31. Curves of the fifth degree, for which a = 4, lie on a surface of the second degree, and are met thrice by the generatrices of one system, and twice by those of the other. The cone whose vertex is any point of the curve has one generatrix for a double edge and the other for a single edge, and consequently meets the surface of the second degree nowhere else save on the curve. Mr. Cayley shews as follows, that such a curve is determined by 11 points on the surface of the second degree, or by 20 conditions. For being given eleven such points, we are given the surface of the second degree, and if through the 11 points, through 4 points on a generating line, and through 4 points not on the surface we describe a surface of the third degree, this must be a proper surface, since the last 4 points are supposed not to be in the same plane; it must contain the assumed generating line, which would otherwise meet the surface in 4 points, and it therefore meets the surface of the second degree in a curve of the fifth degree. Two curves of the fifth degree can be described according as the assumed generatrix is of one or the other system; but all generatrices of the same system give rise to the same curve of the fifth degree. With regard to the class a = 5, I have nothing satisfactory to offer: but the class a = 6 is the reciprocal of the system At + 5Bť + 10 Ct3 + 10Dť2 + 5 Et + F. I have found the equation of the resulting developable, but I do not give it here, as I have not got it in a form to exhibit the cuspidal and nodal edges of that surface. This system is determined by 10 points or 20 constants, and in general the system At+B+Ct-2+ &c. is determined by 4k conditions. For the equation involves m + 1 planes, and has therefore 4m + 3 arbitrary constants. But the equation in its most general form must contain 3 indeterminate constants, for it is not necessary to determine which plane of the system answers to t = 0 or t ; and the system remains A (t - t') Thus the system the same if we substitute for t, = t- t" (Ax + By) + t.(A'x + B'y) is precisely the same system as x + ty. There remain therefore but 4k conditions to be used in determining the system. I have not however been able to ascertain the number of conditions in general necessary to determine a given curve. The characteristics of the system Atk + B+ &c., are m = 3 (k − 2), n=k, r = 2 (k − 1), a = 0, ẞ= 4 (k − 3), (k − 1)(k − 2) 2 July 21, 1849. ON THE DEVELOPABLE SURFACES WHICH ARISE FROM TWO SURFACES OF THE SECOND ORDER. By ARTHUR CAYLEY. ANY two surfaces considered in relation to each other give rise to a curve of intersection, or, as I shall term it, an Intersect and a circumscribed Developable* or Envelope. The Intersect is of course the edge of regression of a certain Developable which may be termed the Intersect-Developable, the Envelope has an edge of regression which may be termed the Envelope-Curve. The order of the Intersect is the product of the orders of the two surfaces, the class of the Envelope is the product of the classes of the two surfaces. When neither the Intersect breaks up into curves of lower order, nor the Envelope into Developables of lower_class, the two surfaces are said to form a proper system. In the case of two surfaces of the second order (and class) the Intersect is of the fourth order and the Envelope of the fourth class. Every proper system of two surfaces of the second order belongs to one of the following three classes:-A. There is no contact between the surfaces; B. There is an ordinary contact; C. There is a singular contact. Or the three classes may be distinguished by reference to the conjugates (conjugate points or planes) of the system. A. The four conjugates are all distinct; B. Two conjugates coincide; C. Three conjugates coincide. To explain this it is necessary to remark that in the general case of two surfaces of the second order not in contact (i. e. for systems of the class A) there is The term 'Developable' is used as a substantive, as the reciprocal to 'Curve,' which means curve of double curvature. The same remark applies to the use of the term in the compound Intersect-Developable. For the signification of the term 'singular contact,' employed lower down, see Mr. Salmon's memoir On the Classification of Curves of Double Curvature,' p. 23. a certain tetrahedron such that with respect to either of the surfaces (or more generally with respect to any surface of the second order passing through the Intersect of the system or inscribed in the Envelope) the angles and faces of the tetrahedron are reciprocals of each other, each angle of its opposite face, and vice versa. The angles of the tetrahedron are termed the conjugate points of the system, and the faces of the tetrahedron are termed the conjugate planes of the system, and the term conjugates may be used to denote indifferently either the conjugate planes or the conjugate points. A conjugate plane and the conjugate point reciprocal to it are said to correspond to each other. Each conjugate point is evidently the point of intersection of the three conjugate planes to which it does not correspond, and in like manner each conjugate plane is the plane through the three conjugate points to which it does not correspond. In the case of a system belonging to the class B, two conjugate points coincide together in the point of contact forming what may be termed a double conjugate point, and in like manner two conjugate planes coincide in the plane of contact (i. e. the tangent plane through the point of contact) forming what may be termed a double conjugate plane. The remaining conjugate points and planes may be distinguished as single conjugate points and single conjugate planes. It is clear that the double conjugate plane passes through the three conjugate points, and that the double conjugate point is the point of intersection of the three conjugate planes: moreover each single conjugate plane passes through the single conjugate point to which it does not correspond and the double conjugate point; and each single conjugate point lies on the line of intersection of the single conjugate plane to which it does not correspond and the double conjugate plane. In the case of a system belonging to the class (C), three conjugate points coincide together in the point of contact forming what may be termed a triple conjugate point, and three conjugate planes coincide together in the plane of contact forming a triple conjugate plane. The remaining conjugate point and conjugate plane may be distinguished as the single conjugate point and single conjugate plane. The triple conjugate plane passes through the two conjugate points and the triple conjugate point lies on the line of intersection of the two conjugate planes, the single conjugate plane passes through the triple conjugate point and the single conjugate point lies on the triple conjugate plane. Suppose now that it is required to find the IntersectDevelopable of two surfaces of the second order. If the equations of the surfaces be T = 0, Y′ = 0 (Y, T' being homogeneous functions of the second order of the coordinates , n,, w), and x, y, z, w represent the coordinates of a point upon the required developable surface: if moreover U, U' are the same functions of x, y, z, w that Y, Y' are of §, n, s, w and X, Y, Z, W; X', Y', Z', W' denote the differential coefficients of U, U' with respect to x, y, z, w; then it is easy to see that the equation of the Intersect-Developable is obtained by eliminating §, n, 5, w between the equations r = 0, r' = 0, Xέ + Yn + ZC + Ww = 0, X' + Y'n + Z'Ç+ W'∞ = 0. If, for shortness, we suppose F = YZ' - Y'Z, G = ZX' - Z'X, Ĩ = XW' L= XW' - X'W, M= YW' - Y'W, H=XY-X'Y, N = ZW - Z'W, (values which give rise to the identical equation LF + MG + NH = 0), then, λ, μ, v, p denoting any indeterminate quantities, the two linear equations in §, n, Š, w are identically satisfied by assuming and, substituting these values in the equations Y = 0, Y′ = 0, we have two equations: Αλ + Β' + Cv + 2Fμν + 2G + 2 Ηλμ + 2Ľλp + 2Mμp + 2Nvp = 0, A'λ2 + B'μ2 + C'v2 + 2Fμv + 2G'vλ + 2H'λμ + 2L'λp + 2M'μp + 2N'vp = 0, which are of course such as to permit the four quantities λ, u, v, p to be simultaneously eliminated. The coefficients of these equations are obviously of the fourth order in x, y, z, w. Suppose for a moment that these coefficients (instead of being such as to permit this simultaneous elimination of λ, μ, v, p) denoted any arbitrary quantities, and suppose that the indeterminates λ, μ, v, p were besides connected by two linear equations, aλ + bμ + cv + dp = 0, a'λ + b'μ + c'v + d'p = 0. b'cf, ad' a'd = l, Then, putting bc' cd' c'd = N, (values which give rise to the identical equation If+mg+nh=0), and effecting the elimination of λ, μ, v, p between the four equations, we should obtain a final equation □ = 0, in which is a homogeneous function of the second order in each of the systems of coefficients A, B, &c. and A', B', &c. and a homogeneous function of the fourth order (indeterminate to a certain extent in its form on account of the identical equation If+mg+nh = 0) in the coefficients f, g, h, l, m, n.* But reestablishing the actual values of the coefficients A, B, &c., A', B', &c. (by which means the function becomes a function of the sixteenth order in x, y, z, w) the quantities f, g, h, l, m, n ought, it is clear, to disappear of themselves; and the way this happens is that the function resolves itself into the product of two factors M and 4, the latter of which is independent of f, g, h, l, m, n. The factor M is consequently a function of the fourth order in these quantities, and it is also of the eighth order in the variables x, y, z, w. The factor is consequently of the eighth order in x, y, z, w. And the result of the elimination being represented by the equation = 0, the Intersect-Developable in the general case, or what is the same thing for systems of the class (A), is of the eighth order. In the case of a system of the class (B) * I believe the result of the elimination is 0 = = where if we write uA + u ́A' A, &c. the quantities P, Q, R are given by the equation (identical with respect to u, u') = Pu2+2Quu + Ru2 (Aa2 + ..) (Aa'2 + ..) — (Aaa' + ..) =u2{(BC—F2) ƒ2+..}+uu'{(BC'+B'C−2FF') ƒ2+..}+u'2{(B'C' — F°2) ƒ2+..} a theorem connected with that given in the second part of my memoir 'On Linear Transformations' (Journal, vol. 1. p. 109). I am not in possession of any verification a posteriori of what is subsequently stated as to the resolution into factors of the function and the forms of these factors. NEW SERIES, VOL. V.- E |