59.- If the centripetal forces be reciprocally as the squares of the distances, the periodic times in ellipses will be in the sesquiplicate ratio of the transverse axes AB, (fig. 14;) or the squares of the periodic times will be as the cubes of the mean distances FD, from the common centre. Put the symbols as in the last, and t, T, for the periodical times. Then, by the nature of the ellipsis, cc = {lr, and c= v lr, and rc =rv{lr. And, for the same reason, RC = RM LR. Also (Art. 51,) the areas described in the same time are as the square roots of the parameters; and, therefore, the whole areas of the ellipses are as the periodical times multiplied by the square roots of the parameters. But the whole areas are also as the rectangles of the axes ; therefore the rectangles of the axes are as the periodical times multiplied by the square roots of the parameters; that is, rc or ry {lr : RC or RV {LR :: tvi:TVL. And squaring, fir3 : {LRS :: ttl : TTL, That is, 73: R::: tt : TT. And t:T::g: R$::27? : 2Ri. 60.-Cor, 1. The areas of the ellipses are as the periodic times multiplied by the square roots of the parameters. 61.Cor. 2. The periodic time in an ellipsis is the same as in a circle, whose diameter is equal to the transverse axis AB, or the radius equal to the mean distance FD. 62.-Cor. 3. The quantities of matter in central attracting bodies, that have others revolving about them in ellipses, are as the cubes of the mean distances, divided by the squares of the periodical times. For (Cor. 2,) the periodic times are the same when the mean distances are equal to the radii ; and the rest follows from Art. 32. Fig. 15. DB ALE C B E 63.-If the centripetal forces be directly as the distances, the periodic times of bodies moving in ellipses round the same centre will be all equal to one another. Let AEL (fig. 15,) be an ellipsis, AGL a circle on the same axis AL, C the centre of both. Draw the tangent AD, and npF pa. rallel to it, and Dn, Bp, parallel to AC; AF being very small. The Dn equal to Bp will be as the centripetal force; and therefore AD and AB, or An and Ap, will be described in the same time, in the circle and ellipsis. Consequently, the areas des cribed in these equal times will be AnC and Apc. But these areas are to one another as nF to PF, or as GC to EC ; that is, as the area of the circle AGL to the area of the ellipsis AEL. Therefore, since parts proportional to the wholes are described in equal times, the wholes will be described in equal times: and therefore the periodic times, in the circle and ellipsis, are equal. But (Art. 26,) the periodic times in all circles are equal in this law of centripetal force; and, therefore, the periodic times in all ellipses are equal. 64.-Cor. The velocity at any point I of an ellipsis, is as the rectangle of the two axes AC, CE, divided by the perpendicular CH, upon the tangent at 1. For the arch I x CH is as the area described in a small given part of time, and that is as the whole area, (because the periodic times are equal,) or ai AC X CE AC X CE; and therefore the arch I, or the velocity, is as CH 65.-The densities of central attracting bodies are reciprocally as the cubes of the parallaxes of the bodies revolving about them, (as seen from these central bodies,) and reciprocally as the squares of the periodic times. For the density, multiplied by the cube of the diameter, is as the quantity of matter; that is (by Art. 62,) as the cube of the mean distance divided by the square of the periodical time of the revolv. ing body: and, therefore, the density is as the cube of the distance, divided by the cube of the diameter, and by the square of the pe‘riodic time. But the diameter, divided by the distance, is as the angle of the parallax; therefore, the density is as 1 divided by the cube of the parallax, and the square of the periodic time. L 2 al B Ñ 66.-1f two bodies A, B, (fig. 16,) revolve about each other they will both of them revolve about their centre of gravity. Let C be the centre of gravity of the bodies A, B, acting upon one another by any centripetal forces; and let AZ be the direction of A's motion; draw BM parallel to AZ, for the direction of B; and let AZ, BH, be described in a very small part of time, so that AZ may be to BH as AC to BC; and then C will be the centre of gravity of Z and H, because the triangles ACZ and BCH are simi. lar. Whence AC : CB :: ZC: CH. But, as the bodies A and B attract one another, the spaces Aa and Bb they are drawn through will be reciprocally as the bodies, or directly as the distances from the centre of gravity; that is, Aa : Bb :: AC : BC. Complete the parallelograms Ac and Bd; and the bodies, instead of being at Z and H, will be at c and d. But since AC : BC :: Aa : Bb. By division AC : BC :: aC: 6C. But AC : BC :: AZ: BH :: ec : bd. Whence aC : 6C :: ac : bd. Therefore the triangles Ca, and dCb, are similar, whence Cc ; Cd :: ac : bd :: AC : BC :: B :A. Therefore C is still the centre of gravity of the bodies at c and d. In like manner, producing Bd and Ac, till dg be equal to Bd, and cq to Ac; and if cf, dh, be the spaces drawn through by their mu. tual attractions; and if the parallelograms ce, di, be completed; then it will be proved, by the same way of reasoning, that C is the centre of gravity of the bodies at q and g, and also at e and i, where A describes the diagonals Ac, ce, &c. and B the diagonals Bd, di, &c. and so on ad infinitum. If one of the bodies B is at rest whilst the other moves along the line AL, then the centre of gravity C will move uniformly along the line Co, parallel to AL. Therefore, if the space the bodies 'move in be supposed to move in direction Co, with the velocity of the centre of gravity, then the centre of gravity will be at rest in that space, and the body B will move in direction BH, parallel to CO or AZ; and then this case comes to the same as the former. Therefore the bodies will always move round the centre of gravity, which is either at rest or moves uniformly in a right line. If the bodies repel one another, by a like reasoning it may be proved that they will constantly move round their centre of gravity. If the lines CA, Cc, Ce, &c. be equal, and CB, Cd, Ci, &c. also equal; then it is the case of two bodies joined by a rod or a string, or of one body composed of two parts. This body or bodies will always move round their common centre of gravity. 67.—Cor. 1. The directions of the bodies, in opposite points of the orbits, are always parallel to one another, For, since AZ: 2c :: BH: Hd; and AZ, Zc, parallel to BH, Hd; therefore the <ZAe= _HBd, and Bd parallel to Ac. And, for the same reason, di is parallel to ce, &c. 68.-Cor. 2. Two bodies, acting upon one another by any forces, describe similar figures about their common centre of gravity. For the particles Ac, Bd, of the curves are parallel to one another, and every where proportional to the distances of the bodies AC, BC. 69.-If the forces be directly as the distances, the bodies will describe concentrical ellipses round the centre of gravity. 70.-Cor. 4. If the forces be reciprocally as the sqnares of the distances, the bodies will describe similar ellipses, or some conic sections, about each other, whose centre of gravity is in ihe focus of both. 71.-If two bodies, S, P, (fig. 17,) attract each other with any forces, and at the same time revolve about their centre of gravity C, then, if either body P, with the same force, describes a similar curve about the other body S at rest, its periodical time will be, to the periodical time of either about the centre of gravity, as the square root of the sum of the bodies ( ✓ + P,) to the square root of the fixed or central body (VS) Let PV be the orbit described about C, and Py that described about S. Draw the tangent Pr, take the arch PQ extremely small, and draw CQR; also draw Sqr parallel to CR, and then PQ and Pq will be similar parts of the curves PV and Po. Now, the times that the bodies are drawn from the tangent through the spaces QR, qr, with the same force, will be as the square roots of the spaces QR, qr; that is, (because of the similar figures CPRQ and SPrq,) as v CP to vSP; that is, (by the nature of the centre of gravity,) as VS to VS + P. But the tinies wherein the bodies are drawn from the tangent through RQ, rq, are the times wherein the similar arches PQ, Pq, are described; and these times are as the whole periodic times. Therefore, the periodic time in PV is, to the periodic time in Pv, as VS to vs 7 P. 72.-Cor. 1. The velocity in the orbit PV about C is, to the velocity in the orbit Pv about S, as v S to S + P. For the velocities are as the spaces divided by the times; therefore, vel. PR P9 CP SP S S + P in PV: vel, in Pv :: vs 757 75:15 p18 ✓S + P : S+P. 73.- Cor. 2. Bodies revolving round their common centre of gravity describe areas proportional to the times. 74.-If the forces be reciprocally as the squares of the distances, and if a body revolves about the centre L, (fig. 17,) in the same periodical time that the bodies S, P, revolve about the centre of gravity C; then will SP : LP :: !/S + P: 5. Let PN be the orbit described about L. Then (Art. 71,) per time in PQ : per. time in Pg :: VS: S+P:: CP : V SP. And (Art. 59.) per. time in Pq : per time in PN :: SP} : LP); supposing PQ,.PN, similar archies. Therefore, per time in PQ : per. time in PN :: VCP x sp} : VSP X LP} :: CP x SP2 ✓LP. But the periodic times are equal; therefore CP X SP2 = VLP', and LP3 = CP x SP?, and LP = {CP X SP2, 774 ASTRONOMY. But LP : SP :: CP x SP? : SP or VP' :: VCP : VSP :: gsps V5: VS + P. 75.-Cor. 1. If the forces be reciprocally as the squares of the distances, the transverse axis of the ellipsis described by P, about the centre of gravity C, is, to the transverse axis described by P about the other body S, at rest, in the same periodical time, as the cube root of the sum of the bodies S + P to the cube root of the fixed or central body S. 76. Cor. 2. If two bodies, attracting each other, move about their centre of gravity, their motions will be the same as if they did not attract one another, but were both · attracted with the same forces, by another body placed in the centre of gravity. 77.--Suppose the centripetal force to be directly as the distance, to determine the orbit which a body will describe, that is projected from a given place P, (fig. 18,) with a given velocity in a given direction PT. By Ex. 2, to Art. 42, the body will move in an ellipsis, whose centre is С the centre of force, and the line of direction PT will be a tangent at the point P. Draw CR perp. to PT; and let the distance CP = d, CR = p, semitransverse axis CA = R, semiconjugate axis CB = C. CG (the semiconjugate to CP,) = B. f= space a body would descend, at P, in a second, by the centripetal force. v = the velocity, at P, the body is projected with, or the space it describes in a second. Then 2df = velocity of a body revolving in a circle at the distance CP. Then (Art. 45,) v : 2df :: B:d, and BV 2df = do, and 2BBdf duv d =ddvv, whence BB = 2f' and B = va But (Conics,) RR vod + CC = BB + dd = + dd. And, again, CR = Bp = po 2f d vod no = Therefore RR + CC. + 2RC = + dd + 2pv v 2f af vod d and R +C= + dd + 2pv v =m. Also RR + CC2f 3f vod d vvd 2RC= R-= 2f 2pvc 2f 21 mt n Therefore R = and C= 2 Then, to find the position of the transverse axis AD. Let F, S, be the foci. Then (by Conics,) we shall bave SC or CF =VRR-CC. Put FP = x; then SP = 2R X, and also SP x PF or 2R.rXX = BB, and RR 2Rx + xx = RR - BB, and R - x = WŘR - BB; whence r =REVRR - BB; that is, the greater part FP = R + VRR - BB, and the lesser part SP = RVRR - BB. Then, in the triangle PCF, or PCs, all the sides are given, to find the angle PCF or PCA. n. 2 1 |