66 KANSAS UNIVERSITY QUARTERLY. 



three points lie on a circle through the triple point. Let these three 

 points be designated by A, B, and C. The lines from the triple point 

 O to the points A, B, C, and the common chord of the osculating cir- 

 cles at two of them form a harmonic pencil. Through one of these 

 points, A, and the triple point draw a circle touching the quartic; the 

 point of contact is on the common chord of the osculating circles at B 

 andC. 



From theorems which we have already proved for a system of cubics 

 having a common node and passing through five others fixed points, 

 we can infer other theorems for a system of quartics having a common 

 triple point and passing through seven other fixed points. For example, 

 any conic through the common double point and two of the fixed 

 points is cut by the cubics in pairs of points which determine at the 

 node a pencil in involution. Hence any cubic having its node at the 

 common triple point and passing through any four of the fixed points 

 is cut by the quartics in pairs of points which determine at the com- 

 mon triple point a pencil in involution. Again, the pairs of tangents 

 to the cubics at the common double point form a pencil in involution, 

 the two cuspidal tangents being the foci of the pencil. Inverting: — the 

 line at infinity (which passes through two of the fixed points, i. e. 

 the circular points) cuts the system of circular quartics in pairs of 

 points in involution. Projecting: — a line through any two of the seven 

 fixed points cuts the system of quartics in pairs of points in involution. 

 Since the line at infinity touches the inverse of a cuspidal cubic, it 

 follows that any line through two of the fixed points will touch two of 

 the quartics of the system; these points of contact are therefore the 

 foci of the involution. 



Other theorems on such a system of quartics will be given in the 

 next section. 



SYSTEMS OF QUARTICS THROUGH SIXTEEN POINTS. 



Let U and V represent a system of quartics through sixteen points. 

 Since the discriminant of quartic is of the 27th degree in the coefficients 

 it follows that there are 27 values of k for which the discriminant 

 vanishes, and hence 2 7 quartics of the system which have double points. 

 As in case of cubics these 27 points are called the critic centres of the 

 system. Let the equation of the system of quartics be written 



U4+U3+U2+Ui+Uo=0. 



In a manner similar to that employed for cubics, we find the equa- 

 tion of the polar cubics of the origin with respect to the system to be 



U3+2U2+3U^+4Uo=0. 



The polar conies of the origin are given by 



