and ds r¬1▼2F,dadßdy, (r>bt)..............(19). Now if έo, no, 。 be the initial velocities of rotation, dňo dy dB - SSS · dy = [[[ (B§. — vñ.) r−3 dadßdy From the last article it follows, that the first triple integral − [[1 (q)ve d£ − fff (ú, — 31q.) r ̄3 dadßy, (r> bt). whence the portion of u, which depends on the initial velocity of rotation, is To obtain the portion of u due to the initial velocities, we must add the right-hand sides of (17) and (20), and must recollect, that in (17) the limits of r are ∞ and at, and in (20) the limits are ∞ and bt; we thus finally obtain fff (31q. — i.) r¬3dadßdy, (bt<r< at)..........(21). PROPAGATION OF AN ARBITRARY DISTURBANCE. 231 This is the portion of u which depends upon the initial velocities, expressed in terms of these quantities. 223. In order to obtain the portion of u due to the initial displacements u。, vo, wo, we must change ., v., w。, q into Uo, Vo, Wo, Po, where P。 = lu2+mv。 + nw。, and differentiate with respect to t. Now u, is some function of the position of a point; hence (wo)at, which is the value of u。 at some point on the surface of a sphere whose centre is 0 and radius at, is some function of The triple integral is taken throughout the space bounded by the two spheres whose common centre is O, and whose radii are respectively equal to at and bt. If therefore we write r2drd for an element of volume, the triple integral may be written and therefore its differential coefficient with respect to t, is Hence the portion of u which depends upon the initial displacements is 1 + 1 4π + duo at ΦΩ = [[[ (3lp. — u,) r3 dadẞdy, (bt<r< at)............. (22). 4.π The complete value of the displacement u, due to the initial displacements and velocities, is obtained by adding the values of u', u' given by (21) and (22). The values of v and w can be written down from symmetry. 224. Sir G. Stokes has applied these results to the solution of two important problems, viz. (i) the determination of the disturbance produced by a given variable force acting in a given direction at a given point of the medium; (ii) the determination of the law of disturbance in a secondary wave of light. We shall now proceed to consider the first problem. Disturbance produced by a given Force. 225. Let P be the point at which the force acts; and let T be a small space described about P, which will ultimately be supposed to vanish, and let O be a point outside T at which the value of the disturbance is sought; also let D be the density of the medium. Let t be the time of observation, measured from some previous epoch; and let t' be the time, which the dilatational wave occupies in travelling from P to 0. Let f (t) be the given force, and F(t) the velocity at P produced by the force during a very small interval of time dt', then the usual equation of motion gives dv Now if we consider the state of things which was going on at P at a time t' ago, we must in this equation write t-t' for t, and dt for dt; also dv = F(t − t), whence This equation gives the value of the velocity communicated during the interval dt' in terms of the force. Let O be the origin, OP = r; also let l, m, n be the direction cosines of OP, and l', m', n' those of the force; and let k be the angle between OP and the direction of the force, so that k = ll' + mm' + nn'. Since the disturbance may by virtue of (23) be regarded as one which is produced by a given initial velocity, the resulting disturbance at O is determined by (21); also since DISTURBANCE PRODUCED BY A GIVEN FORCE. 233 it follows that the first term of (21) becomes Since the force is supposed to have commenced to act an infinitely long time ago, we must integrate this expression with respect to t' between the limits r/a and; but since the force is confined to the indefinitely small volume T, f(t-t) will be insensible except for values of t' comprised between the narrow limits r/a and r/a, where r1, r, are the least and greatest values of the radius vector drawn from 0 to T. We may therefore omit the integral signs, and replace SrdS by T, and we thus obtain for the value of the first term of (21), If we denote by t", the time which a distortional wave occupies in travelling from P to O, and treat the second term of (21) in a similar manner, we shall obtain In order to find what the triple integral in (21) becomes, we see from (17) and (20) that it may be written Since f(t-t') is insensible except throughout the space T, we may write T for dadßdy, and omit the integral signs; we thus obtain and this has to be integrated with respect to t' between the limits. r/a and. This term thus becomes Treating the second term in the same manner, and remembering that the limits of t" are r/b and ∞, and adding we obtain The values of v and w are obtained by putting m, m'; n, n' respectively for l, l'. If therefore we take OP for the axis of a, and the plane passing through OP and the direction of the force as the plane xz, and put & for the inclination of the force to PO, we shall have l = 1, m = 0, n = 0; l' = k = cos 4, m' = 0, n′ = sin 4. 226. In discussing this result Sir G. Stokes says: "The first term in u represents a disturbance which is propagated from P with a velocity a. Since there is no corresponding term in v or w, the displacement, as far as relates to this disturbance, is strictly normal to the front of the wave. The first term in w represents a disturbance which is propagated from P with a velocity b, and as far as relates to this disturbance, the displacement takes place strictly in the front of the wave. The remaining terms in u and w represent a disturbance of the same kind as that which takes place in an incompressible fluid, in consequence of the motion of solid bodies in it. If f (t) represent a force which acts for a short time, and then ceases, ƒ (t−t) will |