density of the earth. Densities of Jupiter, moon, &c. The mass of Jupiter is 332 times that of the earth, and its volume is 1260 times the volume of the earth; therefore the density of Jupiter is 332 1260 =0.264; which is a little more than the density of the sun. The mass of the moon is, and its volume, therefore its density is divided by, or 43=0.6533; about the den23 sity of the earth. From these examples the reader will understand how the densities were found, as expressed in table IV. Gravity on of the other four.d GRAVITY ON THE SURFACE OF SPHERES. (174.) The gravity on the surface of a sphere depends on the surfaces the mass and volume. The attraction on the surface of a planets, how sphere is the same as if its whole mass were collected at its center; and the greater the distance from the center to the surface, the less the attraction, in proportion to the square of the distance but here, as in the last article, some one sphere must be taken for the unit, and we take the earth, as before. The mass of the sun is 354945, and the distance from its center to its surface is 111.6 times the semidiameter of the earth; therefore a pound, on the surface of the earth, is to the pressure of the same mass, if it were on the surface of 354945 1 or as 1 to 28 nearly. That is, one pound on the surface of the earth would be nearly 28 pounds on the surface of the sun, if transported thither. The mass of Jupiter is 332, and its radius, compared to that of the earth, is 11.1 (Art. 131); therefore one pound, on 332 the surface of the earth, would be (11.1) 2' or 2.48 pounds on the surface of Jupiter; and by the same principle, we can compute the pressure on the surface of any other planet. Results will be found in table IV. CHAPTER IV. PROBLEM OF THE THREE BODIES. LUNAR PERTURBATIONS (175.) By the theory of universal gravitation, every body in the universe attracts every other body, in proportion to its mass; and inversely as the square of its distance; but simple and unexceptionable as the law really is, it produces very complicated results, in the motions of the heavenly bodies. CHAP. IV. The theory of gravity. The com results. If there were but two bodies in the universe, their motions would be comparatively simple, and easily traced, for they plexity of would either fall together or circulate around each other in some one undeviating curve; but as it is, when two bodies circulate around each other, every other body causes a deviation or vibration from that primary curve that they would otherwise have. The final result of a multitude of conflicting motions cannot be ascertained by considering the whole in mass; we must take the disturbance of one body at a time, and settle upon its results; then another and another, and so on; and the sum of the results will be the final result sought. We, then, consider two bodies in motion disturbed by a third body; and to find all its results, in general terms, is the famous problem of "the three bodies," but its complete solution surpasses the power of analysis, and the most skillful mathematician is obliged to content himself with approximations and special cases. Happily, however, the masses of most of the planets are so small in comparison with the mass of the sun, and their distances so great, that their influences are insensible. We shall make no attempt to give minute results; but we hope to show general principles in such a manner, that the reader may comprehend the common inequalities of planetary motions. The problem of the three bodies. Abstract Let m, Fig. 35, be the position of a body circulating around another body A, moving in the direction Pm B, and dis- attraction. turbed by the attraction of some distant body D. P* n relations m PL Fig. 35. B A We now propose to show some of the most general effects of the action of D, without paying the least regard to quantity. If A and m were equally attracted by D, and the attraction exerted in parallel lines, then Dwould not disturb the mutual relations of A and m. But while m is nearer to D than A is to D, it must be more strongly attracted, and let the line mp represent this excess of attraction. Decompose this force (see Nat. Phil.) into two others, mn and np, the first along the line Am, the other at right angles to it. The first is a lifting force (called by astronomers the radial force), the other is a tangental force, and affects the motion of m. It will accelerate the motion of m, while acting with it, from P to B; and retard its motion, while acting against it, from B to Q. We must now examine the effect, when the revolving body is at m', a greater distance from D than A is from D. Now A is more strongly attracted than m', and the result of this unequal attraction is the same as though A were not attracted at all, and m' attracted the other way by a force equal to the difference of the attractions of D on the two bodies A and m'. Let this difference be represented by the line m'p', and decompose it into two other forces, m' n' and n'p', the first a lifting force, the other the tangental force. The rationale of this last position may not be perceived by every reader; and to such we suggest, that they conceive A and m' joined together by an inflexible line Am', and both A and m' drawn toward D, but A drawn a greater distance than m'. Then it is plain that the position of the line Am' will be changed: the angle D Am' will become greater, and the angle CAm' less; that is, the motion of m' will be accelerated from Q to C, but from C to P it will be re- CHAP. IV. tarded. constantly urges a revol In short, the motion of m will be accelerated when moving to- The disward the line DB C, and retarded while moving from that line. turbing body That is, retarded from B to Q, accelerated from Q to C, retarded from C to P, and again accelerated from P to B. ving body to If we conceive A to be the earth, m the moon, and D the the line of sun; then DB C is called the line of the syzigies, a term which means the plane in which conjunctions and oppositions take place. At the point B the moon falls in conjunction with the sun, and is new moon; at the point C it is in opposition, or full moon. (176.) Conceive a ring of matter around a sphere, as represented in Fig. 36, and let it be either attached or detached from the sphere, and let D be not in the plane of the ring. Fig. 36. syzigies. Action of B an attracting body on a ring. From what was explained in the last article, the particles of matter at m are constantly urged toward the line DB C, and the particles at m' are constantly urged toward the same line; that is, the attraction of D, on the ring, has a tendency to diminish its inclination to the line DB C; and its position would be changed by such attraction from what it would otherwise be; and if the ring is attached to the sphere, the sphere itself will have a slight motion in consequence of the action on the ring. Now there is, in fact, a broad ring attached to the equatorial part of the earth, giving the whole a spheroidal form; and the plane of the equator is in the plane of the ring. C m When the sun or moon is without the plane of this ring, Cause of that is, without the plane of the equator, their attraction has nutation a tendency to draw the plane of the equator toward the attracting body, and actually does so draw it; which motion is called nutation. How this motion was discovered, and its amount ascertained, will be explained in a subsequent chapter. (177.) We may conceive the line DBC to be in the ring to the CHAP. IV. plane of the ecliptic, D the sun, and the ring around the earth Applica. the moon's orbit, inclined to the plane of the ecliptic with an tion of the angle of about five degrees; then when the sun is out of the lunar orbit. plane of the ring, or moon's orbit, the action of the sun has a constant tendency to bring the moon into the ecliptic, and by this tendency the moon does fall into the ecliptic from either side sooner than it otherwise would. The moon's The point where the moon falls into the ecliptic is called nodes retro- the moon's node; and by this external action of the sun the grade. represented in the figure at D.) moon falls into the ecliptic from its greatest inclination before it describes 90°, and goes from node to node before it describes 180°-and hence we say that the moon's nodes fall backward on the ecliptic. The rate of retrogradation is 19° 19′ in a year, making a whole circle in about 18.6 years. (178.) We are now prepared to be a little more definite, and inquire as to the amount of some of the lunar irregularities. Let S be the mass of the sun, E that of the earth, and m the moon, situated at D. Let a be the mean distance between the earth and sun, z the distance between the sun and moon, and r the mean radius of the lunar orbit. Let the moon have any indefinite position in its orbit. (It is S The attraction of the sun on the earth is the attrac |