While a ship A (fig. 3) moves from a to Ift, let 01110Te through c D motion, while the stm moves round the earth, also uniformly and circularly, what path will the planet actually trace out? To get a notion of the possible speaies of curves, let us simplify the question b7 supposing a circle • a CD moving along a straight line while a point moves round the circle from A.
(C D being equal and parallel to AB), E through EP (Er being parallel to and equal to one-half of A B), o through o n, x through x L, x through N, and let P remain at rest. Then, a spectator in A supposing himself at rest, c will appear to remain at c, E will appear to move through Ef, through o h, x through x 1, m through Mn, and r through rp. The motion of A has been transferred in a contrary direction to each of the other vessels.
When bodies are very distant their changes of distance are not soon perceived, consequently it is only by change of direction that their motion becomes visible. This is the case in all the heavenly bodies ; but we shall now ehow what the apparent motion of a planet, superior and inferior, would be, if changes of distance, as well as direction, could be perceived and estimated.
If the spectator be in motion, an object at rest appears to him to have precisely his own motion, but in a contrary direction; for if the object be o and the spectator move through A B D, no distances would In the first place, if A did not move round at all, the line A o would be described ; if • moved slowly round, the translation of the circle would cause an undulating curve like A n x to be described; if A moved as fast on the circle as the circle itself is moved forward, the undulation would be changed into a curve with cusps like A L o ; while if A move faster on the circle than the circle is carried forward, the circle, so to speak, will not have time to get out of the way, and prevent the formation of loops, aa in ADINMPQRQ.... The faster A moves, the larger and the nearer will be the loops, so that at length no one will be clear of the preceding and following, or the loops will interlace.
if the circle move round another circle, the same appearances will be presented in an inverse order. Let the centre E of the circle ABCD (fig. 7) be carried round the circle ET, whose centre is o. If A did not move at all upon its circle, it would, by the motion of its circle, describe a circle (dotted) equal to if A moved slowly, it would describe a succession of e ose loops enveloping o ; if quicker, the loupe would at last disengage themselves from each other ; while for still more rapid motion of A the loops would become cusps, and afterwards the curve be changed if the spectator were fixed at o, and the object moved through A Is c n, and all directions would only undergo a diametrical change. Consequently the relative motion of the object is represented by allowing it to change places with the spectator, and inverting tho direction of north and sopth, which will have the effect of making the relative motion from west to east, if that of the spectator were from east to west, and rice rend. Let ua suppose now that the earth moves round the sun in a circle, which will be near enough for our present would simply undulate. The character of these curves will be further
discussed under TRocnomAL CURVES, and their astronomical appli cation under PLANETARY 31orioNs. It ie sufficient here to say that the apparent orbits of all the planets (or rather, the orbits as they would be if changes of distance were perceptible) are trochoidal curves of the above-described species, with loops which do not interfere with one another.
The composition of motion has been virtually proved in the preceding paragraphs, combined with the account of the emend LW of motion.
Laws or.] If causes of motion act instantaneously, one of which would make a body describe A a (fig. 8) uniformly, and the other purpose ; it will be immediately obvious that the direction of motion, so far as concerns the order In which constellation will be described, is the earns In the relative motion of the sun round the earth as in the absolute motion of the earth round the sun. For though the absolute directions of motions are opposite , yet a, to a spectator at I:, is seen towards a point of the heavens opposite to that in which E appears from 5. PIOTiON, DIRICVON Or) In giving to the sun the apparent motion which answers to the real motion of the earth, the same motion must be given to the orbits in which the planets are carried round the sun. The question then is as follows a planet move round the sun, my with a uniform circular A C, in the same limeore find in the second law of motion that the body will move so that its distance from A B at the end of any time, measured parallel to A o, is what it would have been if the cause of motion in the direction A II had never existed nor acted. Suppose, for example, that three-flfths of the whole time of motion from A to B has elapsed ; take a n three-fifths of c, and the body mutt be then somewhere in the line it r« Again, take A r three-fifths of A B, and by the same law it follows that the body must be in the line r o, that is, it must be at the point rt, which simple geometry shows to be on the diagonal A R, and by three.fifths of that diagonal distant from A. The same may be shown for any other proportion of the whole time ; consequently the body, impressed with the two motions, describes the diagonal A x uni formly, and in the same time as that in which the separate motions, from A to B, or from A to c, would have been performed. This is pre cisely the course the body would have taken in space, if, while it moved from A to B on the paper, the paper itself had taken the motion A c; but the establishment of the latter assertion must not be confounded with the proof of the composition of velocities impressed on mattes ; the latter requires those considerations which lead to the second law of motion.
There are a great many uses of the word " motion," which are con venient, but require the introduction of arbitrary suppositions. Thus the moon never cuts the ecliptic twice running in the same place, and the intersection of her orbit with the ecliptic being called a node, it is said that the node mores, thus giving this node a sort of imaginary existence in the interval.