211. Hence, if the ratio of the mean radius of a planet to its mean distance from the sun, and the ratio of its angular velocity of rotation to its angular velocity in its orbit, could be ascertained, the point in which the primitive impulse was given, producing its motion in space, might be determined. 212. Were the earth a sphere of uniform density, the ratio R r would be 0.000042665; and the ratio of its rotatory velocity to that in its orbit is known by observation to be 366.25638, whence GF R ; and as the mean radius of the earth is about 4000 miles, 160 the primitive impulse must have been given at the distance of 25 miles from the centre. However, as the density of the earth is not uniform, but decreases from the centre to the surface, the distance of the primitive impulse from its centre of gravity must have been something less. 213. The rotation of the earth has established a relation between time and the arcs of a circle. Every point in the surface of the earth passes through 360° in 24 hours; and as the rotation is uniform, the arcs described are proportional to the time, so that one of these quantities may represent the other. Thus, if a be an arc of any number of degrees, and t the time employed to describe it, 360 360° a :: 24 t: hence a = t; or, if the constant co-effi 24 360 cient be represented by n, ant, and sin a sin nt, cos a 24 = cos nt. In the same manner the periodic time of the moon being 27.3 days nearly, an arc of the moon's orbit would be t, and may also be 360 27.3 expressed by nt. Thus, n may have all values, so that nt is a general expression for any arc that increases uniformly with the time. 214. The motions of the planets are determined by equations of these forms, d'u dt2 but the integral of which is d2u dt2 + n2u = R which are only transformations of the general equation of the motions of a system of bodies. The integrals of both give a value of u in terms of the sines and cosines of circular arcs increasing with the time; the first by approximation, but the integral of the second will be obtained by making u = c2, c being the number whose Napierian logarithm is unity. Whence Hence Let and then or + n2u = 0, and the equation in question becomes d3u = c2(d2x + dx2), dr + dx2 + n2dť2 = 0. Let since the element of the time is constant, which changes the equation to dy + dt (n2 + y2) = 0. If y = m a constant quantity, dm = dy = 0, hence n2 + m2 = 0; whence m = = n√ −1, dxydtndt √ 1, As r has two values, uc gives x = = nt √ = 1. u = bort√=1, and u = b'cnt-1; and because either of these satisfies the conditions of the problem, u = bcri~-1 + b'c=nt√=1, their sum also satisfies the conditions and is the general solution, b and b' being arbitrary constant quantities. But Cat-1 cos nt + = − u (b+b') cos nt + (b b + b' = M sin e ; (b − * 1 sin nt, I sin nt. b') u = M sin (nt + €), b') √1 sin nt. √ − 1 = M cos e ; which is the integral required, because M and are two arbitrary constant quantities. 215. Since a sine or cosine never can exceed the radius, sin. (nt + €) never can exceed unity, however much the time may increase; therefore u is a periodic quantity whose value oscillates between fixed limits which it never can surpass. But that would not be the case were n an imaginary quantity; for let n = a ± B√=1; then the two values of a become x = Bt + at √I x = Bt - at I, consequently, C Cot+at√-1 = cat.cat√√=1 =c%{cos at + c3t—at√=1 = c3t. c- at√=1=c3t{cos at whence uc{(b + b') cos at + (b − b') √ or substituting for b + b' ; (b − b') √ — 1; u = c.M.sin (at + €) ; √−1 sin at} I sin at} I sin at} But C2 = 1 + ßi + } BP2 + 11⁄2 ß3P + &c. 2.3 therefore c increases indefinitely with the time, and u is no longer a periodic function, but would increase to infinity. Were the roots of n2 equal, then x = ßt, and u = C. c, C being constant. Thus it appears that if the roots of n2 be imaginary or equal, the function u would increase without limit. These circumstances are of the highest importance, because the stability of the solar system depends upon them. Rotation of a Solid which turns nearly round one of its principal Axes, as the Earth and the Planets, but not subject to the action of accelerating Forces. 216. Since the axis of rotation oz" is very near oz', fig. 50, the angle 2' o " is so small, that its cosine differs but r √√p2 + q2 + p2 little from unity; hence p and q are so minute that their product may be omitted, which reduces equations (45) to Cdr = 0, Adp (CB) qrdt = 0, (AC) prdt = 0; the first shows the angular velocity to be uniform, and the two last give d'q (A-C) dp 7" dt2 B dt hence if the constant quantity M' = M. + n2q = 0; r2 = n3, d q dt2 q = M' cos (nt+g). p = M sin (nt+g); = VACA (B-C) A A(A — C). * qr; 217. If oz" the real axis of rotation coincides with oz', the principal axis in the beginning of the motion, then q and p are zero; hence also, M = 0, and M' = 0. It follows therefore, that in this case q and P will always be zero, and the axis oz" will always coincide with oz'; whence, if the body begins to turn round one of its principal axes, it will continue to rotate uniformly about that axis for ever. On account of this remarkable property these are called the natural axes of rotation; it belongs to them exclusively, for if the position of the real axis of rotation oz" be invariable on the surface of the body, the angular velocity will be constant ; hence dp 0, dq= 0, dr = 0, = and (CB) qrdt = 0, (AC) rpdt = 0, (B — A) pqdt = 0. 218. If A, B, C, be unequal, these equations will only be zero in every case when two of the quantities p, q, r, are zero; but then, the real axis coincides with one of the principal axes. If two of the moments of inertia be equal, as A = B, the three equations are reduced to rp = 0, qr = 0; both of which will be satisfied, that is, they will both be zero for every value of q and P, if r0. The axis of rotation is, therefore, in a plane at right angles to the third principal axis; but as the body is then a solid of revolution, every axis in that plane is a principal axis. 219. When ABC, the three preceding equations are zero, whatever may be the values of p, q, r, then all the axes of the body will be principal axes. Thus the principal axes alone have the property of permanent rotation, though they do not possess that property in the same degree. 220. Suppose the real axis of rotation oz", fig. 50, to deviate by an indefinitely small quantity from oz', the third principal axis, the coefficients M and M' will then be indefinitely small, since q = M' × cos (nt + g), and p = M sin (nt + g) are indefinitely small. Now if n be a real quantity, sin (nt + g), cos (nt + g), will never exceed very narrow limits, therefore q and p will remain indefinitely small; so that the real axis oz" will make indefinitely small oscillations about the third principal axis. But if n be imaginary, by article 215, sin (nt+g), cos (nt + g), will be changed into quantities which increase with the time, and the real axis of rotation will deviate more and more from the third principal axis, so that the motion will have no stability. The value of n will decide that important point. Since (AB) (B-C), it will be a real quantity when C the moment of inertia with regard to oz', is either the greatest or the least of the three moments of inertia A, B, C, for then the product (AC) (BC) will be positive; but if C have a value that is between those of A and B, that product will be negative, and n imaginary. Hence the rotation will be stable about the greatest and least of the principal axes, but unstable about the third. n=r ds dt 221. Having determined the rotation of the solid, it only remains to ascertain the position of the principal axis with regard to quiescent space, that is, with regard to the fixed axes ox, oy, oz. That evidently depends on the angles p, 4, and 0. If the third principal axis oz', fig. 50, be assumed to be nearly at right angles to the plane roy, the angle zoz', or e, will be so very small that its square may be omitted, and its cosine assumed equal to unity; then the equations (41) give dp - dy = rdt; or if ra, be a constant quantity, the integral is, Y = 8 at + €. = 8, If sine cos sin sinu, the two first of equations (41), after the elimination of dy, give + all = The integrals of these two quantities are obtained by the method in article 214, and are |