WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 55 



or the next greater integer. When v = 3 or 4 we saw that a = 1, so 

 that the expression (A) that enters into eq. (3) vanishes for v = 3, as it 

 should, and is equal to unity for v — 4 ; therefore eq. (4) gives the 

 correct value of v for all twisted curves on the cone. If then we take 

 the value of v given by eq. (4), any twisted curve a a can be cut out by an 

 Ml"l a sucn tDafc fc he residual will consist entirely of edges or multiple 

 edges of the cone, and therefore formula (1) holds for all twisted curves 

 on the quartic cones. In determining the above value of v, no account 

 was taken of the actual multiple points, not at the vertex, that a a may 

 have. We have seen (p. 24) that an /rc-tuple point reduces by m — 1 the 

 number of points of a a through which it is necessary to make M* v 2 a pass 

 in order for it to contain a a , if this m-tuple point be taken as one of 

 them ; therefore, if a (a — 2 a + 4) + 4 = 1 (mod. 4) and a a has a 

 double point not at the vertex, the value of v may be taken one less 

 than that given by eq. (4) ; if a (a — 2 a + 4) + 4 = 2 (mod. 4) 

 and a a has two double points or one triple point, not at the vertex, the 

 value of v from eq. (4) is reduced by unity, and so on. If a a has four 

 double points, or two triple points, or two double points and a triple 

 point, or a double point and a 4-tuple point, or one 5-tuple, not at the 

 vertex, then the value of v is always one less than that given by eq. (4). 

 We have shown, then, that formula (1) gives the number of inter- 

 sections, aside from those at the vertex, of any two curves on any quartic 

 cone. It is to be observed that any branch of a a through the vertex has 

 an edge of the cone as its tangent at that point and that one of the two 

 consecutive points, in which the edge meets the branch there, is one of 

 the a points of a a that lie on this edge ; if a double or cuspidal edge is 

 tangent to a branch of a a at the vertex, then two of the 2 a points of 

 a a that lie on it are consecutive to the vertex ; and if a triple edge is 

 tangent to a branch of a a at the vertex, then three of the 3 a points of 

 a a that lie on it are consecutive to the vertex. If, therefore, a a and b$ 

 each have a branch through the vertex tangent to the same edge, i. e. 

 a branch of a a tangent to a branch of bp at the vertex, then one of these 

 two intersections of the curves at this point is included in the number of 

 intersections given by formula (1). In like manner, if a a and bp each 

 have a branch through the vertex, and these branches have there a com- 

 mon inflectional or cuspidal tangent or have any peculiar relation to one 

 another, the excess of the number of intersections that these branches 

 have there over the number of intersections that two arbitrary branches 

 would have there is included in the number of intersections given by 

 formula (1). 



