the initial position, the time of this plane's turning through two right angles is the same, namely, π a 198. If we eliminate t between (7) and (8), we find This is of the same form as the polar equation to an ellipse about the center. The projection of the particle's path on a horizontal plane is therefore approximately an ellipse, its semiaxes being aa, aß. 199. To determine approximately the apsidal angle. At an apse z is of course a maximum or minimum, and dz therefore =0. This gives, by § 195 (4), dt (a2 — z2) { V2 — 2g (k − z)} — h2 = 0 ....................... (1), ......... whose two positive roots are the alternate values of z at the apses. Since we have supposed the particle to have been projected horizontally, the point of projection is an apse; and therefore k is a root of this equation. Substituting k for z, we get h2 = V2 (a2 — k2) ; therefore (1) becomes after reduction (k − z) {(k+ z) V2 — 2g (a2 — z2)} = 0 ......... And, if I be the other positive root of (1) or (2), we have (2). Hence the apsidal angle or the value of from z = k to 2 = l, is ra-k = To get rid of y put zaw, the integral becomes da ☛ (2a–∞) | {(a−k) − ∞} {∞ − (a−1)} {((a+ (a+k) (a+1) and, expanding in powers of those factors whose variation is small compared with themselves, we have finally The integration may easily be carried on farther, all the terms being evidently positive, but we have enough to shew that the apsidal angle is greater than and that therefore in π 2 the approximate elliptic path considered in last article the apse continually progredes. In the case of this orbit, if p and q be its semiaxes, we have, by the properties of the sphere, and the rate of progression of the apse therefore varies as the area of the projected orbit nearly. 200. To determine the nature of the small oscillations executed under the action of gravity, on a smooth surface, by a particle about a position of stable equilibrium. The tangent plane at the position of equilibrium must be horizontal, and the surface must evidently lie above it in order that the equilibrium may be stable. If p, p, be the radii of curvature of the principal normal sections, and if the axes of x and y be tangents to these sections respectively, at the point of contact with the horizontal plane, we know by Analytical Geometry that the equation to the surface in the immediate neighbourhood of the origin is The equations of motion of the particle are, as in § 191, where A, μ, v are the direction-cosines of the normal to the surface at the point x, y, z. Since x and y are very small, z is of the second order of small quantities by (1) and may there d2z dt2. y μ v=1, approximately. Elimi P1 nating R from equations (2), we have which show (§ 173) that the motion consists of superposed simple pendulum oscillations in the principal planes, the lengths of the pendulums being the corresponding radii of curvature. The annexed cut shows a very simple arrangement, due to Prof. Blackburn of Glasgow, by which this species of con B straint may easily be produced. Three strings are knotted together at the point C, the other ends A and B of two of them are attached to fixed points, and the third supports the heavy particle D. Suppose CE to be vertical, then the small oscillations of D will evidently be executed as if on a smooth surface whose principal planes of curvature at D are in, and perpendicular to, the plane of the paper. The radii of curvature in these planes are CD and DE respectively. 9 If we put =n, and =n, the integrals of (3) are P x= A cos (nt + B), Į y = Дcos (n ̧t+B1). ...(4). The curves corresponding to these equations are very interesting, but we cannot enter at length on the consideration of them. We may take, as a special case, that in which DE 4CD; in which therefore = x = A cos (nt +B), ....... (5). The circumstances of projection determine in each case the particular curve described-a few of the principal forms are sketched below, one of which is a portion of a parabola. 8888< When n1 is nearly, but not exactly, equal to 2n, the curve described is always for a short time approximately one of the above figures, but its form slowly passes in succession from one member of the series to the next, completing the round when one pendulum has executed one more or less than twice as many complete oscillations as the other. 201. To find the Brachistochrone for a particle constrained to move on a given smooth surface, gravity being the only impressed force. |