WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 57 



edge meets the quadric twice and no more ; then every double edge on 

 which an apparent double point of the original quartic lies is met by the 

 original quartic in two distinct points, and therefore the residual quartic 

 must cross this double edge at the same two points, since there are two 

 and only two branches of the complete intersection at these two points 

 where the double edge meets the quadric ; therefore, for every apparent 

 double point of the original quartic there is an apparent double point of 

 the residual quartic. The double or cuspidal edge, on which an actual 

 double point or cusp of the original quartic lies, meets the quadric only 

 once there; and must meet it at one more point, which is, therefore, a 

 double point or cusp on the residual quartic. A cuspidal edge due to an 

 apparent cusp on the original quartic meets the quadric in two consecu- 

 tive points, and the residual quartic must pass through these two con- 

 secutive points, i. e. it has this cuspidal edge as a tangent and therefore 

 the residual quartic also has an apparent cusp at this point. The re- 

 sidual quartic is therefore a " quartic of the first kind " similar to the 

 original quartic. Each of these two "quartics of the first kind" consid- 

 ered as lying on the quadric, meets the generators of each system of the 

 quadric in two points, and is therefore a 4., on the quadric ; the formula 

 for the number of intersections of two curves a a and b$ on a quadric 

 being (rt a , bp) = a /3 + b a — 2 a ft, the two quartics in question, con- 

 sidered as lying on the quadric, intersect in (4 2 , 4 2 ) = 8 + 8 — 8 = 8 

 points ; but on the cone these quartics are 4/s, and, since they do not 

 go through the vertex, formula (1) gives the total number of their 

 intersections, so that regarding these quartics as lying on the cone they 

 intersect in (4 1? 4 t ) = 4 -f 4 — 4 = 4 points. This illustrates what 

 was said (p. 25) in reference to the point of crossing of two curves on a 

 multiple line not counting as a point of intersection of those curves, con- 

 sidered as lying on the surface having the multiple line, when those 

 curves lie on different sheets at this point of crossing. In the present 

 case we have four such points, two on each of the two double or cus- 

 pidal edges on which the apparent double points or cusps lie ; these four 

 points count as points of intersection of the quartics considered as lying 

 on the quadric, because the quartics lie on the same sheet of the quadric, 

 but they do not count as points of intersection of the quartics, considered 

 as lying on the cone, because these curves lie on different sheets of the 

 cone at these points ; these four points make up the difference in the 

 number of intersections that the quartics have on the two surfaces. 



The "quartic of the first kind" may lie on a cone with a tacnodal 

 edge formed by the union of two double or cuspidal edges ; the quar- 



