piece of Faraday's heavy glass. But in this case the glass was inserted in a hole in a cork, to which also the mirror was attached. On examining the cork it was found to be magnetic. accounted for all the phenomena observed.

This

But in the previous session this would not account for the phenomena. There was no cork employed; and experiments made to test this by altering the position of the magnet showed that there was no magnetism in the arrangement.

It was the doubt that hung over the last winter's experiments that made me wish to delay publishing any results until I should have finally settled the matter. I have been unable to do so hitherto, and offer the original experiments in the meantime.

Note on the preceding paper.

[The first statement is that a rotation of the plane of polarised light might be produced by rotating a transparent body about the ray as an axis.

It is improbable that no such effect would be produced, but that the question is by no means a simple one may be seen by looking at Sir W. Thomson's paper on this subject (Proc. R. S. Lond., 1856).

I have also tried a great many hypotheses besides those which I have published, and have been astonished at the way in which conditions likely to produce rotation are exactly neutralised by others not seen at first.

There can be no doubt, however, that a rotation of some kind is going on in a diamagnetic medium under magnetic force, and this may be of the molecules of the glass of the ether or what not, and this probably goes on in all media whether transparent or not.

This rotation, as Prof. George Forbes says, stops as soon as the magnetic force is removed. He supposes that it is stopped by friction, and therefore, that energy is being dissipated at all times as long as magnetic force acts on a medium.

But we know that a magnet will retain its magnetism for a long time, and it has never been shown that a magnet must necessarily lose its magnetism. Hence we must admit that the molecular rotation is not accompanied with friction, but that it is set up by

electro-motive force, and exerts electro-motive force when it is stopped, like a rotating body having inertia.

(a) If the friction supposed by Prof. Forbes exists, it would act as an accelerating force on the glass, so that if free it would rotate faster and faster up to a certain great velocity, and if suspended by a fibre, it would rotate till the moment of friction was balanced by the moment of tortion of the fibre.

(B) If there is no friction the only effects possible would be those due, not to the maintenance, but to the starting and stopping of the molecular rotation.

To investigate (a) experimentally we must observe the elongations of the oscillation as follows:

Make + ve observe three turning points A B C, break for nearly half a complete vibration. Make ve observe three turning points DE F, break again, and make +, and so on. obtained by taking

1

8n

ve

{A+2 B+C - (D + 2 E + F) }

Then the result is

when n represents the number of repetitions of the series of six observations.

To investigate (B) experimentally we must make and break when the mirror is passing the point of equilibrium.

In Prof. Forbes's experiments there is a disturbing effect due to the ordinary diamagnetic action of the electro-magnet on the tube, which, if the tube is not perfectly symmetrised about the axis of the fibre, will tend to produce rotation. This force, however, is the same whether the current be + or

tube is the same. Hence, if the + and

provided the position of the

it may be possible to distinguish this effect from the effect sought

by Prof. Forbes.

J. CLERK MAXWELL.]

4. On the Linear Differential Equation of the Second

Order.

By Professor Tait.

(Abstract.)

This paper contains the substance of investigations made for the most part many years ago, but recalled to me during last summer by a question started by Sir W. Thomson, connected with Laplace's theory of the tides.

A comparison is instituted between the results of various processes employed to reduce the general linear differential equation of the second order to a non-linear equation of the first order. The relation between these equations seems to be most easily shown by the following obvious process, which I lit upon while seeking to integrate the reduced equation by finding how the arbitrary constant ought to be involved in its integral.

Let u and v be any functions of x,

where B and A, and therefore their ratio C, are arbitrary constants.

Now we have, by adding and subtracting multiples of (1), &c.,

whence, if u and v are independent integrals of the equation

and the process above shows why it takes this particular form.

But (2) gives

y= Au + Bv

(2),

Various classes of cases in which this form is integrable are given, of which the following is one :

Let έ=√, then the equation becomes integrable in the form

The next subject treated is the effect of the alteration of sign of P or Q in (2). This is illustrated by the equation

y" ±xy' ±y=0,

which is integrable or at least reducible to quadratures for any of the four combinations of sign.

Another portion of the investigation deals with certain infinite but convergent series, whose sums can always be expressed in terms of the integral of a linear differential equation of the second order. Consider, for instance, the expansion

Eliminating P, between (5) and (6), we obtain

This equation is thus true for all positive integral values of n, and its form at once shows that it is true for negative integral values also. It is very singular that such a series of equations of all orders should have a common solution. But it depends upon the fact, which I do not recollect having seen in print, that

This can be verified at once by applying it to "; as can also the companion formula

we should find the same equation (7) for Q as for Po. In fact, as

We thus get the two distinct particular integrals of each of the corresponding differential equations.