301 



If two rays of a group coincide, we have'). 



ka^ -\- Ib^ -\- mc^ = 



ka^ -\- lb, + "'^2 = 0. 

 We lliid, therefore, for the curiw of coinculences 



<h ^1 ^i I 

 rt, h, c\ = 0, 

 X, X, .>; I 

 i.e. a y' with quadruple point S^. 



Tills result was lo 1)6 foreseen : for the net [c*] lias moreover 

 4 single base-points *S^; the Jacobian has consequently 4 nodes Sk 

 and an octuple point \, breaks up, therefore, ijilo four rays S^S^ 

 and a y^. 



If the thiee rajs of a group of the involution /,* coincide, we have 



rtjj 0,1 C,j 



a,, 6i, c,, =0. 



n h c 



'■'■it '^ii '-jj 



There are consequently tiiree groups of the /, in which the three 

 points coincide; their lines t are stationary tangents of the cui-ve r. 



As {Fy has now a triple point in S^, P bears only /cwr straight 

 lines t. The curve t is consequently of class 4; as it must be of 

 the genus null and lias 3 stationary tangents, it is a curve of order 

 three. 



The /j' has a neutral pair; these two sti-aight lines form a c* with 

 the conic that passes through the five singular points. 



13. The net determined by 



= 



has 12 base points, consequently produces an /^. If', however, the 

 6 conies corres[)onding to the 6 qiuidratic functions, all pass thiongh 

 a point aS'o, the curves of [c.^] have a node in S^ and pass fui'ther 

 through 9 fixed points besides. The variable base-points of the pencils 

 (c") form now an I ^. This triple involution of the tkird class I have 

 fullly investigated in a paper, printed in volume XVII, p. 134 of 

 these "Proceedings". In a paper |)ublished in volume XVII, p. 105, 

 a triple involation of the second class is to be found; its groups 

 are arrived at by intei'secting any conic of a pencil with any curve 

 of a pencil (c*); the two pencils viz. have three base-points in common. 



1) By flji-is meant .by aki the form 



d.r/, d.rj^dxi 



20 



Proceedings Royal Acad. Amsterdam. Vol. XXI. 



