the additive constant being zero if the axis of X be taken coincident with, and not merely parallel to, the axis of the impulse I. The exact solution of (43) involves the use of elliptic functions. The nature of the motion, in the various cases that may arise, is however readily seen from the theory of the simple pendulum. For a full discussion of it we refer to Thomson and Tait, Arts. 333, et seq. It appears from (43) that the motion of the solid parallel to its axis is stable or unstable according as AB. Since A denotes twice the kinetic energy of the solid moving with unit velocity parallel to its axis, and similarly for B, it is tolerably obvious that if the solid resemble a prolate ellipsoid of revolution A < B, whilst the reverse is the case if it resemble an oblate ellipsoid. Compare Art. 111. The above analysis applies equally well to the somewhat more general case (b) of a body with two mutually perpendicular planes of symmetry, when the motion is altogether parallel to one of these planes. If this plane be that of xy we must suppose the if it be that of xz, 0, 0); origin transferred to the point (M", 0, 0); B 118. The question of the stability of the motion of a body moving parallel to an axis of symmetry is more simply treated by approximate methods. Thus, in the case (d) of a body with three planes of symmetry, and slightly disturbed from a state of steady motion parallel to x, we have, writing u=c+u', and assuming u', v, w, p, q, r to be all small, dt dt pdp=0, qdq= (C-A) cw, Rdx = (A–B) cv. dr dt with a similar equation for q. The motion is therefore stable only if A be greater than either B or C. It appears from Art. 111 that the only direction of stable motion of an ellipsoid is that of its least axis. For practical illustrations of this result see Thomson and Tait, Art. 336. 119. If in (24) we write T=T+T', and separate the terms due to T and T' respectively, we obtain expressions for the forces exerted on the moving solid by the pressure of the surrounding fluid; viz. we have for the total component (X, say,) of the fluid pressure parallel to x and for the moment (L) of the same pressures about x, d dT dT dT dT dT L == +w +r dt dp dv dw q- The forms of these expressions being known, it is not difficult to verify them by direct calculation from the formula (18). We should thus obtain an independent though somewhat tedious proof of the general equations of motion (24). If the body be constrained to move with a uniform velocity of translation, the components of which, relatively to the axes of Art. 113, are u, v, w, we have X, Y, Z = 0, so that the effect of the fluid pressure is represented by a couple whose components are L= (B− C) vw, M=(C– A) wu, £= (A − B) uv.......(45). The coefficients A, B, C in the expression for 27 differ from those in the expression for 2 only by the addition of the mass of the solid, so that it is immaterial in (21) which set of coefficients we understand by these symbols. If we draw in the ellipsoid Ax2 + By3 + Cz3 = const.. ..(46), a radius-vector r in the direction of the velocity (u, v, w) and erect the perpendicular h from the centre to the tangent plane at the extremity of r, the plane of the above couple is that of h and r, and its magnitude is proportional to sin hr directly, and to h inversely. Its tendency is to turn the body from r to h. Let us suppose that A, B, C are in order of magnitude, and that the direction of the velocity (u, v, w) deviates but slightly from that of one of the principal axes of (46). If this axis be that of x, the tendency of the above couple is to diminish, and if that of z, to increase the deviation; whilst in the case of a slight deviation from the axis of y the tendency of the couple depends on the position of r relative to the principal circular sections of (46). Compare Art. 118. Case of a Perforated Solid. 120. If the moving solid have one or more apertures or perforations, so that the space external to it is multiply-connected, the fluid may have a motion independent of that of the solid, viz. a cyclic motion in which the circulations in the various non-evanescible circuits which can be drawn through the apertures may have any values whatever. We will briefly indicate how the foregoing methods may be adapted to this case. Let K1, K....... be the values of the circulations in the above-mentioned circuits, and let do, do,,... be surface-elements of the corresponding barriers necessary (as explained in Art. 54) to reduce the region occupied by the fluid to a simply-connected one. Further, let l, m, n denote the direction-cosines of the normal drawn towards the fluid at any point of the surface of the solid, or drawn on the positive side at any point of a barrier. We may now write $ = u$, +v¢ ̧+wQs + PX1 + 9X2 + rxs + x,w1 + x2w2 + ...(47). The functions 4, x are determined by the same conditions as before. To determine w, we have the conditions (a) that it must satisfy y3w, = 0 throughout the fluid; (b) that its derivatives must vanish at infinity; (c) that dw, dn =0 at the surface of the solid; and (d) that w, must be a monocyclic function, the cyclic constant being unity; viz. the increment of the function must be unity when the point to which it refers describes a circuit cutting the first barrier once and once only, and zero when the point describes a circuit not cutting this barrier. It appears from Art. 62 that these conditions completely determine w1, save as to an additive constant. The energy of motion of the fluid is given by Art. 67, viz. we have αφ Substituting the values of 4, from (47) we obtain a homogeneous expression of the second degree in u, v, w, ..., K1, K2, .... K2) This expression consists of three parts. The first is a homogeneous quadratic function of u, v, w, p, q, r, the coefficients in which are given by the same formulæ as in Art. 110; the second part consists of products of u, v, w,... into 1, ...; whilst the third part is a quadratic function of the coefficients . The coefficients of the second part all vanish. Thus the coefficient of u is к. and to see that the value of this expression is in fact zero, we have only to compare (30) and (31) of Art. 66, writing = = w1, and therefore 1 =,=... =0, x,' = 1, K, K The coefficients of the third part are found as follows. = =1, We have do1, rdw, do, by another simple application of Thomson's extension of Green's theorem. Hence the total energy is obtained by adding to the right-hand side of (21) an expression of the form 121. The impulsive forces necessary to produce from rest the actual motion at any instant now consist partly of impulsive forces applied to the solid, and partly (as explained in Art. 61) of impulsive pressures р, р,, &c. uniform over the several membranes which are supposed for a moment to occupy the positions of the barriers above-mentioned. The components of the force- and couple-resultants of the first set, we denote by 1, ~1, 1, and λ1, μ1, 1, respectively; those of the force and couple equivalent to the second set by 2, 72, 2, and λ, H2, V. By the 'impulse' of the motion at any instant we shall understand the force and couple equivalent to both these sets combined, so that if §, n, ; λ, μ, v be its components, we have ין If we use the term 'impulse' in this sense, the reasoning of Art. 108 and consequently the equations of motion (20) will still hold. The formulæ (23), however, connecting §, n, S, &c. with T require correction. By the same reasoning, and with the same notation as in Art. 112, we have |
