712 MECHANICS orbit is (at any instant) an ellipse, but in which the ellipse gradually changes its form and position, so as to be always inscribed in a definite rectangle. This experimental arrange ment is exceedingly instructive. To avoid as far as may be the effects of resistance of the air, the vibrations should be slow, i.e., the wires should be as long as possible. The bob should be a ball of lead, containing a tube full of ink which slowly escapes from a fine orifice at its lower end, so as to make a permanent record of the path on a sheet of paper placed below the plane of motion of the bob, but parallel and very close to it. Or, the bob may be furnished with a spike at its lower end, from which induction sparks may be taken so as to pierce a sheet of paper laid on a copper plate below it, By mere alterations of the point of suspension A, the ratio of o>, w may be varied at pleasure, provided that AC and BC are long enough compared with CP. Lissa- 157. Lissajoux produced similar curves by attaching joux s plane mirrors to the legs of tuning-forks, and allowing a forks g ~ ra y ^ ^ nt) a ^ ter success i ve reflexions from two such mirrors, to fall on a screen. But it seems to have been first pointed out much earlier by Sang, and afterwards developed by Wheatstone, that the same result is obtained by fixing firmly one end of a steel rod, and setting the free end in vibra tion. There are two planes of greatest and least flexural rigidity ( 274) in all wires, however carefully drawn. These are at right angles to one another ; and the motion of the free end of the wire when slightly disturbed is there fore precisely that of the bob of the Blackburn pendulum. Another interesting mode of producing the same result is by causing a ray of sunlight to be reflected in succession from four mirrors, all attached, nearly at right angles, to parallel axes. One pair is made to rotate, the two in opposite directions, with one angular velocity. A ray reflected in succession from these is ( 65) made to oscillate according to the simple harmonic law, in a plane which can be varied at pleasure by altering the relative position of the normals to the two mirrors. The other pair of mirrors supplies the other simple harmonic motion, also in any desired plane. Foucault 158. We must next consider the effect of the earth s pendu- ro tation upon the motion of a simple pendulum. Strange to say it was left for Foucault to point out, in February 1851, that the plane of vibration of a simple pendulum suspended at either pole would appear to turn through four right angles in twenty-four hours, the plane, in fact, remaining constant in position while objects beneath the pendulum were carried round by the diurnal rota tion. At the equator, it was pretty obvious that no such effect would occur, at least if the original plane of vibration was east and west. By some process, of which he gives no account, Foucault arrived at the result that the plane of oscillation must, in any latitude, appear to make a complete revolution in 24* x cosec latitude. This curious result has been amply verified by experiment. The equations of motion of the pendulum, referred to rectangular axes fixed in direction in space and drawn from the earth s centre, the polar axis being that of z, are obviously with similar expressions in y and z (a, b, c being the coordinates of the point of suspension, T the tension, Z the length of the string, and X, Y, Z the components of gravity). The equations of motion referred to a new set of axes parallel to the former, but drawn through the point of suspension, are &c. =&c. X- cPa df 2 (1). Let us now refer the motion to axes turning with the earth, but drawn from the point of suspension. If the axis of | be drawn vertically, and the axes of 77, respectively southwards and east wards ; and if <ai be the angle at time t between the planes of xz and |TJ, A being the co-latitude of the point of suspension, we have (assuming that | intersects z} cos # = sin A cos cat , , &c. By means of these expressions we can at once find the values of x-a, y -b, z - c in terms of |, rj, , t, as follows : y-b = |s z - c = I cos A - f] sill A . Let 7 be the acceleration due to the attraction of gravity alone., and v the angle (nearly equal to A) which its direction makes with the polar axis. [We have above in effect assumed that its direction lies in the plane of s|, as we have assumed that the axis of intersects the polar axis, while we know that the centrifugal force- lies in their common plane.] Let r be the distance of the point of suspension from the earth s centre, p the angle its direction with the polar axis. Then With these data we transform equations (1) from x, y, z to , 17, The equations immediately obtained are inconveniently long for our columns. But they are easily simplified as follows. We contemplate small vibrations only ; so we may treat | as- being practically equal to - I, and omit its differential coefficients. We also omit powers and products of rj, and all terms in s , except those in which it is multiplied by a large quantity. For it is known that the centrifugal force at the equator is about l/289tlx of gravity, or that approximately rw 2 - 0/289. With these considerations, and the condition that to the degree of approximation desired we have T = my, we still further simplify our equations. We are led to recognize that 7003 v = g cos A ; and thus we have finally dt I * These are the equations of the motion of the bob, referred to n horizontal plane fixed to the earth. The middle terms obviously depend upon the earth s rotation. To interpret equations (2) it is convenient to employ a second change of coordinates to refer the motion to axes revolving uni formly in the plane of i}, , with angular velocity ft. If p, ? be the coordinates referred to the new axes, we have by analytical geometry i = J) cos fit - $ sin tit, = p sin nt + ? cos nt, the substitution of which in (2) leads to the equations provided we take H=-a>cosA , ...... (4), and omit as before terms of the order &> 2 . (4) shows that the new axes rotate, in the opposite direction to that of the earth, with the component of the earth s angular velocity about the vertical at the place. And, in the plane so revolving, we see by (3) that the bob of the pendulum describes an approxi mately elliptic orbit, of which a straight line is a particular case. A circular path being obviously possible, let us assume as par ticular integrals of (2) The substitution of these values gives the same result in each of equations (2). Put g/l = n 2 , then the values of p are, to the degree of approxi mation above employed, t WCOSA, so that the (apparent) angular velocity of a conical pendulum is increased or diminished by OICOSA according as its direction of rotation is negative or positive. 159. The preceding problem is a particular case of the Vary in following general one. To find the motion of a particle con -. subjected to the action of given forces and under varying stuimt constraint. It would lead us to details incompatible with our limits to enter upon a full discussion of so wide a question, but we give one or two simple and useful cases to siiow the commoner forms of procedure. A particle under any forces, and resting on a smooth horizontal
plane, is attached by an incxtcnsible string to a point which mores