Surface reciprocal to a given one. 465 



a (?? — 2) = A •+ p +'2a, 



b {n-2)= /» + 2fi+37 + 3/f, (A) 



c (?i-2) = 2o- + 4/3+ 7. 



I am not prepared to give a satisfactory explanation a priori of the numerical 

 coefficients in these formulas. I have obtained them by induction from an exa- 

 mination of a variety of particular surfaces. In particular, the surface which 

 is the reciprocal of a surface of the third degree, and whose singularities can, 

 without difficulty, be determined, furnished all the formulas except the coeffi- 

 cient of 7, there being no points 7 on this surface. 

 We derive, then, from these equations (A), 



K={a-h-c) (n -2) 4-6/3-1- 47-}- 3<. 



But since, if the surface had no multiple lines, the number of cuspidal edges in 

 the tangent cone would be {a + 1b + 3c) (n - 2), the diminution in these caused 

 by the double lines is 



(3i-|-46') (w-2)-6/3-47-3<. 



Next, to find the number of double edges in the cone a. I use the symbol 

 [a6] to denote the ntimber of apparent intersections of a and b, that is to say, 

 the number of points where these two lines, seen from any point of space, appear 

 to intersect, though they do not actually do so. The following formulas, then, 

 contain an analysis of the intersections of a, 6, c with ^U : 



a (n - 2) (w - 3) = 28 -f 3 [ac] -f 2 [aJ], 



i.(n-2) (w-3) = 4^--f- [ai]-f3[k], (B) 



c{n-2){n-Z) = &h+ [ac] + 2[hc]. 

 Hence, 



28 = (a-2i-3c) ((i-2) (»^ - 3) -f SA-f 18A -f 12[6c]. 



But the number of apparent intersections of two curves is at once deduced 

 from the number of their actual intersections. For if cones be described hav- 

 ing a common vertex, and standing on the two curves, the common sides of 

 these cones must answer either to apparent or actual intersections. Hence, 



