PROF. W. K. CLIFFORD ON MR. SPOTTISWOODE’S CONTACT PROBLEMS. 717 
made up of two lines, passing each through two of the six points, and a quartic having 
nodes at their intersection and the remaining two points and passing through those 
four. Let a , b, c, d, e, f be the six points, p the intersection of ab, cd ; then the sextic 
is in this case made up of 
lines ab, cd, 
quartic p 2 e 2 f 2 abcd. 
If, however, in the transformed figure the node at a principal point was the intersec- 
tion of a line with the conic, the original sextic was made up of a line, a conic, and a 
nodal cubic, viz. if p be the intersection of the line ab and the conic bcdef, the nodal 
cubic is pPacdef. 
Now we know that we can draw a singly infinite number of quartics with 3 fixed 
nodes and 4 fixed points, or of cubics with a fixed node and 5 fixed points. Hence in 
both these cases the sextic includes two representatives of straight lines in the cubic, 
together with another curve which may be arbitrarily chosen from a pencil. 
