the symbol having reference to the sum of similar expressions, taken for all the particles whose action on P is sensible. By expansion we obtain F() being the differential coefficient of (r) taken with respect to r, but (r+p')2= (8x+da)2 + (dy +§ ß)2 + (d≈+dy)2; .. r2+2rp' = r2+2 (dxda+dydß+dzdy), omitting powers of p and da, dß, dy; and by substituting this value in the above equation it gives d2 a dt2 Fr = Σ. {pr + (dxda + dysß + d≈dy) + .........} (dx+da) but Ep (r).dx is manifestly the accelerating force, resolved parallel to x, on the particle P in its state of rest, and consequently is equal to zero; we have then 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 and similar expressions for 83 and dy. But it is evident that the sum of a series of terms of which one factor in each is (r) and the other Sa". Sy". Sz", 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, dy, dz, two equal values of x, 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 da, Sß, dy in equation (1), and omitting quantities which vanish identically; we get ф (2), and again Σ. (r) dx2 =Σp(r)dy2 =Σp(r)dz2 = 2n2, writing n2 for abbreviation; three equations of remarkable simplicity and elegance; of which the following are evidently solutions: a=a cos k {nt-(ex+fy+gz)}, B=b cos k {nt-(ex +fy+gx)}, y=c.cosk {nt-(ex+fy+gz)}, 2 subject to the restriction that e+f+g=1. We can easily get rid of this restriction by writing cos e for e, cos o for f, and cos for g, cosy 0, and being the angles which a straight line makes with the co-ordinate axes of x, y, z. For simplicity put x cose + y cos + cos y = p, p being the projection on the line making angles 0, 4, with the axes, of the distance OP of the point P from the origin: the above equations then become It is true a more general solution of these equations would have been obtained by assuming the values of k, 0, 0, & different for each of the three equations, but as the complete solution will consist of a series of terms similar in form to the above, it is sufficient for our purpose to exhibit that term above which has the same period for each of the three directions, and which consequently corresponds to one and the same undulation. From these equations it is manifest that the same state of motion recurs when kp is increased by 2nπ, and consequently is the length of 2 π k a wave; also the motion at the end of t + At is the same at the point p+ Ap as at the time t at transmission parallel to p = p, when n▲t = Ap, whence the velocity of dp dt = n. It may be worth while to notice here that the proposition which we have considered assumes the velocity of transmission to be the same in all directions: in general, however, this will not be the case, the direction of transmission being defined by the simultaneous transmission of a system of waves, and the velocity will have reference to that de dt direction only; but as is independent of any particular direction, and depends only on the nature of the substance, it must be either the velocity of transmission itself, or the velocity in a direction making a constant angle with that of transmission, and consequently varies as that velocity. And for the same reason p must be either in the direction of transmission, or making a constant angle with it, and as the introduction of a constant cannot in any manner affect our results, we dp may consider P and respectively as defining the place and velocity of the wave at the end of the time t. Another remark is also important, that since from the constitution of the medium it is indifferent in what direction the axes of co-ordinates are taken, all the functions which we may introduce involving Sp must finally turn out independent of 0, p and y, so that we might at once suppose the direction of transmission to be the axis of y, and put dy for Sp; this, however, I shall not do, as it does not appear necessary, and it is convenient to retain the symbol p, on other accounts to be noticed hereafter. The above remark will be mainly useful in pointing out to us what are the quantities to be rejected in our equations of motion. Let us now take as the solution of equation (1) the form we have obtained from equation (2), which is perfectly allowable, since the latter is only a particular case of the former: the quantities n and k are of course not necessarily the same for both. Put the solution under the form a = a cos (ct-kp); .. da=a cos (ct-kp-ksp)-a cos (ct-kp) =-a cos (ct-kp). (1-cos kop) + a sin (ct-kp) sin kdp, where Sp=da cos 0+dy cos + dx cos y is the projection of the distance PQ on the line OP. By the substitution of this expression for da, and analogous ones for 83 and dy; our equation (1) becomes now it is manifest that Σp(r) sin kdp=0; and also because sin (ct – kp) F(r) is independent of 2, and the term Σ Sady sin kop is of an odd order, in this wherever there is a positive term, there will be a corresponding negative one: the whole expression denoted by the symbol term will therefore be identically equal to zero. Σ Precisely the same reasoning applies to all similar expressions. F(r) Ιδρ Further as regards the term Σ Sady sin kop, bearing in mind F(r) r sin? kop 2 бу r 2 the remark above made with respect to p, it is manifest that the part Sy will have one and the same value, for two equal values of da with different signs: thus, to assist the conception referring to the cube as at the commencement of this paper, the point P being at its centre, suppose a particle at each of the two upper corners of the face on which you are looking, and y vertical, then the expression Σ sin Sy is the same for each of them, but dx in F(r) Ιδρ 2 one corresponds to Sa in the other, and the sum of the above function for two such particles vanishes. It is clear, therefore, that this expression Exdy (r) sin kop, and all analogous ones are identically Σδάδη r equal to zero. 2 Our equations then become much reduced, and assume the remarkably simple form, |