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 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. SECTION I. Interpretation of the Equations corresponding to Vibratory 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 x, y, z be its co-ordinates when at rest, x + dx, y + dy, ≈≈ those of another particle of the same medium, x + ▲ x, y + Ay, ≈ + As those of a particle of B; ?= √dx2 + dy2 + 8≈2, R = √Ax2 + Ay2 + Ax2. Let the same quantities at the end of the time t become x + a, Y, Z. x + a + dx + da, y + dy, ≈ + dz. x + a + ▲ x + ▲a, y + ▲y, ≈ + AZ. R+AR. (the motion being in the direction of the axis of x) and let the function which expresses the force be rør. The action of the particles of A on the particle in question parallel to x, is evidently the sum of all such expressions as the following, P2. ø (r + dr) (dx + da); and that of the particles of B on the same particle the sum of M. ø (R + ▲R) (^x + ▲a), hence, σ and d2 a df M Pp (r+dr). (dx+da) – £ p (R+R) (Ax + Aa); 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 da compared with da: 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 da is the approach of two particles to each other, whilst the corresponding da 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 da 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. 3. Now (r + dr) pr Fr. dr nearly, where F(r) stands for the differential coefficient of p(r), taken with respect to r, Similarly, (R + ▲R). (^x + ▲a) = &R.▲x + (øR + but P. opr. 8x + M.Σ&R. Ax is the force which acts on the particle at rest; and consequently is identically equal to zero; In the same manner, if both particles vibrate in the direction of transmission, 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. =- ca suppose, where a = a cos (ct — kx). It is clear that, in order to effect this, we must suppose 1 a negative quantity. Now if ør = or the force vary as the in verse nth power of the distance, this can be accomplished; k8x Now on expanding the sine, it is clear that when becomes λ 2 greater than unity, or which is the same thing, when dr becomes greater. than we must put the supplement instead of the arc in the expansion; 2 but we saw reason in the case of light to suppose that the expression for the whole force has the same sign, as the expression for the force exerted by those particles only which lie within the range of the first half wave. In fact, if the different half waves give different signs, it is evident that they must give them alternately; and thus the above hypothesis would be confirmed; we shall therefore retain only the first term in the k8x and reduce our investigation to the consideration 9 2 28y2 - ndx2 of the sign of o expansion of sin2 half wave. Sa2, taken within the range of the first For every value of dy within this range there is a corresponding equal value of 8x, and vice versá: so that we may write the above expression as follows; |