Tide-predicting Machine. with one end hanging down and carrying a weight, pass alternately over and under the pulleys in order, and vertically upwards or downwards (according as the number of pulleys is even or odd) from the last pulley to a fixed point. The weight is to be properly guided for vertical motion by a geometrical slide. Turn the machine now, and the wire will remain undisturbed with all its free parts vertical and the hanging weight unmoved. But now set the axis of any one of the pulleys to a distance 1⁄2 T from its shaft's axis and turn the machine. If the distance of this pulley from the two on each side of it in the other row is a considerable multiple of T, the hanging weight will now (if the machine is turned uniformly) move up and down with a simple harmonic motion of amplitude (or semi-range) equal to T in the period of its shaft. If, next, a second pulley is displaced to a distanceT", a third to a distance 7", and so on, the hanging weight will now perform a complex harmonic motion equal to the sum of the several harmonic motions, each in its proper period, which would be produced separately by the displacements T, T', T". Thus, if the machine was made on a large scale, with T, T',... equal respectively to the actual semi-ranges of the several constituent tides, and if it was turned round slowly (by clockwork, for example), each shaft going once round in the actual period of the tide which it represents, the hanging weight would rise and fall exactly with the water-level as affected by the whole tidal action. This, of course, could be of no use, and is only suggested by way of illustration. The actual machine is made of such magnitude, that it can be set to give a motion to the hanging weight equal to the actual motion of the water-level reduced to any convenient scale: and provided the whole range does not exceed about 30 centimetres, the geometrical error due to the deviation from perfect parallelism in the successive free parts of the wire is not so great as to be practically objectionable. The proper order for the shafts is the order of magnitude of the constituent tides which they produce, the greatest next the hanging weight, and the least next the fixed end of the wire: this so that the greatest constituent may have only one pulley to move, the second in magnitude only two pulleys, and so on. One machine of this kind has already been constructed for the British Association, and another (with a greater number of shafts to include a greater number of tidal constituents) is being con structed for the Indian Government. The British Association Tide-predicting machine, which is kept available for general use, under charge Machine. of the Science and Art Department in South Kensington, has ten shafts, which taken in order, from the hanging weight, give respectively the following tidal constituents*: 7. 8. The smaller elliptic semi-diurnal. The solar diurnal declinational. 9. The lunar quarter-diurnal, or first shallow-water tide of 10. The luni-solar quarter-diurnal, shallow-water tide. The hanging weight consists of an ink-bottle with a glass tubular pen, which marks the tide level in a continuous curve on a long band of paper, moved horizontally across the line of motion of the pen, by a vertical cylinder geared to the revolving shafts of the machine. One of the five sliding points of the geometrical slide is the point of the pen sliding on the paper stretched on the cylinder, and the couple formed by the normal pressure on this point, and on another of the five, which is about four centimetres above its level and one and a half centimetres from the paper, balances the couple due to gravity of the inkbottle and the vertical component of the pull of the bearing wire, which is in a line about a millimetre or two farther from the paper than that in which the centre of gravity moves. Thus is ensured, notwithstanding small inequalities on the paper, a pressure of the pen on the paper very approximately constant and as small as is desired. Hour marks are made on the curve by a small horizontal movement of the ink-bottle's lateral guides, made once an hour; a somewhat greater movement, giving a deeper notch, serves to mark the noon of every day. The machine may be turned so rapidly as to run off a year's tides for any port in about four hours. Each crank should carry an adjustable counterpoise, to be * See Report for 1876 of the British Association's Tidal Committee. VOL. I. 31 Tide-predicting Machine. adjusted so that when the crank is not vertical the pulls of the approximately vertical portions of wire acting on it through the pulley which it carries shall, as exactly as may be, balance on the axis of the shaft, and the motion of the shaft should be resisted by a slight weight hanging on a thread wrapped once round it and attached at its other end to a fixed point. This part of the design, planned to secure against "lost time" or "back lash" in the gearings, and to preserve uniformity of pressure between teeth and teeth, teeth and screws, and ends of axles and "end-plates," was not carried out in the British Association machine. Equation- II. MACHINE FOR THE SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS*. Let B, B,... B be n bodies each supported on a fixed axis (in practice each is to be supported on knife-edges like the beam of a balance). Let P1, P2, P... P be n pulleys each pivoted on B1; P. C1, Cg Ca... C, be n cords passing over the pulleys; 3 n ,,,... be the lengths of the cords between D,, E, and D, E, ... and D, E, along the courses stated above, when B1, B,,... B, are in particular positions which will be called their zero positions; 11 + e1, 4 + e... 1+e be their lengths between the same fixed points, when B, B,, ... B are turned through angles x,, ,,... x from their zero positions; (11), (12), (13),... (ln), (21), (22), (23), ... (2n), (31), (32), (33), ... (3n), * Sir W. Thomson, Proceedings of the Royal Society, Vol. XXVIII., 1878. ... (I). We shall suppose x1, „ x to be each so small that (11), (12),...(21), etc., do not vary sensibly from the values which. they have when x,, x,..., are each infinitely small. In practice it will be convenient to so place the axes of B1, B,,... B2 and the mountings of the pulleys on B1, B,, ... B, and the fixed points D1, E,, D2, etc., that when x,, x,, ... x, are infinitely small, the straight parts of each cord and the lines of infinitesimal motion of the centres of the pulleys round which it passes shall be all parallel. Then (11), † (21), ... § (nl) will be simply equal to the distances of the centres of the pulleys P11, P217 P.,, from the axis of B1; § (12), § (22) ....... § (n2) the distances of P, from the axis of B,; and so on. ... ... P P n2 In practice the mountings of the pulleys are to be adjustable by proper geometrical slides, to allow any prescribed positive or negative value to be given to each of the quantities (11), (12), ... (21), etc. Suppose this to be done, and each of the bodies B1, B1, ... B 29... B of the forces re * The idea of force here first introduced is not essential, indeed is not technically admissible to the purely kinematic and algebraic part of the subject proposed. But it is not merely an ideal kinematic construction of the algebraic problem that is intended; and the design of a kinematic machine, for success in practice, essentially involves dynamical considerations. In the present case some of the most important of the purely algebraic questions concerned are very interestingly illustrated by these dynamical considerations. Equation. stretched. The principle of "virtual velocities," just as it came from Lagrange (or the principle of "work"), gives immediately, in virtue of (I), G1 = (11) T,+ (21) T ̧ + ... + (nl) T, G ̧ = (12) T', + (22) T ̧+ ... + (n2) T ̧ (II). G=(ln) T,+(2n) T ̧+ ... + (nn) T2) 2 B. Apply and keep applied to each of the bodies, B ̧, B ̧‚ ... (in practice by the weights of the pulleys, and by counter-pulling springs), such forces as shall have for their moments the values G1, G.... G, calculated from equations (II) with whatever values seem desirable for the tensions T,, T,,... T. (In practice, the straight parts of the cords are to be approximately vertical, and the bodies B,, B,, are to be each balanced on its axis when the pulleys belonging to it are removed, and it is advisable to make the tensions each equal to half the weight of one of the pulleys with its adjustable frame.) The machine is now ready for use. To use it, pull the cords simultaneously or successively till lengths equal to e,, e,,...e are passed through the rings E, E,... E, respectively. The pulls required to do this may be positive or negative; in practice, they will be infinitesimal downward or upward pressures applied by hand to the stretching weights which remain permanently hanging on the cords. Observe the angles through which the bodies B1, B.,... B. are turned by this given movement of the cords. These angles are the required values of the unknown x,, x,,... x, satisfying the simultaneous equations (I). The actual construction of a practically useful machine for calculating as many as eight or ten or more of unknowns from the same number of linear equations does not promise to be either difficult or over-elaborate. A fair approximation having been found by a first application of the machine, a very moderate amount of straightforward arithmetical work (aided very advantageously by Crelle's multiplication tables) suffices to calculate the residual errors, and allow the machines (with the setting of the pulleys unchanged) to be re-applied to calculate the corrections (which may be treated decimally, for convenience): thus, 100 times the amount of the correction on each of the original unknowns may be made the new unknowns, if the magnitudes thus |