Page images

equal or such as to transmit equal centrifugal forces to the particles initially considered, it follows that the luminiferous motions are only components of the whole motion; and that a less luminiferous component in one direction, compounded with a motion existing in the medium when transmitting no light, gives an equal resultant to that of a greater luminiferous motion in the contrary direction compounded with the same non-luminous motion. I think it is not only impossible to conceive any other than this dynamical explanation of the fact that circularly polarized light transmitted through magnetized glass parallel to the lines of magnetizing force, with the same quality, right-handed always, or left-handed always, is propagated at different rates according as its course is in the direction or is contrary to the direction in which a north magnetic pole is drawn; but I believe it can be demonstrated that no other explanation of that fact is possible. Hence it appears that Faraday's optical discovery affords a demonstration of the reality of Ampère's explanation of the ultimate nature of magnetism ; and gives a definition of magnetization in the dynamical theory of heat. The introduction of the principle of moments of momenta (“the conservation of areas "') into the mechanical treatment of Mr. Rankine's hypothesis of " molecular vortices," appears to indicate a line perpendicular to the plane of resultant rotatory momentum ("the invariable plane”) of the thermal motions as the magnetic axis of a magnetized body, and suggests the resultant moment of momenta of these motions as the definite measure of the “magnetic moment.” The explanation of all phenomena of electro-magnetic attraction or repulsion, and of electro-magnetic induction, is to be looked for simply in the inertia and pressure of the matter of which the motions constitute heat. Whether this matter is or is not electricity, whether it is a continuous fuid interpermeating the spaces between molecular nuclei, or is itself molecularly grouped; or whether all matter is continuous, and molecular heterogeneousness consists in finite vortical or other relative motions of contiguous parts of a body ; it is impossible to decide, and perhaps in vain to speculate, in the present state of science.

I append the solution of a dynamical problem for the sake of the illustrations it suggests for the two kinds of effect on the plane of polarization referred to above.

Let the two ends of a cord of any length be attached to two

points at the ends of a horizontal arm made to rotate round a vertical axis through its middle point at a constant angular velocity, w, and let a second cord bearing a weight be attached to the middle of the first cord. The two cords being each perfectly light and flexible, and the weight a material point, it is required to deternine its motion when infinitely little disturbed from its position of equilibrium*.

Let l be the length of the second cord, and m the distance from the weight to the middle point of the arm bearing the first. Let x and y be, at any time t, the rectangular coordinates of the position of the weight, referred to the position of equilibrium 0, and two rectangular lines OX, OY, revolving uniformly in a horizontal plane in the same direction, and with the same angular velocity as the bearing arm; then, if we choose OX parallel to this arm, and if the rotation be in the direction with OY preceding Ox, we have, for the equations of motion,

[merged small][ocr errors]
[ocr errors]

dx dt



If for brevity we assume

=n', and

9 g

=l, 217

217 we find, by the usual methods, the following solution :x=A cos {[wo+no+(+*+4n*w*)/]bt+a}

+B cos {[w'+n?—(19+4n?u?)]]t+B}, 2w2–1? +(\*+4n*w*)?

2w2—12–(14+4n’wa) y=2w[w*+no+(\*+4n°w°)}]}

2w[w9+ n°– (**+ 4n*w*) 177B sin 4, where A, a, B, ß are arbitrary constants, and 9 and 4 are used for brevity to denote the arguments of the cosines appearing in the expression for x.

The interpretation of this solution, when w is taken equal to the component of the earth's angular velocity round a vertical at the

A sin

* By means of this arrangement, but without the rotation of the bearing arm, a very beautiful experiment, due to Professor Blackburn, may be made by attaching to the weight a bag of sand discharging its contents through a fine aperture.

[ocr errors]
[ocr errors]

locality, affords a full explanation of curious phenomena which have been observed by many in failing to repeat Foucault's admirable pendulum experiment. When the mode of suspension is perfect, we have 1=0; but in many attempts to obtain Foucault's result, there has been an asymmetry in the mode of attachment of the head of the cord or wire used, or there has been a slight lateral unsteadiness in the bearings of the point of suspension, which has made the observed motion be the same as that expressed by the preceding solution, where has some small value either greater than or less than w, and n has the value

The only case, however, that need be considered as illustrative of the subject of the present communication is that in which w is very great in comparison with n. To obtain a form of solution readily interpreted in this case, let [wa+no+('+4n’?)}]}=w+p, [w*+ n°–(\*+4n°w°)}]}=w-o, 2w2-1° +1+4n’wa)

202-2-(18+4n’wa) 2w1wo+n? +(\*+4nov)] ° (=1

2w[w*+n*--(^'+4now)!]1=1-f. The preceding solution becomes

x=A cos {(w+pt+a} + B cos {(w-o)t+B}
y=-A sin {(w+pt+a}-B sin {(

- A sin {(w+pt+a}+fB sin {(W—0)t+B}.

express the result in terms of coordinates &, n, with reference to fixed axes, instead of the revolving axes OX, OY, we may assume &=x cos wt - y sin wt,

n=x sin wt + y cos wt. Then we have &=A cos (pt +a) + B cos (ot-)

(eA sin {(w+pt+a} - fB sin {(w—o)t+B}) sin wt n=-A sin (otta) + B sin (ot-B)

+(-eA sin {(w+pt+a} +fB sin {(w—0)t +B})cos wt. is

very large, e and f are both very small, and the last two terms of each of these equations become very small periodic terms, of very rapidly recurring periods, indicating a slight tremor in the resultant motion. Neglecting this, and taking a=0 and B=0, as we may do without loss of generality, by properly choosing the axes

When w

of reference, and the era of reckoning for the time, we have finally, for an approximate solution of a suitable kind,

&=A cos pt + B cos ot,

n=-A sin pt +B sin ot. The terms B, in this expression, represent a circular motion of


period 27, in the positive direction (that is, from the positive axis

[ocr errors]

P р

of Ę to the positive axis of n), or in the same direction as that of the rotation w; and the terms A represent a circular motion, of period 22

in the contrary direction. Now, w being very great, p and o are р very nearly equal to one another ; but p is rather less than o, as the following approximate expressions derived from their exact values expressed above, show:


1 * 14

o=nt + 8 win 8 WK

8 win' 8w3" Hence the form of solution simply expresses that circular vibrations of the pendulum in the contrary directions have slightly different

27 periods, the shorter, when the motion of the pendulum follows

2. that of the arm supporting it, and the longer, -*, when it is in the contrary direction. The equivalent statement, that if the pendulum be simply drawn aside from its position of equilibrium, and let go without initial velocity, the vertical plane of its motion will rotate


2 taking A=B, and reducing the preceding solution to the form

&=2A cos at cos n't,

n=2A sin at cos n't, where

, n'=n+

8 wn' and


8 It is a curious part of the conclusion thus expressed, that the faster the bearing arm is carried round, the slower does the plane of a simple vibration of the pendulum follow it. When the bearing arm is carried round infinitely fast, the plane of a vibration of the

slowly at the angular rate (0–2), is expressed most shortly by

a= (-p), or, approximately,

pendulum will remain steady, and the period will be n; in other words, the motion of the pendulum will be the same as that of a

2 simple pendulum whose length is or a harmonic mean be

1 1


[ocr errors]
[blocks in formation]
[ocr errors]

tween the effective lengths in the two principal planes of the actual pendulum.

It is easy to prove from this, that if a long straight rod, or a stretched cord possessing some rigidity, unequally elastic or of unequal dimensions, in different transverse directions, be made to rotate very rapidly round its axis, and if vibrations be maintained in a line at right angles to it through any point, there will result, running along the rod or cord, waves of sensibly rectilineal transverse vibrations, in a plane which in the forward progress of the wave, turns at a uniform rate in the same direction as the rotation of the substance ; and that if 24 be the period of rotation of the substance, and I and m the lengths of simple pendulums respectively isochronous with the vibrations of two plane waves of the same length, a, in the planes of maximum and of minimum elasticity of the substance, when destitute of rotation, the period of vibration in a wave of the same length in the substance when made to rotate will be

[ocr errors]
[ocr errors][ocr errors]

and the angle through which the plane of vibration turns, in the propagation through a wave length, will be



4 nws
or the number of wave lengths through which the wave is propagated
before its plane turns once round, will be


; ; where, as before,

[merged small][merged small][merged small][merged small][ocr errors][merged small]

and w denotes the angular velocity with which the substance is made to rotate.

« PreviousContinue »