For, if a be contained an even number of times in 180, then the number of images in the second series, formed by reflections at the surface 180 2 AB, is ; and the distance EOQ (2ma), of ihe last image from E, is 2a 1800. In the same manner, t'e distance EIV, of the last image, in the second series, formed by reflections at AC is 180°, therefore the two images, Q and V, coincide in EA proluced. If a be contained an odd number of times in 180, then the number of iniages in the first series, 180 +0 formed ly reflections at AB, is and the distance EOK of the 2u 180 ta last of these images from E, is X 2a - 2c; or 180° + a - 2c. 2a Also the distance EVP, of the last image in the first series formed by reflections at AC, is 180° + n - 26; therefore EOK + EMP=360 +2a - 20 - 2b = 360°, that is, K and P coincide. ASTRONOMY. THE LAWS OF CENTRIPETAL AND CENTRIFUGAL FORCE: Showing the Motions of Bodies in Circular Orbits, and in the Conic Sections, and other Curves. In the following chapter is explained and demonstrated the laws of Centripetal Forces; a doctrine upon which all Astronomy is grounded, and without the knowledge of which po rational account can be given of the motions of any of the celestial bodies, as the Comets, the Planets, and their Satellites. Io the first section are given the centripetal forces of bodies re. volving in circles; their velocities, periodic times, and distances, conpared together; their relations and proportions to each other, and that when they either revolve about the same centre, or about different ones. The different notions caused by different forces, or by different central acting bodies, are here shown. In the second section are shown the notion of bodies in the ellipsis, byperbola, and parabola, and in other curves ; the proportion of the centripetal forces, and velocities in different parts of the same curve; the law of centripetal force to describe a given curve, and the velocity in any point of it; and more particularly with re. spect to that law of centripetal force that is reciprocally as the square of the distance, which is the grand law of nature in regard to the action of bodies upon one another at a distance; and accordiing to this law is shown the motion of bodies round one another, and round their common centre of gravity, and the orbits they will describe. Definitions.-1. The centre of attraction, is the point towards which any body is attracted or impelled. 2. Centripetal force, is that force by which a body is impelled to a certain point, as a centre. Here all the particles of the body are equally acted on by the force. 3. Centrifugal force, is the resistance a moving body makes to pre. vent its being turned out of its direct course. This is opposite and equal to the centripetal force; for action and re-action are equal and contrary. 4. Angular velocity, is the quantity of the angle a body describes in a given time about a certain point, as a centre. Apparent velocity is the same thing. 5. Periodical time, is the time of revolution of a body round a centre. J. On the Motion of Bodies in Circular Orbits. 1. The centripetal forces, whereby equal bodies, at equal distances from the centres of force, are drawn towards these centres, are us the quantities of matter in the central bodies. For, since all attraction is made towards bodies, every part of the attracting body must contribute its share in that effect. Therefore, a body twice as great will attract the same body twice as much; and one thrice as great, thrice as much, and so on. Tbere fore, the attraction of the central body, that is, the centripetal force, is as the quantity of matter in the attracting or central body. 2.-Cor. 1. Any body, whether great or little, placed at the same distance, is attracted throngh equal spaces in the same time, by the central body. For, though a body, twice or thrice as great as another, is drawn with twice or ihrice the force, yet it will acquire no greater velocity, nor pass through a greater space. For (by Mechanics,) the velocity generated iu a given time is as the force directly, and qnantity of matter reciprocally; and the force, which is the weight of the body, being as the quantity of matter, therefore the velocity generated is as the quantity of matter directly, and quantity of matter reciprocally, and therefore is a given quantity. 3.-Cor. 2. Therefore, the centripetal force, or force towards the centre is not to be measured by the quantity of the falling body, but by the space: falls through in a given time. And, therefore, it is sometimes called an 7 celerative force. Fig. 1. 4.-If a body revolves in a circle, (fig. 1,) and is retained in it, by a centripetal force tending to the centre of it; put R = radius of the circle or orbit described, AC. F = absolute force, at the distance R. 1 : = the space a falling body could descend through, by the force at A; and t = lime of the descent. * = 3.1416. Then its periodic time, or the time of one revolution will be * V2R. And the velocity, or s ace it describes in the time t, Also, by the laws of uniform motion, t v4D, will be ✓2Rs. For, let AB be a tangent to the circle at A ; take AF an ipfi. nitely small arch, and draw FB perp. to AB, and FD perp. to the radius AC. Let the body descend through the infinitely small height AD or BF, by the centripetal force in the time 1. Now that the body may be kept in the circular orbit AFE, it ought to describe the arch AF in the same time 1. The circumference of the circle AE is 2R, and the arch AF = V2R X AD. AD By the laws of falling bodies 18:t:: VAD:tv = time of moving through AD or AF. And, by uniform motion, as AF, to the time of its description :: circumference AFEA, to the time of AD one revolution; that is, V2R X AD:t :: 2*R : periodic 2tR 2R time = V2Rs AD or time of describig AF: AF or 2R X AD ::t: V2Rs = the velocity of the body, or space described in time t. 5.-Cor. 1. The velocity of the revolving body is equal to that which a falling body acquires in descending through half ihe radius AC, by the force at A uniformly continued. For V8 (height): 28 (the velocity) :: ✓ ĮR (the height): V 2Rs, the velo. city acquired by falling through { R. 6.-Cor. 2. Hence, if a body revolves uniformly in a circle, by means of a given centripetal force, the arch, which it describes in any time, is a mean proportional between the diameter of the circle and the space which the body would descend through in the same time, and with ihe same given force. For 2R (diameter): ✓2Rs :: ✓ R$: s; where ✓2Rs is the arcb de. scribed, and s the space descended through, in the time t. 7.-Cor 3. If a body revolves in any carve AFQ, (fig. 2,) about the centre of force 8; and if AC or R be the radius of curvature in any point A; 8 = space descended by the force directed to C; then the velocity in A will be V Rs. For this is the velocity in the cirele; and therefore in the curve, which coincides with it. 8.- If several bodies revolve in circles round the same or different centres, the periodic times will be as the square roots of the radii directly, and the square roots of the centripetal forces reciprocally. S S Let F = centripetal force at A, (fig. 1,) tending to the centre C of the circle. V = velocity of the body. P = periodic time. 2R it; whence P = IVF ; and, since 2, , and t, are given quan R tities, therefore VF 9.-Cor. 1. The periodic times are as the railii directly, and the velocities reciprocally. OR For (Art 4,) V=V2Rs = V2RF, and V2 — 2RF, and P=#tv 2R and PP = 7? 2 x therefore P2 V2 = ?? (2 x 4R?, and p = F q? t? X 4R? at X PR R V 10.-Cor. 2. The periodic times are as the velocities directly, and the centripetal forces reciprocally. VV R V For V2 = 2Rs = 2RF; and R= and But (Cor. 1.) V 2F 10.--Cor. 3. If the periodic times are eqnal, the velocities, and also the centripetal forces, will be as the radii. R R V For, if P be given, then and and F are all given ratios. 12.-Cor. 4. If the periodic times are as the square roots of the radii, the velocities will be as the square roots of the radii, and the centripetal forces equal. RR For (Art.8,) putting ✓ R for P, we bave v RAVE Therefore, 1 a and VRA V, and V F is a given quantity. 2F' 13.-Cor. 5. If the periodic times are as the radii, the velocities will be equal, and the centripetal forces reciprocally as the radii. R R and 1 For, putting R for P, we have Rava; whence vra ; that is, R Q , or the centripetal force is reciprocally as the radins; and V is a given quantity, 14. Cor. 6.-If the periodic times are in the sesquiplicate ratio of the radii, the velocities will be reciprocally as the square roois of the radii, and the centripetal forces reciprocally as the squares of the radii. Put Ri for P, then Rt av R R an and VR 15.-Cor. 7. If the periodic times be as the nth power of the radius, then the velocities will be reciprocally as the n-1th power of the radii, and centripetal forces reciprocally as the 2n-1th power of the radii. Ꭱ Ꭱ R Pui Ra for P, tben R" (Vaj. Whence Rin a and Rs Also R an 16.-If several bodies revolve in circles round the same or different centres, the velocities are as the radii directly, and periodic times reciprocally. For, putting the same letters as in Art. 8, we have (Art. 4,) V= V v ?Rs = V2RF; and P a (Art. 10,) and PF «V, and Fap F 2RV Ifhence V = V2RF = 2R X and V? = and V = P P 2RR р 17.-Cor. 1.. The vclocities are as the periodical times, and the centripetal forces. For we had PF o V. 18.-Cor. 2. The squares of the velocities are as the radii and the centri. petal forces. For V=V2RF. 19.-Cor. 3. If the velocities are equal, the periodic times are as the radii, and the radii reciprocally as the centripetal forces. R for, if V be given, its equal 3 is a given ratio ; and ✓ RP is given, whence R« 20.-Cor. 4. If the velocities be as the radii, the periodical times will be the same, and the centripetal forces as the radii. R For then V or Ra P 21.-Cor. 5. If the velocities be reciprocally as the radii, the centripetal forces are reciprocally as the cubes of the radii, and the periodic times as the squares of the radii. = 2RF, whence Fas R KR R: R and P Q RR. and i di la Alo R=v2kf, whence RQ F. 1 1 For, put Ã for V, then(-Art. 18,) 1 =vxUF, |