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for dispersion, at the same time I think that simple as they may appear at a first glance, and satisfactory as they undoubtedly are to a certain extent, it will be found a difficult task to pursue them into detail, even in the case of sound. We know little or nothing of the laws which regulate the developement of heat, which affect the velocity of light, at least if we adopt the hypothesis of molecular radiation, and have thus only shifted our difficulty without removing it. If on the other hand, we choose to regard heat as an effect consequent on the alteration of the positions of the attractive or repulsive particles within a medium (which seems reasonable from some recent experiments on the Polarization, &c. of Heat), then, by analogy, Mr Airy's hypothesis amounts in fact to supposing the particles endued with attractive or repulsive energies, influenced by the particular positions into which they are thrown, and varying with the change of these positions, to the action of which all the effects are assigned.
The great obstacle to a simple explanation of this subject appears to have arisen from the fact, that theorists generally have not divested themselves of the idea of motion directly: velocity has been substituted for force, and wave for change of force.
It occurred to me about two years since, that if we could deduce a simple equation of motion on the supposition that the particles of a medium are at a finite distance from each other, we might arrive at results very different in form from those usually adopted. In fact it appeared probable that the velocity might depend on the positions into which the particles should arrange themselves, and thus might be affected by the length of a wave.
Such a formula I actually obtained, and deduced from it the necessary result, that the square of the velocity is represented by a series
6 in my analysis which I regarded as fatal to the whole; namely, that having a function involving the distance between two consecutive particles, and the space through which a particle is disturbed, I had expanded it in terms of the ratio of the latter quantity to the former.
- 2 of terms of which c (". o is a type. There was, however, one point It appeared to me at the time doubtful whether this series might not be a diverging one, and thence it became extremely probable that the existence of the function in form above, was owing to the absence of terms omitted in this expansion. Lately M. Cauchy's Memoir on the same subject has fallen into my hands, and an opportunity has been afforded me of comparing his results with my own. The comparison has shewn me that although in some points we differ, in the essentials of principle, at least, we coincide.
Whatever difficulty may attach itself to my hypothesis as to the sphere of action of the particles, will attach itself equally to his, as they are identical : at least I have reduced mine to the same state as his.
Although there were many points of coincidence in our processes, there were not a few of difference, to many of which the present Memoir is indebted; as I have not scrupled to adopt anything which would tend either to simplify or generalize the results; my object being by no means to regard my formula as a re-discovery of what M. Cauchy had published in 1830, but rather to attempt an improvement on what is already known. I may be allowed to add, that M. Cauchy’s equations, owing to his proceeding with great generality at first, and only adding new hypotheses to simplify them when they became perfectly unwieldy, are so buried in symbols, that a person must possess no ordinary sagacity to give to them any interpretation. And further, there are some points in which the result is more general than the hypothesis would render necessary.
The plan which I have pursued is to simplify the equations as I proceed, and not to retain any result which admits of reduction.
THE problem about which we are to occupy ourselves, is the motion of any system of material particles, exerting on each other forces varying according to any function of the distance. It would be a useless generalization, in the present state of analysis, to proceed at once to the solution of this problem without any further restrictions, for even should we succeed in integrating the resulting equations, whether by approximation or any other method, we should at length be obliged to have recourse to particular hypotheses in order to interpret our results.
I propose then to make the following hypotheses:
1. That the distance between the particles is sufficiently large compared with their sphere of motion, to allow the square of the latter quantity to be omitted compared with that of the former.
2. That the disposition of the particles is a disposition of symmetry. It may serve to fix our ideas, if we consider them symmetrically situated with respect to the three co-ordinate planes; as, for instance, arranged in the angular summits of cubes, whose edges are parallel to the co-ordinate axes, and whose centre is the origin. This is, however, merely stated as something to guide us, since we must suppose, in whatever manner it can be accomplished, that the disposition is perfectly symmetrical. On these two hypotheses, which, virtually at least, are M. Cauchy's hypotheses, I shall now proceed to determine the equations of motion.
Let a, y, x be the co-ordinates of any particle P in its state of rest, the origin being taken at pleasure, and the axes any axes of symmetry. a + 3a, y + 3 y, z + 3x those of any other particle Q, which lies within the sphere of sensible attraction to P; r the distance PQ; a + a, y + 3, x + y the co-ordinates of P after any time t from the beginning of the motion; a + a + 3a + 3a, y + 3 + 3 y + 33, x + y + 38 + 3-y those of Q at the same time; r + p' the corresponding value of the distance PQ. Let the accelerating force of Q on P at the distance PQ be represented by the function (r. 4 p"). p(r + p.). Resolving this attraction parallel to the axis of a, it gives p (r + p.). (84. 4- 3a), whence we obtain
the symbol X having reference to the sum of similar expressions, taken for all the particles whose action on P is sensible. By expansion we obtain
‘p (r--p') = p(r) + F (r). p' + ............ F'(r) being the differential coefficient of p(r) taken with respect to r,
and by substituting this value in the above equation it gives
# = x. (or ++Great varior......... } (3a +3a) but Sq (r).3a is manifestly the accelerating force, resolved parallel to a', on the particle P in its state of rest, and consequently is equal to zero; we have then
which we will call equation (1).
Previous to the solution of this equation in its general form, let us examine what it becomes in that particular case where those particles only which are in the immediate vicinity of P sensibly affect its motion, an hypothesis which is tantamount to supposing all the particles very near each other, as it is manifest that on the latter supposition the sum of the forces exerted by those particles nearly in contact with P, is beyond all comparison greater than those of particles at a finite interval from it. Proceeding on this supposition, we obtain
da § da
da * =#3, ##; y + i. 33
But it is evident that the sum of a series of terms of which one factor in each is p(r) and the other 3a". 3 y”. 3x", where m + n + p is an odd integer, will be identically equal to zero; since if m, for instance, is odd, we shall have, for any particular values of r, 39, 8x, two equal values of 8a, the one positive, and the other negative, and one of the quantities m, n, p must be an odd number. Substituting, therefore, the above values of 3a, 33, 3-y in equation (1), and omitting quantities which vanish identically; we get
three equations of remarkable simplicity and elegance; of which the following are evidently solutions:
subject to the restriction that e^+f +g" = 1. We can easily get rid of this restriction by writing cos 0 for e, cos p for f. and cos N, for g,