Equation. stretched. The principle of "virtual velocities," just as it came 2 Apply and keep applied to each of the bodies, B1, 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 E1, 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 B,, 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 זי struct of the the or Equation falling to be dealt with are convenient for the machine. There Solver. is, of course, no limit to the accuracy thus obtainable by successive approximations. The exceeding easiness of each application of the machine promises well for its real usefulness, whether for cases in which a single application suffices, or for others in which the requisite accuracy is reached after two, three, or more, of successive approximations. The accompanying drawings represent a machine for finding six* unknowns from six equations. Fig. 1 represents in elevation and plan one of the six bodies B,, B,, etc. Fig. 2 shows in elevation and plan one of the thirty-six pulleys P, with its cradle on geometrical slide (§ 198). Fig. 3 shows in front-elevation the general disposition of the instrument. This number has been chosen for the first practical machine to be constructed, because a chief application of the machine may be to the calculation of the corrections on approximate values already found of the six elements of the orbit of a comet or asteroid. Elevation. Plan. Equation- Front elevation. Plan. FIG. 2. One of the thirty-six pulleys, P, with its sliding cradle. In Fig. 3 only one of the six cords, and the six pulleys over which it passes, is shown, not any of the other thirty. The three pulleys seen at the top of the sketch are three out of eighteen pivoted on immoveable bearings above the machine, for the purpose of counterpoising the weights of the pulleys P, with their sliding cradles. Each of the counterpoises is equal to twice the weight of one of the pulleys P with its sliding cradle. Thus if the bodies B are balanced on their knife-edges with each sliding cradle in its central position, they remain balanced when one or all of the cradles are shifted to either side; and the tension of each of the thirty-six essential cords is exactly equal to half the weight of one of the pulleys with its adjustable frame, as specified above (the deviations from exact verticality of all the free portions of the thirty-six essential cords and the eighteen counterpoising cords being neglected). Side e vation Disk, III. AN INTEGRATING MACHINE HAVING A NEW KINEMATIC PRINCIPLE*. The kinematic principle for integrating ydx, which is used in the instruments well known as Morin's Dynamometer† and Sang's Planimeter, admirable as it is in many respects, involves one element of imperfection which cannot but prevent our contemplating it with full satisfaction. This imperfection consists in the sliding action which the edge wheel or roller is required to take in conjunction with its rolling action, which alone is desirable for exact communication of motion from the disk or cone to the edge roller. The very ingenious, simple, and practically useful instrument well known as Amsler's Polar Planimeter, although different in its main features of principle and mode of action from the instruments just referred to, ranks along with them in involving the like imperfection of requiring to have a sidewise sliding action of its edge rolling wheel, besides the desirable rolling action on the surface which imparts to it its revolving motion-a surface * Professor James Thomson, Proceedings of the Royal Society, Vol. XXIV., 1876, p. 262. 66 + Instruments of this kind, and any others for measuring mechanical work, may better in future be called Ergometers than Dynamometers. The name dynamometer" has been and continues to be in common use for signifying a spring instrument for measuring force; but an instrument for measuring work, being distinct in its nature and object, ought to have a different and more suitable designation. The name " dynamometer," besides, appears to be badly formed from the Greek; and for designating an instrument for measurement of force, I would suggest that the name may with advantage be changed to dynamimeter. In respect to the mode of forming words in such cases, reference may be made to Curtius's Grammar, Dr Smith's English edition, § 354, p. 220.— J. T., 26th February, 1876. Sang's Planimeter is very clearly described and figured in a paper by its inventor, in the Transactions of the Royal Scottish Society of Arts, Vol. IV. January 12, 1852. |