into a definite point P'. If now we 

 choose four other points A",B",C",D", as the corresponding points 

 to A',B',C',D', we would have obtained a projective transformation 

 T, transforming P directly to P". It is clear that two transforma- 

 tions T and T^ together produce the same effect as Tg. Thus it 

 may be shown in general that any two projective transformations 

 of the plane are together equivalent to some third. Therefore all 

 the projective transformations of the plane form a Continuous 

 Group of Transformations. 



The number of projective transforiuations in the plane is like- 

 wise determined from the same considerations. Having given four 

 points A,B,C,D, a transformation is determined when their corres- 

 ponding points are chosen; and there are as many transformations 

 of the plane as there are sets of four points in a plane. Since the 

 plane contains 00- points, we easily see that there are oo** such sets 

 of points and hence there are co'* projective transformations in the 

 plane. 



Another method of determining the number of projective trans- 

 formations in the plane leads to the same result. From the method 



(81) KAN. UNIV. QUAR., VOL. V, NO. a, OCTOBER, 1896, 



