WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 



59 



quartic is a 4 X , and these two quartics intersect in only 4 points, 

 (4ij 4 X ) = 4. The six points, two on each double or cuspidal edge, 

 where the branches of the two curves cross, do not count as points of 

 intersection of the two quartics considered as lying on the cone, for the 

 branches lie on different sheets of the cone at these six points, and this 

 accounts for the difference in the number of intersections that the curves 

 have on the two surfaces, — a further illustration of the principle already 

 stated. 



The different species of twisted quartics that lie on the different cones 

 can be tabulated as follows, where 8 is the number of actual double 

 points, k the number of actual cusps, h the number of apparent double 

 points, k the number of apparent cusps, and T the number of apparent 

 triple points. 



5. Salmon* has divided twisted quintics into four groups, viz. group 

 (1), having four apparent double points, group (II), having five appar- 



* Geom. of Three Dimensions, § 352. 



