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exceedingly small amount of constraint before it will be relieved from its state of tension by its molecules assuming new positions of equilibrium. Consequently the same oblique pressures can be called into action in a fluid as in a solid, provided the amount of relative displacement of the parts be exceedingly small. All we know for certain is that the effect of elasticity in fluids, (elasticity of the second kind be it remembered,) is quite insensible in cases of equilibrium, and it is probably insensible in all ordinary cases of fluid motion. Should it be otherwise, equations (8) and (12) will not be true, or only approximately true. But a little consideration will shew that the property of elasticity may be quite insensible in ordinary cases of fluid motion, and may yet. be that on which the phenomena of light entirely depend. When we find a vibrating string, the small extent of vibration of which can be actually seen, filling a whole room with sound, and remember how rapidly the intensity of the vibrations of the air must diminish as the distance from the string increases*, we may easily conceive how small in general must be the amount of the relative motion of adjacent particles of air in the case of sound. Now the extent of the vibration of the ether, in the case of light, may be as small compared with the length of a wave of light as that of the air is compared with the length of a wave of sound: we have no reason to suppose it otherwise. When we remember then that the length of a wave of sound in air varies from some inches to several feet, while the greatest length of a wave of light is about 00003 of an inch, it is easy to imagine that the relative displacement of the particles of ether may be so small as not to reach, nor even come near to the greatest relative displacement which could exist without the molecules of the medium assuming new positions of equilibrium, or, to keep clear of the idea of molecules, without the medium assuming a new arrangement which might be permanent.

It has been supposed by some that air, like the luminiferous ether, ought to admit of transversal vibrations. According to the views of this article such would, mathematically speaking, be the case; but the extent of such vibrations would be necessarily so very small as to render them utterly insensible, unless we had

[In all ordinary cases it is to the vibrations of the sounding-board, or of the supporting body acting as a sounding-board, and not to those of the string directly, that the sound is almost wholly due.]

organs with a delicacy equal to that of the retina adapted to receive them.

It has been shewn to be highly probable that the ratio of A to B increases rapidly according as the medium considered is softer and more plastic. For fluids therefore, and among them for the luminiferous ether, we should expect the ratio of A to B to be extremely great. Now if N be the velocity of propagation of normal vibrations in the medium considered in Sect. III., and T that of transversal vibrations, it may be shewn from equations (32) that

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This is very easily shewn in the simplest case of plane waves: for if ẞ=y=0, a= f(x), the equations (32) give

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whence a=(Nt − x) + † (Nt + x); and if a=y=0, B=ƒ(x),

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Consequently we should expect to find the ratio of N to T extremely great. This agrees with a conclusion of the late Mr Green's*. Since the equilibrium of any medium would be unstable if either A or B were negative, the least possible value of the ratio of N2 to T2 is 4, a result at which Mr Green also arrived. As however it has been shewn to be highly probable that A> 5B even for the hardest solids, while for the softer ones A/B is much greater than 5, it is probable that N/T is greater than √3 for the hardest solids, and much greater for the softer ones.

If we suppose that in the luminiferous ether A/B may be considered infinite, the equations of motion admit of a simplification. da dß, dy For if we put mA + + dx dy dz

p in equations (32), and

suppose mA to become infinite while p remains finite, the equations become

Cambridge Philosophical Transactions, Vol. vii. Part I. p. 2.

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When a vibratory motion is propagated in a medium of which (33) are the equations of motion, it may be shewn that p=√(t) if the medium be indefinitely extended, or else if there be no motion at its boundaries. In considering therefore the transmission of light in an uninterrupted vacuum the terms involving p will disappear from equations (33); but these terms are, I believe, important in explaining Diffraction, which is the principal phenomenon the laws of which depend only on the equations of motion of the luminiferous ether in vacuum. It will be observed that putting A = ∞o comes to the same thing as regarding the ether as incompressible with respect to those motions which constitute Light.

9

ON THE PROOF OF THE PROPOSITION THAT (Mx+ Ny)1 IS AN INTEGRATING FACTOR OF THE HOMOGENEOUS DIFFERENTIAL EQUATION M + N dy/dx = 0.

[From the Cambridge Mathematical Journal, Vol. IV. p. 241. (May, 1845.)]

A FALLACIOUS proof is sometimes given of this proposition, which ought to be examined. The substance of the proof is as follows.

Let us see whether it is possible to find a multiplier V, a homogeneous function of x and y, which shall render Mdx + Ndy an exact differential. Let M and N be of n, and V of p dimensions; let

dU=V (Mdx+Ndy).

........(1);

then, on properly choosing the arbitrary constant in U, U will be a homogeneous function of n + p + 1 dimensions,) whence, by a known theorem,

(A),

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and the first side of this equation

being an exact differential, it

follows that the second side is so also, and consequently that (Mx+Ny)1 is an integrating factor.

Now the factor so found is of n-1 dimensions; so that the first side of (2) is zero. In fact, we shall see that the statement (A) is not true as applied to the case in question, unless

Mx + Ny = 0.

The general form of a function of x of n dimensions is Ax”. The general form of a homogeneous function of x and y of n di

mensions is a"↓ (?). x"

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The integral of the first is in general Ax+1/(n + 1), omitting the arbitrary constant; and consequently the dimensions of the function are increased by unity by integration. But in the particular case in which n = 1, the integral is A logx, which is not a quantity of 0 dimensions, at least according to the definition just given, according to which definition only is the proposition with reference to homogeneous functions assumed in (2) true. Let us now examine in what cases U will be of n+p+1 dimensions.

Putting M= M ̧x”, N=N ̧x”, y=xz, M, and N will be functions of z alone, and we shall have

Mdx+Ndy=x" {(M ̧+N ̧2) dx + N ̧x dz}.

-n-1

If M+ N≈ = 0, i.e. if Mx + Ny = 0, we see that will be an integrating factor. The integral, being a function of z, will be of 0 dimensions, and both sides of (2) will be zero.

If Mx + Ny is not equal to 0, we may multiply and divide by (M ̧+ N ̧≈) x, and we have

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n+

0

dx

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Hence we see that {x2+1 (M+N2)}1 or (Mx+Ny) is an integrating factor. For this factor we have

U = log (x) + 4 (2),

o denoting the function arising from the integration with respect.

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It may be of some interest to enquire in what cases an exact differential of any number of independent variables, in which the differential coefficients are homogeneous functions of n dimensions, has an integral which is a homogeneous function of n+1 dimensions.

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