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XXXV. — On the Hessian. By Professor Chrystal. 



(Read 18th May 1885.) 



1. Let 



TJ=u z u + u l z n - 1 + u, 2 z n ~ 2 + . . . . = 



be the equation to an algebraical curve of the nth degree, the co-ordinates of 

 any point on which in a system of linear co-ordinates are (x, y, z), u , u x , u 2 . . . . 

 being homogeneous functions of x and y of degrees indicated by the attached 

 suffixes ; then 



H=U„U yy U„ + 2U„U«U, 1 , - U,JV - U yy U^ - U„IV = 



is the equation to its Hessian, which is a curve of the 3(rc-2)th degree. 



Every one of the 3n(n — 2) points of intersection of H and U is a point of 

 inflexion on U if it be not a multiple point on U. In this last case the inter- 

 section may or may not be a point of inflexion on some one of the branches of 

 U; but in any case where H passes through a multiple point the total number 

 Sn(n — 2) of inflexions suffers a reduction. It is therefore a problem of great 

 geometrical interest to calculate the number of the intersections of H and U 

 which are absorbed at a multiple point on the latter. This problem has never 

 been solved directly in any but a few simple cases. It has been shown, for 

 example, that at an ordinary double point on U, H has also a double point the 

 tangents at which are the same as the tangents at the double point on U, and 



that such a point absorbs 2x2 + 2 = 6 of the intersections HU; also that a 

 multiple point of order k, all of whose tangents are distinct, is a multiple point 

 of order 3&— 4 on H, k of whose tangents are tangents to U, and that such a 

 point absorbs k(3k-4:) +k = Qx ^k(k-l) intersections, — in other words, has the 

 same effect as the ^k(k — l) ordinary double points to which it may be regarded 

 as equivalent. It has also been shown that a point which is a cusp of the 

 ordinary kind on U is a triple point on H, two of whose tangents coincide with 



the cuspidal tangent of U ; this cusp counting for 2x3 + 2 = 8 among HU. 

 Finally, Cayley has laid down that every singularity of an algebraical curve 

 can be regarded, for our present purpose, as equivalent to a certain number S 

 of ordinary double points, and a certain number k of ordinary cusps. But the 

 proofs which have been given of this theory by Nother, Zeuthen, Stolz, 

 Henry Smith, and the methods given for ascertaining the indices S and k, are 

 of an indirect nature, and it has been doubted whether any proof of this theory 

 can be given by methods appropriate to co-ordinate geometry. 



VOL. XXXII. PART III. 5 Q 



