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Mechanical

integrator's disk through an angle always = ["Pdz, a given Integration

of

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 9,, 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.de 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

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which shows that g, put for u satisfies the differential equa tion (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 G1, 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

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dg

that is, we secure arbitrary values for 91 and by the arbitrari

dx

ness of the two initial positions G1, G, of the globes.

Mechanical
Integration

of General
Linear
Differential

Equation of
Any Order.

VI. MECHANICAL INTEGRATION OF THE GENERAL LINEAR DIFFERENTIAL EQUATION OF ANY ORDER WITH VARIABLE COEFFICIENTS*.

Take any number i of my brother's disk-, globe-, and cylinderintegrators, and make an integrating chain of them thus:Connect the cylinder of the first so as to give a motion equal to its own to the fork of the second. Similarly connect the cylinder of the second with the fork of the third, and so on. Let 9,, 99, up to g, be the positions of the globes at any time. Let infinitesimal motions P, dx, P,dx, P.dx, ... be given simultaneously to all the disks (de denoting an infinitesimal motion of some part of the mechanism whose displacement it is convenient to take as independent variable). The motions (dê dê ... dк) of the cylinders thus produced are

d=g,P,dx, dк= g ̧P ̧dx, ... d=g, P.dx......(1). But, by the connexions between the cylinders and forks which move the globes, dÊ ̧ = dg, dÊ ̧= dg ̧ dk dg; and there

...

1-1

=

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Suppose, now, for the moment that we couple the last cylinder with the first fork, so that their motions shall be equal-that is to say, κ=g1. Then, putting u to denote the common value of these variables, we have

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* Sir W. Thomson, Proceedings of the Royal Society, Vol. XXIV., 1876, p. 271. + For brevity, the motion of the circumference of the cylinder is called the cylinder's motion.

For brevity, the term "position" of any one of the globes is used to denote its distance, positive or negative, from the axial line of the rotating disk on which it presses.

Integration

Linear

Thus an endless chain or cycle of integrators with disks moved Mechanical as specified above gives to each fork a motion fulfilling a dif- of General ferential equation, which for the case of the fork of the ith inte- Differential Equation of grator is equation (4). The differential equations of the displace- Any Order. ments of the second fork, third fork, ... (i-1)th fork may of course be written out by inspection from equation (4).

This seems to me an exceedingly interesting result; but though P, P, P... P, may be any given functions whatever of x, the differential equations so solved by the simple cycle of integrators cannot, except for the case of i=2, be regarded as the general linear equation of the order i, because, so far as I know, it has not been proved for any value of i greater than 2 that the general equation, which in its usual form is as follows,

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can be reduced to the form (4).

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The general equation of the form (5), where Q1, Q,,... Q, are any given forms of x, may be integrated mechanically by a chain of connected integrators thus:

First take an open chain of i simple integrators as described above, and simplify the movement by taking

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so that the speeds of all the disks are equal, and dx denotes an infinitesimal angular motion of each. Then by (2) we have

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Now establish connexions between the i forks and the ith cylinder, so that

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Q1 I1 + Q2I 1⁄2 + ... + Qi−19i−) + Q¿J; = K¡ ....

Putting in this for g1, 9,, etc. their values by (6), we find an equation the same as (5), except that κ, appears instead of u. Hence the mechanism, when moved so as to fulfil the condition (7), performs by the motion of its last cylinder an integration of the equation (5). This mechanical solution is complete; for we may give arbitrarily any initial values to K, 9v I-19. Is Ig that is to say, to

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...

Mechanical
Integration

of General

Linear
Differential
Equation of
Any Order.

Until it is desired actually to construct a machine for thus integrating differential equations of the third or any higher order, it is not necessary to go into details as to plans for the mechanical fulfilment of condition (7); it is enough to know that it can be fulfilled by pure mechanism working continuously in connexion with the rotating disks of the train of integrators.

Mechanical
Integration
of any
Differential
Equation of
Any Order.

ADDENDUM.

The integrator may be applied to integrate any differential equation of any order. Let there be i simple integrators; let ,, 91, K, be the displacements of disk, globe, and cylinder of the first, and so for the others. We have

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This will leave just one degree of freedom; and thus we have 21 simultaneous equations solved. As one particular case of relations take

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Or again, take 2i double integrators. Let the disks of all be connected so as to move with the same speed, and let t be the

displacement of any one of them from any particular position. Mechanical

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be the displacements of the second cylinders of the several double integrators. Then (the second globe-frame of each being connected to its first cylinder) the displacements of the first globe-frames will be

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Let now X, Y, X', Y', etc. be each a given function of

x, y, x, y, x', etc.

By proper mechanism make the first globe of the first double integrator-frame move so that its displacement shall be equal to X, and so on. The machine then solves the equations

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Integration
of any
Differential
Equation of
Any Order.

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Construct in (frictionless) steel the surface whose equation is

a = $f( +r)

(and repetitions of it, for practical convenience, though one theoretically suffices). By aid of it (used as if it were a cam, but for two independent variables) arrange that one moving auxiliary piece (an a-auxiliary I shall call it), capable of moving to and fro in a straight line, shall have displacement always equal to

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that another (a y-auxiliary) shall have displacement always equal to

(y'-y)ƒ{(x-x)2 + (y' - y)"},

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