Surface reciprocal to a (jiven one. 485 



planes, with respect to the two surfaces, shall meet an arbitrary right Hue, aud 

 this is immediately found to be a surface of the degree (;« + n— 2). Now the 

 curve IX meets this surface either in points in which the two given surfaces 

 have common tangent planes, or in the p points, the tangent to fx at which 

 meets the arbitrary right line.* Hence, in the case we are discussing, the 

 curve fx intersects the curve (n — 1)- - /j in fx (2n - 4) — /> points. But of 

 these p are the points the tangents to jx at which meet the arbitrary line 

 through which we are seeking how many tangent planes can be drawn to the 

 surface, and the remaining /i ( 2?i — 4 ) — 2/) are cuspidal points. And the formula 

 lor the intersection of the three surfaces is 



nS(/i-l)=-;u|-2/,-3JM(2«-4)-2pi, 



or the diminution in the degree of the reciprocal is /^ ( 7n — 1 2 ) — Ap. 



The surface is intersected by any polar surface in the curve fx (reckoned 

 twice), and in a curve w(7i — 1) — 2/i, which meets fx in the cuspidal points, 

 and in the /x(rt — 2) other points, where the curve meets the second polar 

 surface. 



In like manner, if the curve fx be of the order p of multiplicity on the given 

 surface, the points c of special contact will be in number 



2/i (/*-!) (tt-p) -p{p-'^)p; 



the points b will still be /^ {n — p), and the edges a, where the cone of simple 

 contact intersects the cone standing on the multiple line, will be 



fx{n-p) {n-p - 1) - fx{ix- I) p(p- 1) +p(p- I) p. 



Hence, proceeding precisely as before, we obtain for the reduction in the de- 

 gree of the reciprocal, 



ix{p - I) {3p + 1)71 - 2p^p(p' - 1) -j/{p - 1) p^ 



for the reduction in the number of cuspidal edges of the cone of simple contact, 



,x\3ip-iyn-p{p-l) (2p-l)\-p{p-l) {p-2)p; 



* I owe to Mr. Cayley this demonstration of a theorem, of which I have given a less satis- 

 factory proof (" Canib. and Dub. Math. Journal," vol. v. p. 35). 

 VOL. XXIII. 3 S 



