484 The Rev. George Salmon on the Degree of the 



The diminution, therefore, in the number of cuspidal edges of the cone is, 



?,{p-\yn-p{iy-\) (2p-l). 



To investigate the diminution of the number of double points: we have 

 already seen that the cone of simple contact intersects the plane through the 

 multiple line in three distinct classes of edges, viz. : a, tangents to the plane 

 section ; ft, lines to the points where the multiple line meets that section; and 

 c, lines to the points of special contact. Now, I have satisfied myself that the 

 formula for the intersections of the curve of contact with the curve which 

 determines the point of contact of double edges of the tangent cone is 



(tt-2) (n-i)\n{n-\)-p{p-\)\-p{2y-l)a-2{p-l) {p-2)b 



-{p-2)(p-3)c. 

 Putting in this formula the values 



a = (n-p) (n-p-l), l) = n-p, c = 2(p-l) (n-p), 



we obtain for twice the reduction in the number of double edges, 



2p(p-l)n'-{p-l) {Up-8)n-p(p-l) (p'-9p + 2). 



Now since the degree of the tangent cone is reduced from n{7i - I) to n{n- I) 

 -.p(p-l)^ the degree of its reciprocal is reduced for this reason alone, by 



2p{p-l)n'-2p{p-l)n-p{p-l) (p^-p+l). 

 Subtract from this twice the reduction in the number of double edges, and 

 three times the reduction in the number of cuspidal edges, and we get the same 

 value as before for the reduction in the degree of the reciprocal. 



I now proceed to examine the effect on the degree of the reciprocal produced 

 by a multiple curve in general, and commence with the case of a double curve, 

 which is supposed to be of the degree /x, and rank p (the rank being the degree 

 of the developable generated by its tangents). The intersection of the two polar 

 surfaces consists of the curve ju, and of another curve of the degree (n — iy-fj., 

 and the question is, in how many points do these two curves intersect. Now, 

 I say, in general that if two surfaces of degrees wi and n have the curve fx 

 common, it will intersect the remaining curve of inter section in yu(m + ?i — 2) — /> 

 points. In fact, we might seek the points on the curve /x where the surfaces 

 touch, by first finding the locus of points such that the intersection of its polar 



