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X. On the Motion of a System of Particles, considered with reference to the Phenomena of Sound and Heat. By PHILIP KELLAND, B.A. Fellow and Tutor of Queens College, Cambridge.
[Read May 16, 1836.]
IN a former Memoir, it was my endeavour to simplify the equations of motion of a system of particles attracting each other with forces varying according to any law. The discussion of these equations was restricted to their bearing on the phenomena of Light, on which account one of the three was left untouched.
It appeared that the hypothesis of attractive forces led to the result that two of the equations corresponding to the motion in a plane perpendicular to the direction of transmission, indicated vibratory motion, whilst the third assumed a form altogether different, shewing that, as far as it was concerned, the motion was not vibratory.
On the other hand, the hypothesis of repulsive forces would give the motion in the direction of transmission vibratory, whilst the contrary would be the case in a plane perpendicular to this direction.
The discussion of the equations corresponding to motion in the direction of transmission is the object of the present memoir.
It is not improbable that to the action of forces, such as those of which we are treating, a considerable number of the phenomena of nature may be referred; but on account of our imperfect knowledge of the analogies subsisting between phenomena which apparently differ widely from each other in some essential points, we are obliged to restrict ourselves to the most simple, or to those which have been the most carefully examined.
Vol. VI. PART II. H H
Instances offer themselves in the cases of sound and light, since both have, for a long period, been referred to vibrations, though the difference in the nature of these vibrations had to be pointed out before it was admitted that a complete parallel was not to be expected between them. The same observation is applicable to the theories of light and heat. . Remarkable as are the analogies between them, demanding as it would seem from their very nature the same mode of explanation in each, there are nevertheless peculiarities in the latter which seem to strike at the very foundation of the theory, and to require the construction of another on totally new principles.
On whatever grounds then a theory be raised, we must not be discouraged if some succeeding facts appear for the moment to militate against it, and particularly, when that theory is one in which the action of force in its different modifications plays a conspicuous part, for there we are presented with a range so wide that facts, almost antagonist to each other, are brought together in the interpretations of the various kinds of motion which occur.
In the present memoir, I have ventured to push these interpretations to a considerable extent, from a conviction that the explanation of many phenomena is contained in them, and the hope that in some cases at least the real explanation may coincide with, or at any rate bear a close resemblance to, those which I have attempted.
I have adopted the hypothesis that the medium, whose motion we have under consideration, is not composed of particles of one nature, but of a regularly distributed series of particles of two kinds, of which each is endued with forces and inertia differing from those of the other. For the sake of distinction, in forming the equations, I have called these media A and B, which when applied to sound signify air and vapour, when applied to light and heat, ether and caloric or those substances, by whatever names they may be designated, which serve for the propagation of these respectively.
We will assume that the law of force is the same in both cases, an assumption which there appears no reason to suspect.
Interpretation of the Equations corresponding to Pibratory Motion.
1. LET the mass of a particle of medium A, estimated by its repulsion at the distance unity, be represented by P; that of a particle of medium B, estimated in like manner, by Q; and the moving force of a particle of A on a particle of B by M, which is also the moving force of a particle of B on one of A.
We will first consider the motion of a particle of medium A.
Let ar, y, z be its co-ordinates when at rest, a + &r, y + 3 y, z + 3x those of another particle of the same medium, a + A ar, y + A y, z + Ax those of a particle of B; R = voya” + Ayo -- Ax". Let the same quantities at the end of the time t become a + a, y, x. a + a + 3a 4- 3a, y + 3 y, z + 3x. a + a + A a + Aa, y + A y, x + Az. r + &r. R + A R. (the motion being in the direction of the axis of a and let the function which expresses the force be rq r.
The action of the particles of A on the particle in question parallel to w, is evidently the sum of all such expressions as the following, P. p (r + 3r) (3a + 3a); and that of the particles of B on the same particle the sum of M. p (R + AR) (Aa + Aa),
a and > indicating the respective sums taken for all the particles which are in motion.
2. In order to reduce these expressions to an integrable form, it is requisite to adopt some process of approximation. Suppose then we omit 8a compared with 3a : this appears at the first glance a doubtful process, for we cannot here suppose, as we did in the case of light, that the particles have a very small motion; we know, in fact, that this is not the case for sound; but all scruple will be removed when we reflect that any particular 3a is the approach of two particles to each other, whilst the corresponding 3a is their original distance; and from the fact of the repulsive nature of the forces, we cannot, even in the most unfavourable case, suppose the approach other than a small fraction of the original distance. Or again, if a be the amount of recess of the particles, since any recess must, in the case of vibrations, have a corresponding approach, the same reasoning applies.
but P. a pr. 3r + M. Sop R. Aa is the force which acts on the particle at rest; and consequently is identically equal to zero;
- - . . kõ = a sin (ct — kar). sin kóa – 2 a. sin" o: In the same manner, if both particles vibrate in the direction of transmission, AE Aar
and, by the hypothesis that each medium is a medium of symmetry, we shall, by reasoning precisely the same as that which I adopted on a previous occasion, [Part I. p. 156. of this Vol.], arrive at the following result:
4. It is clear that, in order to effect this, we must suppose
- - - , I a negative quantity. Now if pr= ;: , or the force vary as the in
verse n” power of the distance, this can be accomplished;