On two pieces of paper draw the curves

y=["+(x)dx, and y=4(x).

Attach these pieces of paper to the circumference of two circular cylinders, or to different parts of the circumference of one cylinder, with the axis of x in each in the direction perpendicular to the axis of the cylinder. Let the two cylinders (if there are two) be geared together so as that their circumferences shall move with equal velocities. Attached to the framework let there be, close to the circumference of each cylinder, a slide or guide-rod to guide a moveable point, moved by the hand of an operator, so as always to touch the curve on the surface of the cylinder, while the two cylinders are moved round.

Two operators will be required, as one operator could not move the two points so as to fulfil this condition-at all events unless the motion were very slow. One of these points, by proper mechanism, gives an angular motion to the rotating disk equal to its own linear motion, the other gives a linear motion equal to its own to the centre of the rolling globe.

The machine thus described is immediately applicable to calculate the values H1, H,, H, etc. of the harmonic constituents of a function (x) in the splendid generalization of Fourier's simple harmonic analysis, which he initiated himself in his solutions for the conduction of heat in the sphere and the cylinder, and which was worked out so ably and beautifully by Poisson*, and by Sturm and Liouville in their memorable papers on this subject published in the first volume of Liouville's Journal des Mathématiques. Thus if

¥ (x) = H ̧¢ ̧ (x) + H ̧‡ ̧ (x) + H ̧‡ ̧ (x) + etc.

3

be the expression for an arbitrary function x, in terms of the generalized harmonic functions 4, (x), 4, (x), (x), etc., these functions being such that

['4, (x) 4, (x) dx = 0, ['4, (x) 4, (x) dx = 0, ['$, (x) 4, (x) = 0, etc

Φι

etc.,

His general demonstration of the reality of the roots of transcendental equations essential to this analysis (an exceedingly important step in advance from Fourier's position), which he first gave in the Bulletin de la Société Philomathique for 1828, is reproduced in his Théorie Mathématique de la Chaleur, § 90.

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

X

eft

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 différences 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

2іп

and the integration is performed through a range equal to n
(i any integer) that gives this application. In this case the
addition of a simple crank mechanism, to give a simple harmonic
angular motion to the rotating disk in the proper period

2π

when the cylinder bearing the curve y=(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 Func

tions.

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*.

津

Every linear differential equation of the second order may, as Mechanical is known, be reduced to the form

d 1 du dx P dx

(1 day)

= 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) (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, is any function of x, to begin with, as for example ux; then u, u,, etc. are successive approximations converging to that one of the solutions of (1) which vanishes when x =

=

0.

Now let my brother's integrator be applied to find C – -["u,dx,

and let its result feed, as it were, continuously a second machine,
which shall find the integral of the product of its result into
Pdx. The second machine will give out continuously the value
Use again the same process with u, instead of
instead of u,, and

of de then u

and so on.

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 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