On the Group of 216 Collineations in 

 the Plane. 



BY H. B. NEWSON. 



§1. Introduction. 



The group of 216 collineations in the plane was discovered by C. 

 Jordan and treated by him in Crelle, Band^^, pp. 89-215; and dis- 

 cussed again by him in A//i dcUa Rcale Accademia di Napoli, Tome 

 8 (1879). This group has been further studied by Maschke in 

 Math. Amia/en, Band 33, pp. 324-330. This paper by Maschke is 

 the standard reference on the subject. 



The object of the present paper is to study the geometric prop- 

 erties of the group and its sub-groups with respect to a pencil of 

 cubic curves through nine points of inflection; to determine the 

 types of collineations entering into the group; to determine the 

 order of each transformation and the position of the invariant tri- 

 angle in each case. 



§2. The Pencil of Cubics, x^-^y^-^z^-j-emxj'zr^o. 



The theory of the group of 216 collineations in the plane is so 

 intimately related to the theory of a pencil of cubics through nine 

 points of inflection, that a resume of certain properties of such a 

 pencil is a necessary preliminary to. the study of the group. 



If m is a variable parameter, the equation, 



x3_|_y3_j_23_[_6p^XyZ=0, (i) 



represents cc^ cubics having nine points of intersection which are 

 points of inflection on all cubics of the pencil. 



From any point P on a cubic C four tangents can be drawn to C 

 exclusive of the tangent at P. The cross-ratio of these tangents is 

 constant for all points on the curve and is different for different 

 curves of the pencil. This cross-ratio k is absolutely unaltered by 

 projection, and hence two cubics can not be linearly transformed 

 into each other unless they have the same absolute invariant k. 

 The value of k in terms of m is given by the equation 



(13) KAN. UNIV. QUAR., VOL. X, NO. 1, JAN., 1001, SERIES A. 



