by trigonometry it will be easy to determine values of y, p, 0, with regard to any fixed plane whatever, which will introduce two new arbitrary quantities, making in all six, which are requisite for the complete solution of the problem. 206. These two new arbitrary quantities are the inclination of the invariable plane on the fixed plane in question, and the angular distance of the line of intersection of these two planes from a line arbitrarily assumed on the fixed plane; and as the initial position of the fixed plane is supposed to be given, the two arbitrary quantities are known. If the position of the three principal axes with regard to the invariable plane be known at the origin of the motion, p', e', will be given, and therefore p, q, r, will be known at that time; and then equation (46) will give the value of k. The constant quantity arising from the integration of dt depends on the arbitrary origin or instant whence the time is estimated, and that introduced by the integration of dy' depends on the origin of the angle ', which may be assumed at pleasure on the invariable plane. 207. The determination of the sixth constant quantity h is very interesting, as it affords the means of ascertaining the point in which the sun and planets may be supposed to have received a primitive impulse, capable of communicating to them at once their rectilinear and rotatory motions. The sum of the areas described round the centre of gravity of the solid by the radius of each particle projected on a fixed plane, and respectively multiplied by the particles, is proportional to the moment of the primitive force projected on the same plane; but this moment is a maximum relatively to the plane which passes through the point of primitive impulse and centre of gravity, hence it is the invariable plane. B 208. Let G, fig. 52, be the centre of gravity of a body of which ABC is a section, and suppose that it has received an impulse in the plane ABC fig. 52. at the distance GF, from its centre of gravity; it will move forward in space at the same time that it will rotate about an axis perpendicular to the plane ABC. Let v be the velocity generated in the centre of gravity by the primitive impulse; then if m be the mass of the body, m.v. GF will be the moment of this F A impulse, and multiplying it by t, the product will be equal to the sum of the areas described during the time t; but this sum was shown to be t√2+12+1112; hence √√ 12 + 1/2 + 1/12 = m.v.GFh; which determines the sixth arbitrary constant quantity h. Were the angular velocity of rotation, the mass of the body and the velocity of its centre of gravity known, the distance GF, the point of primitive impulse, might be determined. 209. It is not probable that the primitive impulse of the planets and other bodies of the system passed exactly through their centres of gravity; most of them are observed to have a rotatory motion, though in others it has not been ascertained, on account of their immense distances, and the smallness of their volumes. As the sun rotates about an axis, he must have received a primitive impulse not passing through his centre of gravity, and therefore it would cause him to move forward in space accompanied by the planetary system, unless an impulse in the contrary direction had destroyed that motion, which is by no means likely. Thus the sun's rotation leads us to presume that the solar system may be in motion. 210. Suppose a planet of uniform density, whose radius is R, to be a sphere revolving round the sun in S, at the distance SG or 7, with an angular velocity represented by u, then the velocity of the centre of gravity will be v = ur. Imagine the planet to be put in motion by a primitive impulse, passing through the point F, fig. 53, then the sphere will fig. 53. at right angles to that plane will be zero; √√12 + 1/2 + 1/12 = m.v.GF. becomes "mur.FG; and l" rC. S G The centre of gyration is that point of a body in rotation, into which, if all the particles were condensed, it would retain the same degree of rotatory power. It is found that the square of the radius of gyration in a sphere, is equal to % of the square of its semi-diameter; 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°: a :: 24 t: hence a = t; or, if the constant co-effi 360 360 cient be represented by n, a = nt, and sin a = sin nt, cos a = 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 360 t, and may also be 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 + n2u = R dt2 d'u + n2u = 0, dt2 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 c, c being the number whose Napierian logarithm is unity. Whence = d3u = c2(d2x + dx2), and the equation in question becomes 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, x = = nt √ = 1. As a has two values, uc gives u = bet~=1, and u = b'c-nt~=1; and because either of these satisfies the conditions of the problem, also satisfies the conditions and is the general solution, b and b' being arbitrary constant quantities. Let and then or u = (b+b') cos nt + (b - b') √√ 1 sin nt. b + b' = M sin e; (b − b') √ − 1 = M cos e ; u = M{sin e cos nt + cos e sin nt) u = M sin (nt + €), * which is the integral required, because M and e are two arbitrary constant quantities. 215. Since a sine or cosine never can exceed the radius, sin. (nt + e) 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 r become x = Bt + at √ I x = Bt - at √1, consequently, C cot+at√=1 = cut, cai~1 = c{ cos at +1 sin at} whence uct { (b + b') cos at + (b − b') √ — I sin at} 2.3 therefore ct 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 = Bt, and u = C. cst, C being constant. Thus it appears that if the roots of n 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 'o " is so small, that its cosine √p2 + q2 + r2 differs but 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, |
