point on the curve; that is, the tangent at a point of inflexion is stationary; therefore the ratios § : ( and ʼn : « do not vary, if x, y, z vary; let us take the x-, y-, z-partial derived-functions of (115) and we have and this is the Hessian of F (x, y, z) = 0. As to its dimensions it is to be observed that as each of the second partial derived functions is of the (n-2)th degree, so is the equation of the 3 (n-2)th degree; and the Hessian curve therefore has 3n (n-2) points common with the original curve: hence we infer that 3n(n-2) is the number of points on a curve at which the tangent is stationary. 247.] Stationary tangents are also a particular form of certain straight lines which touch a curve at many different points. Such lines are called generally Multiple Tangents; of which particular species are double, triple, ... tangents, which are so named according as they touch the curve at two, three, ... points. Let us consider the most simple case in which a curve exhibits this property; that, viz. in which the axis of x is a double tangent. In the equation to the curve let y = 0; then, if the curve is of the nth degree in terms of x, we have an expression of the form, (119) which may be resolved into its factors; and we have If all the roots of this equation are real and unequal the curve cuts the axis of x in n different points, and the curve is such as that drawn in fig. 23. But let us suppose two of these roots to be equal; then two of the points of section with the axis of a coincide, and the axis of x touches the curve at the point. Let two pairs of roots be equal; that is, suppose a1 = ɑ2, and a3 = ɑ4, so that (82) becomes (x-α1)2 (x-α3)2 (x —α5) ... (x−a,) = 0; (120) then the axis of x touches the curve at x = a1, and at x = aз; and we have a double tangent. Similarly, if three pairs of roots are equal, the axis of x is a triple tangent: and similarly as many pairs of equal roots as there are, such is the order of the multiplicity of the tangent. It is evident that a double tangent cannot occur in a curve, the order of which is lower than the fourth and a triple tangent cannot be in a curve whose order is lower than the sixth; and so on for multiple tangents of a higher order. Suppose, however, three roots of (118) or (119) to be equal; say a1 = α2 = α3; then (x —α1)3 (x —αş) ... (x —α,) = 0; (121) and thus the axis of a meets the curve in three consecutive points; and thus two consecutive tangents, those, viz. which pass through the first and second, and the second and third points become coincident; for this reason the tangent to the curve at such a point is called a stationary tangent, and the point is evidently a point of inflexion; for if, in fig. 23, A1, A2, A3 become consecutive points, the intermediate points of maximum and minimum ordinates will become coincident with them; the curve will have two tangents coincident with the axis of x, and the curve having been below the axis of a will intersect it, and pass to the upper side of it. It is evident also that under these circumstances dy becomes stationary, and therefore dx d dy dx dx = d2y As a straight line cannot generally, omitting infinite values, cut in three points a curve of any degree lower than the 3rd, so points of inflexion do not generally exist on curves of degrees lower than the third. I say generally, because the hyperbola, which is of the second degree, has a rectilinear asymptote, with which the curve at infinity coincides, and in this respect more than two points of the curve are on the tangent, and there is a point of inflexion. If four roots of (118) are equal then we have PRICE, VOL. I. 3 с (x — α1)1 (x — α5) ....... (x—an) = 0: ... (122) and thus the axis of x meets the curve in four consecutive points, and three tangents become coincident. A point at which these circumstances occur is called a point of undulation. Into this subject however it is unnecessary to enter further, as the principle of the preceding inquiry can be easily extended to any other number of points. I may observe however that if the number of consecutive points common to the curve and the tangent is even, the curve is on the same side of the tangent on both sides of the point of contact, and there is no change of curvature; whereas, if the number of points which are common to the tangent and the curve is odd, the curvature changes, and the curve, having been on one side of the tangent, intersects it, and comes to the other; that is, the curve having been convex, say, downwards, becomes concave, or vice versa. SECTION 4.-On multiple points of plane curves. 248.] Thus far we have considered the geometrical properties which belong to the first and second derived-functions of an equation to a curve when they have determinate values; but dy suppose that at any point on a curve, assumes the inde terminate form, a question occurs, and which has to be inquired into, What is the meaning of such indeterminateness? In the following discussion we shall find it most convenient to use the implicit form of the equation to the curve; viz. Suppose that at a certain point on the curve conditions of which are that (d) = 0, = 0; and let the dy dr expression be evaluated according to the method of Art. 139; then, see (45) of that Article, we have And suppose that this quantity does not become at the point in question; that is, suppose that all the second partial differential coefficients do not vanish or become infinite; then, multiplying and reducing, we have dy (126) which is a quadratic in d; and gives therefore two values for dy dai dx and thereby shews that two branches of the curve pass through the point. This is called a double point, admitting of several varieties, according as the roots of (126) are real and unequal, real and equal, or imaginary; and according as the curve extends or not in the plane of reference on both sides of the point in question. Now the roots of (126) are real and unequal, real and equal, imaginary, d2F according as (1,3) (d) is {-}-(171)". dy2 = dx dy (127) Let us first consider the case of the two roots being unequal; then we have, dy dx − (drdy) ± { (drdy)" — (d23) (dy')}* d2F dy2 (128) Now although in algebraical expressions, so far as our knowledge goes, a quantity does not pass from + to or vice versá without passing through zero or infinity, yet we may be needlessly restricting the geometrical properties of equations, and our means of interpreting them, if we assume that in all equations, transcendental as well as algebraical, no change of sign can take place unless the quantity passes through zero or infinity. I shall suppose therefore that the sign of d2F 12 d2F d2r dx dy dx2 dy2 may be changed and yet that the quantity may not become zero or infinity; that is, that the both sides of the point in question, then the curve extends in the plane of reference on both sides, as at Po in fig. 57, and has the same tangents, and the point is called a real double point. dy dx If the two values of are unequal, and are real on one side of the point in question and imaginary on the other, the point is such as that indicated in fig. 58: the curve is in the plane of reference on one side, and in another plane on the other, and the tangents are similarly so; such a point is called a salient point: evidently no algebraical equation can express a curve with a salient point, because the curve will be discontinuous at dy dx are imaginary on both sides. the point. If the two values of of the point, that is, if the two roots of (126) are imaginary, then the curve is out of the plane of reference on both sides of the point, and thus two branches of the curve, both of which are out of the plane of reference, pierce the plane at the point in two different directions, and the point is called a conjugate point. Again, if two roots of (126) are equal, we have and thus two branches pass through the point and have the same tangent. If these branches are in the plane of reference on both sides of the point, the curve is such as one or the other of those delineated in fig. 59, and the points where the curve meets the tangent are called points of osculation, and by French writers |