In the physical applications of this theory the integrals which constitute the denominators of the formulas for H1, H„, etc. are always to be evaluated in finite terms by an extension of Fourier's formula for the xu, dx of his problem of the cylinder* made by Sturm in equation (10), § iv. of his Mémoire sur une Classe d'Équations à differences partielles in Liouville's Journal, Vol. I. (1836). The integrals in the numerators are calculated with great ease by aid of the machine worked in the manner described above. The great practical use of this machine will be to perform the simple harmonic Fourier-analysis for tidal, meteorological, and perhaps even astronomical, observations. It is the case in which sin 2in and the integration is performed through a range equal to n 2π angular motion to the rotating disk in the proper period n when the cylinder bearing the curve y=4(x) moves uniformly, supersedes the necessity for a cylinder with the curve y = $(x) traced on it, and an operator keeping a point always on this curve in the manner described above. Thus one operator will be enough to carry on the process; and I believe that in the application of it to the tidal harmonic analysis he will be able in an * Fourier's Théorie Analytique de la Chaleur, § 319, p. 391 (Paris, 1822). Machine to calculate Integral of Product of two Functions. Machine to calculate Integral of Product of two Functions. hour or two to find by aid of the machine any one of the simple harmonic elements of a year's tides recorded in curves in the usual manner by an ordinary tide-gauge—a result which hitherto has required not less than twenty hours of calculation by skilled arithmeticians. I believe this instrument will be of great value also in determining the diurnal, semi-diurnal, ter-diurnal, and quarter-diurnal constituents of the daily variations of temperature, barometric pressure, east and west components of the velocity of the wind, north and south components of the same; also of the three components of the terrestrial magnetic force; also of the electric potential of the air at the point where the stream of water breaks into drops in atmospheric electrometers, and of other subjects of ordinary meteorological or magnetic observations; also to estimate precisely the variation of terrestrial magnetism in the eleven years sun-spot period, and of sun-spots themselves in this period; also to disprove (or prove, as the case may be) supposed relations between sun-spots and planetary positions and conjunctions; also to investigate lunar influence on the height of the barometer, and on the components of the terrestrial magnetic force, and to find if lunar influence is sensible on any other meteorological phenomena—and if so, to determine precisely its character and amount. From the description given above it will be seen that the mechanism required for the instrument is exceedingly simple and easy. Its accuracy will depend essentially on the accuracy of the circular cylinder, of the globe, and of the plane of the rotating disk used in it. For each of the three surfaces a much less elaborate application of the method of scraping than that by which Sir Joseph Whitworth has given a true plane with such marvellous accuracy will no doubt suffice for the practical requirements of the instrument now proposed. V. MECHANICAL INTEGRATION OF LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER WITH VARIABLE COEFFICIENTS*. Every linear differential equation of the second order may, as Mechanical is known, be reduced to the form d 1 du dx P dx/ = u where P is any given function of x. (1), On account of the great importance of this equation in mathematical physics (vibrations of a non-uniform stretched cord, of a hanging chain, of water in a canal of non-uniform breadth and depth, of air in a pipe of non-uniform sectional area, conduction of heat along a bar of non-uniform section or nonuniform conductivity, Laplace's differential equation of the tides, etc. etc.), I have long endeavoured to obtain a means of facilitating its practical solution. Methods of calculation such as those used by Laplace himself are exceedingly valuable, but are very laborious, too laborious unless a serious object is to be attained by calculating out results with minute accuracy. A ready means of obtaining approximate results which shall show the general character of the solutions, such as those so well worked out by Sturmt, has always seemed to me a desideratum. Therefore I have made many attempts to plan a mechanical integrator which should give solutions by successive approximations. This is clearly done now, when we have the instrument for calculating f (x) 4 (x) dx, founded on my brother's disk-, globe-, and cylinder-integrator, and described in a previous communication to the Royal Society; for it is easily proved that if * Sir W. Thomson, Proceedings of the Royal Society, Vol. xxiv., 1876, p. 269. + Mémoire sur les équations différentielles linéaires du second ordre, Liouville's Journal, Vol. 1. 1836. 1874. Cambridge Senate-House Examination, Thursday afternoon, January 22nd, Integration of Linear Differential Equations of Second Order. VOL. I. 32 where u Now let my brother's integrator be applied to find C- [" and let its result feed, as it were, continuously a second machine, After thus altering, as it were, u, into u, by passing it through the machine, then u, into u, by a second passage through the machine, and so on, the thing will, as it were, become refined into a solution which will be more and more nearly rigorously correct the oftener we pass it through the machine. If u+1 does not sensibly differ from u,, then each is sensibly a solution. So far I had gone and was satisfied, feeling I had done what I wished to do for many years. But then came a pleasing surprise. Compel agreement between the function fed into the double machine and that given out by it. This is to be done by establishing a connexion which shall cause the motion of the centre of the globe of the first integrator of the double machine to be the same as that of the surface of the second integrator's cylinder. The motion of each will thus be necessarily a solution of (1). Thus I was led to a conclusion which was quite unexpected; and it seems to me very remarkable that the general differential equation of the second order with variable coefficients may be rigorously, continuously, and in a single process solved by a machine. Take up the whole matter ab initio here it is. Take two of my brother's disk-, globe-, and cylinder-integrators, and connect the fork which guides the motion of the globe of each of the integrators, by proper mechanical means, with the circumference of the other integrator's cylinder. Then move one integrator's disk through an angle = x, and simultaneously move the other Mechanical integrator's disk through an angle always = ["Pda, a given Integration of Linear Differential of Second function of x. The circumference of the second integrator's Equations cylinder and the centre of the first integrator's globe move each Order. of them through a space which satisfies the differential equation (1). To prove this, let at any time g,, 9, be the displacements of the centres of the two globes from the axial lines of the disks; and let dx, Pdx be infinitesimal angles turned through by the two disks. The infinitesimal motions produced in the circumferences of two cylinders will be g,dx and g,Pdx. But the connexions pull the second and first globes through spaces respectively equal to those moved through by the circumferences of the first and second cylinders. Hence which shows that g, put for u satisfies the differential equation (1). The machine gives the complete integral of the equation with its two arbitrary constants. For, for any particular value of x, give arbitrary values G,, G. [That is to say mechanically; disconnect the forks from the cylinders, shift the forks till the globes' centres are at distances G,, G. from the axial lines, then connect, and move the machine.] 2 dg dx that is, we secure arbitrary values for 91 and by the arbitrari ness of the two initial positions G,, G, of the globes. |