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and equator seen edgewise, then the orbits of his four satellites seen edgewise will have the positions J 1, J 2, J3, J4. These are extremely near to one another, for the angle E JO is only 3° 5' 30".

NOTE 88, p. 28. In consequence of the satellites moving so nearly in the plane of Jupiter's equator, when seen from the earth, they appear to be always very nearly in a straight line, however much they may change their positions with regard to one another and to their primary. For example, on the evenings of the 3rd, 4th, 5th, and 6th of January, 1835, the satellites had the configurations given in fig. 23, where O is Jupiter, and 1, 2, 3, 4, are the first, second, third, and fourth satellites. The

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satellite is supposed to be moving in a direction from the figure towards the point. On the sixth evening the second satellite was seen on the disc of the planet.

NOTE 89, p. 28. Angular motion or velocity is the swiftness with which a body revolves-a sling, for example; or the speed with which the surface of the earth performs its daily rotation about its axis.

NOTE 90, p. 29. Displacement of Jupiter's orbit. The action of the planets occasions secular variations in the position of Jupiter's orbit J O, fig. 22, without affecting the plane of his equator JE. Again, the sun and satellites themselves, by attracting the protuberant matter at Jupiter's equator, change the position of the plane J E without affecting JO. Both of these cause perturbations in the motions of the satellites.

NOTE 91, p. 29. Precession, with regard to Jupiter, is a retrograde motion of the point where the lines J O, JE, intersect fig. 22.

NOTE 92, p. 30. Synodic motion

of a satellite. Its motion during the interval between two of its consecutive eclipses.

A

NOTE 93, p. 30. Opposition. body is said to be in opposition when its longitude differs from that of the sun by 180°. If S, fig. 24, be the sun, and E the earth, then Jupiter is in opposition when at O, and in conjunction when at C. In these positions the three bodies are in the same straight line.

NOTE 94, p. 30. Eclipses of the

Fig. 24.

satellites. Let S, fig. 25, be the sun, J Jupiter, and a Bb his shadow. Let the earth be moving in its orbit, in the direction E ART H, and the

T

Fig. 25. B

third satellite in the direction a bm n. When the earth is at E, the satellite, in moving through the arc a b, will vanish at a, and reappear at b, on the same side of Jupiter. If the earth be in R, Jupiter will be in opposition; and then the satellite, in moving through the arc a b, will vanish close to the disc of the planet, and will reappear on the other side of it. But, if the satellite be moving through the arc mn, it will appear to pass over the disc, and eclipse the planet.

NOTE 95, pp. 30, 43. Meridian. A terrestrial meridian is a line passing round the earth and through both poles. In every part of it noon happens at the same instant. In figures 1 and 3, the lines NQS and NGS are meridians, C being the centre of the earth, and NS its axis of rotation. The meridian passing through the Observatory at Greenwich is assumed by the British as a fixed origin from whence terrestrial longitudes are measured. And as each point on the surface of the earth passes through 360°, or a complete circle, in twentyfour hours, at the rate of 15° in an hour, time becomes a representative of angular motion. Hence, if the eclipse of a satellite happens at any place at eight o'clock in the evening, and the Nautical Almanac shows that the same phenomenon will take place at Greenwich at nine, the place of observation will be in the 15° of west longitude.

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Conjunction. Let S be the sun, fig. 24, E the earth, and JO J'C' the orbit of Jupiter. Then the eclipses which happen when Jupiter is in O are seen 16m 26* sooner than those which take place when the planet is in C. Jupiter is in conjunction when at C, and in opposition when in O.

NOTE 97, p. 31. In the diagonal, &c. Were the line A S, fig. 26, 100,000 times longer than AB, Jupiter's true place would be in the direction A S', the diagonal of the figure AB S'S, which is, of course, out of proportion.

NOTE 98, p. 31. Aberration of light. The celestial bodies are so distant that the rays of light coming from them may be reckoned parallel. Therefore, let S A, S' B, fig. 26, be two rays of light

coming from the sun, or a planet, to the earth moving in its orbit in the direction A B. If a telescope be held in the direction A S, the ray S A, instead of going down the tube, will impinge on its side, and be lost in consequence of the telescope being carried with the earth in the direction A B. But, if the tube be held in the position A E, so that AB is to A S as the velocity of the earth to the velocity of light, the ray will pass through S'E A. The star appears to be in the direction A S', when it really is in the direction AS; hence the angle SA S' is the angle of aberration.

NOTE 99, p. 32. Density proportional to elasticity. The more a fluid, such as atmospheric air, is reduced in dimensions by pressure, the more it resists the pressure.

NOTE 100, p. 32. Oscillations of pendulum retarded. If a clock be carried from the pole to the equator, its rate will be gradually diminished, that is, it will go slower and slower because the centrifugal force, which increases from the pole to the equator, diminishes the force of gravity.

NOTE 101, p. 34. Disturbing action. The disturbing force acts here in the very same manner as in note 63; only that the disturbing body d, fig. 14, is the sun, S the earth, and p the moon.

NOTE 102, pp. 35, 36, 86. Perigee. A Greek word, signifying round the earth. The perigee of the lunar orbit is the point P, fig. 6, where the moon is nearest to the earth. It corresponds to the perihelion of a planet. Sometimes the word is used to denote the point where the sun is nearest to the earth.

NOTE 103, p. 35. Erection. The evection is produced by the action of the radial force in the direction Sp, fig. 14, which sometimes increases and sometimes diminishes the earth's attraction to the moon. It produces a corresponding temporary change in the excentricity, which varies with the position of the major axis of the lunar orbit in respect of the line Sd, joining the centres of the earth and sun.

NOTE 104, p. 35. Variation. The lunar perturbation called the variation is the alternate acceleration and retardation of the moon in longitude, from the action of the tangential force. She is accelerated in going from quadratures in Q and D, fig. 14, to the points C and O, called syzygies, and is retarded in going from the syzygies C and O to Q and D again.

NOTE 105, p. 36. Square of time. If the times increase at the rate of 1, 2, 3, 4, &c., years or hundreds of years, the squares of the times will be 1, 4, 9, 16, &c., years or hundreds of years.

NOTE 106, p. 37. In all investigations hitherto made with regard to the acceleration, it was tacitly assumed that the areas described by the radius vector of the moon were not permanently altered; that is to say, that the tangential disturbing force produced no permanent effect. But Mr. Adams has discovered that, in consequence of the constant decrease in the excentricity of the earth's orbit, there is a gradual change in the central disturbing force which affects the areal velocity, and consequently it alters the amount of the acceleration by a very small quantity, as well as the variation and other periodical inequalities of the moon. On the

latter, however, it has no permanent effect, because it affects them in opposite directions in very moderate intervals of time, whereas a very small error in the amount of the acceleration goes on increasing as long as the excentricity of the earth's orbit diminishes, so that it would ultimately vitiate calculations of the moon's place for distant periods of time. This shows how complicated the moon's motions are, and what rigorous accuracy is required in their determination.

To give an idea of the labour requisite merely to perfect or correct the lunar tables, the moon's place was determined by observation at the Greenwich Observatory in 6000 different points of her orbit, each of which was compared with the same points calculated from Baron Plana's formulæ, and to do that sixteen computers were constantly employed for eight years. Since the longitude is determined by the motions of the moon, the lunar tables are of the greatest importance.

NOTE 107, p. 37. Mean anomaly. The mean anomaly of a planet is its angular distance from the perihelion, supposing it to move in a circle. The true anomaly is its angular distance from the perihelion in its elliptical orbit. For example, in fig. 10, the mean anomaly is PC m, and the true anomaly is P Sp.

NOTE 108, pp. 38, 68. Many circumferences. There are 360 degrees or 1,296,000 seconds in a circumference; and, as the acceleration of the moon only increases at the rate of eleven seconds in a century, it must be a prodigious number of ages before it accumulates to many circumferences.

NOTE 109, p. 39. Phases of the moon. The periodical changes in the enlightened part of her disc, from a crescent to a circle, depending upon her position with regard to the sun and earth.

NOTE 110, p. 39. Lunar eclipse. Let S, fig. 27, be the sun, E the earth, and m the moon. The space a Ab is a section of the shadow, which

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has the form of a cone or sugar-loaf, and the spaces A ac, Abd, are the penumbra. The axis of the cone passes through A, and through E and S, the centres of the sun and earth, and n m n' is the path of the moon through the shadow.

NOTE 111, p. 39. Apparent diameter. The diameter of a celestial body as seen from the earth.

NOTE 112, p. 40. Penumbra. The shadow or imperfect darkness which precedes and follows an eclipse.

NOTE 113, p. 40. Synodic revolution of the moon. The time between two consecutive new or full moons.

NOTE 114, p. 40. Horizontal refraction. The light, in coming from a celestial object, is bent into a curve as soon as it enters our atmosphere; and that bending is greatest when the object is in the horizon.

NOTE 115, p. 40. Solar eclipse. Let S, fig. 28, be the sun, m the moon, and E the earth. Then a Eb is the moon's shadow, which some

Fig. 28.

P

a m

times eclipses a small portion of the earth's surface at e, and sometimes falls short of it. To a person at e, in the centre of the shadow, the eclipse may be total or annular; to a person not in the centre of the shadow a part of the sun will be eclipsed; and to one at the edge of the shadow there will be no eclipse at all. The spaces Pb E, P'a E, are the penumbra.

Fig. 29.

NOTE 116, p. 43. From the extremities, &c. If the length of the line ab, fig. 29, be measured, in feet or fathoms, the angles Sba, Sab, can be measured, and then the angle a Sb is known, whence the length of the line S C may be computed. a Sb is the parallax of the object S; and it is clear that, the greater the distance of S, the less the base ab will appear, because the angle a S'b is less than a Sb.

NOTE 117, p. 44. Every particle will describe a circle, &c. If N S, fig. 3, be the axis about which the body revolves, then particles at B, Q, &c., will whirl in the circles B G A a, QE q d, whose centres are in the axis N S, and their planes parallel to one another. They are, in fact, parallels of latitude, Q E qd being the equator.

NOTE 118, p. 44. The force of gravity, &c, Gravity at the equator acts in the direction QC, fig. 30. Whereas the direction of the centrifugal force is exactly contrary, being in the direction C Q; hence the difference of the two is the force called gravitation, which makes bodies fall to the surface of the earth. At any point, m, not at the equator, the direction of gravity is m b, perpendicular a to the surface, but the centrifugal force acts perpendicularly to N S, the axis of rotation. Now the effect of the centrifugal force

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