Geometry on the Cubic Scroll of the First Kind. 13 



belong to the residual intersection, V,,^ gives similar pairs 

 belonging to the curve cp = 0, andU' and V'_^^ give single 

 points on the double director which belong to the residual 

 intersection, while U and V,_9^ give corresponding single 

 points belonging to the curve, which has single branches at 

 such points of the double director. 



Since k and \i enter alike in the formulae of the sub- 

 stitution it will in general be possible to interchange X and 

 }x at will in o), and also to exchange either or both \ and [x , 

 as many times as they occur in co , for arbitrary linear 

 homogeneous functions of both X and [x. However, in the 

 particular case just considered where tu ^ (Dq. U'^ or æ^. V „t 

 such changes cannot be made. 



The residual intersection arising from the passing of 

 the surface S through the curve (p , q) on 2 will consist of 

 the double director v = occurring as many times as it is 

 contained as a factor in a> and counting twice for each 

 occurrence; of the pinch-point generators X = and |x = 

 occurring as many times as they are contained as factors 

 respectively in œ and of any other generators a^X -j- b^jx = 0, 

 a.^X -j- \\i = 0, etc. as many times as they occur respectively 

 as factors in a>; but in accordance with the statement of 

 the last paragraph these residual generators may be inter- 

 changed at pleasure or may be changed in general for any 

 other generator or generators occurring all together the same 

 number of times as the degree of (u in X and [x together. 

 In no case will it be found necessary to introduce into to 

 such a factor as should, when equated to zero, represent an 

 irreducible curve on 2. Hence the double director and any 

 generator or generators may occur as a part of the residual 

 intersection, but in general a> will involve nothing more. 

 The linear director may also occur as a part of the residual 



