Hence the series (57) Σ a,(\,— w)ˆ e−1,5= Σ Am+v (\m+v— w)^ e−\m+v8 λιω = σ+ Here we is summable (μ, x), where μ, λm+v, for s = σ ̧ + d. may put s= = 0. Therefore the series R* (w) = Σ ~,(^,− w)* = Σ Am+v (Xm+v — w)* λv>w (59) 0 where Also put and R* (w) = Σ a,(^„ -- w)^ e-λvo elvσ = £ b„e“vo, by = αm+v (λm + v — w)^e-\m+v°, μv=λm+vi Then we have Since the series (57) is summable (u, x) for s =σ =σ,+8, **B*(Q) tends to a finite limit, as . Therefore, corresponding to any prescribed positive number e, there exists a positive number K, independent of 7, such that for T. Integrate (60) x times by parts, observing that for all sufficiently large values of w, say for w> w. Observe that the expression e(+2) is independent of 2. Now consider the integral From (62), (65), (66), (68) and (69), we obtain, for w> w | £ ̄* C* (N) | <o(1) + e(o+2ɛ)w + Σ o (1) = e(o+2■)w + 0(1). Let 2 tend to infinity, then R* (w)|=| lim * C* (12) |≤e(o+2e)w (w>wo). 8 and e being any prescribed positive numbers. Hence we have ON THE HESSIAN, THE STEINERIAN, AND THE CAYLEYAN. * By A. B. BASSET, M.A., F.R.S. 1. THE properties of the Hessian and the Steinerian of a plane curve and a surface have been investigated by Steiner, Clebsch† and other writers; but their methods are not very intelligible, which is partly due to the fact that they have neglected to employ those valuable pieces of machinery the triangle and the tetrahedron of reference. I have always considered that the late Dr. Ferrers rendered an important service to Analytical Geometry by placing trilinear coordinates on a sound geometrical basis, thereby superseding Salinon's unsatisfactory methods of "abridged notation" and the "linear unit z." The advantages of trilinear coordinates, as expounded by Ferrers, are that the elements of the triangle of reference can be chosen at pleasure so as to suit the special problem under consideration; and many geometrical investigations are unnecessarily complicated by the neglect of its use. Similar observations apply to quadriplanar coordinates in the Theory of Surfaces, in which a tetrahedron is substituted for a triangle of reference. The object of this paper is to give simple proofs of the principal theorems by the preceding methods. References to iny treatises on Cubic and Quartic Curves and on the Geometry of Surfaces will be denoted by the letters C.Q. and G.S. respectively. 2. The definitions of the curves under consideration are as follows; and since their properties depend upon one another it will be convenient to consider them together. The Hessian is the locus of points P, whose polar conics degrade into a pair of straight lines. The Steinerian is the locus of the points of intersection Q of these straight lines. The points P and Q are called corresponding points; and the envelope of PQ is the Cayleyan. 3. Let the triangle of reference be chosen so that A is a point on the Hessian, C the corresponding point on the Ibid, vol. lxiv., p. 288. *Crelle, vol. xlvii., p. 1. Steinerian, whilst B is arbitrary. Then since the polar conic of A must degrade into a pair of straight lines intersecting at C, the equation of the primitive curve is 11-3 F= (La+Maß + Nß3) a" ́'+a"-3 u ̧+ ... u, = 0.... (1) where u,=(P, Q, R, SXẞ, y)3. Let A = d'F|da', A'= d'F|dß dy, &c., then we obtain from (1) the following equations in which only the highest powers of a are written down: An (n − 1) La", B=2Na”, C=2 (RB + 3 Sy) a”-3 A' = 2 (QB + Ry) a” ̃3, B' = (n − 3) ( Qẞ2+2Rßy + 3Sy2) a"−, C'=(n−1) Mɑ"-2 ...(2). ABC+2A'B'C' - AA"- BB-CC" 0......(3), and its degree is therefore 3 (n − 2). 4. The second polar of a point on the Steinerian touches the Hessian at the corresponding point. Since A is a point on the Hessian, the highest power of a is the (3n-7)th, and the terms which contain it are ABC-CC"; hence its coefficient is 2(n-1) {2n LN− (n − 1) M'} (RB + 3 Sy).............. (4), which shows that the line RB + 3 Sy=0 is the tangent to the Hessian at A. But the equation of the second polar of C is daF/dy3=2a" ̄3 (Rß + 3Sy) + ..., which shows that the second polar of the corresponding point C touches the Hessian at A. 5. Since the first polar of C is dF/dy=0, it obviously has a node at the corresponding point A; and we shall now show that: The tangent at a point A on the Hessian is the harmonic conjugate with respect to the line AC, joining two corresponding points, and the two nodal tangents at A to the first polar of C. Since AB is arbitrary, let it be the tangent to the Hessian at A; then R=0. Also the nodal tangents at A to the first polar of Care Q8+3Sy=0, which together with AB and AC form a harmonic pencil. |