Let PV be the orbit described about C, and Pv that described about S. Draw the tangent Pr, take the arch PQ extremely small, and draw CQR; also draw Sgr parallel to CR, and then PQ and Pq will be similar parts of the curves PV and Pv. 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 v Sto VS + P. But the times 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 + P. 72.-Cor. 1. The velocity in the orbit PV about C is, to the velocity in the orbit Pv about S, as S to $ + P. For the velocities are as the spaces divided by the times; therefore, vel. P9 CP SP S S + P in PV: vel, in Pv :: VS :S + P. 73.-Cor, 2. Bodies revolving romd their common centre of gravity describe areas proportional to the times. PQ 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: S. Let PN be the orbit described about L. Then (Art. 71,) per time in PQ : per. time in Pq :: VS: 15 FP :: VCP : VSP. 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 ✓LP3. But the periodic times are equal; therefore CP SP2 = V LP, and LP3 = CP * SP2, and LP = {CP X SP2, а But LP: SP :: CP X SP? : SP or VP :: V CP: VSP :: VS: 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 abont 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. 1 Then (Art. 45,) v : 2df :: B;d, and BV 2df = dv, and 2BBd; duv = ddvv, whence BB = and B =ON But (Conics,) RR 2f' 2f vod + CC = BB + dd = + dd. And, again, CR = Bp = pr 2f vod d and R+C = 12 R C + dd + 2pv v =m. Also RR +CC 29 d d 2RC = 2pvos 2f 21 mt n Therefore R = 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 2RxXx = BB, and RR — 2Rx + xx = RR - BB, and R - x= WŘR - BB; whence x = R + VRR – BB; that is, the greater part FP = R + RR - BB, and the lesser part SP = R- VRR - BB. Then, in the triangle PCF, or PCS, all the sides are given, to find the angle PCF or PCA. d + dd + 2po veoma 2f + dd—2pv vaf and R-C=T 2 + dd mn = n. and C="?* C 2 2 ad's 78.-Cor. The periodical time in seconds, is 3.1416 V v} For arch vždy: time 1" :: circumference 3.1416 + 2d : 3.1416 V , the periodical time in a circle whose radins is d. And, by Art. 63, the periodica time is the same in all circles and ellipses. Fig. 19. . D 79.-Supposing the centripetal force reciprocally as the square of the distance; to determine the orbit which a body will describe, that is, projected from a given place P, (fig. 19,) with a given velocity, in a given direction PT. By Art. 42, the body will move in a conic section, whose focus is S, the centre of force; and the line of direction PT will be a tangent at the point P. Let the distance SP = d, transverse axis AD = 2. f = space a body will descend at P, in a second, by the centri. petal force. v = the velocity the body is projected with from P, or the space it describes in a second. Then 2df is the velocity of a body revolving in a circle, at the distance SP. Then (Art. 44,)v: v 2df :: vz-div{z. Whence v vir= : V Vadfz - 2ddf, and vvz = 4df2 — 4ddf; and 4df3 — vvz = 4ddf, whence % = 4ddf duv ब 4df - 00 Therefore, if 4df is greater than vv, z is affirmative, and the orbit is an ellipsis : but if lesser, z is negative, and the curve is a byperbola; and if equal, it is a parabola. Draw SR perp. to PT, and let SR = p. Alsu draw from the other focus H, HF perp. to PT. Then (Conics,) the angle SPR = angle HPF, whence the triangles SPR, HPF, are similar; there d d-dHE p. 71,) SR HF or 2 - d = rectangle DHA or CB', the square of half the conjugate axis; therefore CB =p In the triangle SPH, the angle SPH and the sides SP, PH, are given, to find the angle PSH, the position of the transverse axis. d = fore SP (d): SR (p) :: HP (s—d): HF = 7 p; and (Art. x 4d 80.-- Cor.3. The periodical time in the ellipsis APDB = 3.1416 X 4df-tels 2d For 3.1416 V = periodic time in the circle whose radius is d. And s ed (Art. 59,) zd? : 3.1416 v ī 21 : periodical time in the ellipsis = 3.1416 12 2 z 4ddf FX zd = 3.1416 X 4uf -- vr1 81.-Cor. 2. The latus rectum of the axis AD is = dds 82.-Cor. 3. Hence the transverse axis and the periodic time will remain the same, whatever be the angle of direction SPT. For no quantities but d, f, and v, are concertied; all which are given. SCHOLIUM.-The laws of centripetal and centrifugal forces are the foundation of the theory of astronomy. For it is ascertained, by experience, that the primary planets revolve round the sun in elliptical orbits, the sun being situated in one of the foci. ' he se. condary planets, as the moon, the satellites of Jupiter, &c. also revolve round their primaries in orbits of an elliptical farm. The comets are likewise known to revolve round the sun in ellipses, having so great a degree of eccentricity as to approach very nearly to parabolic orbits. The existence of any distant sympathy between matter producing the phenomena of Attraction, Repulsion, and Gravitation, is an appeal to faith founded on the phenomena and ou the difficulty in accounting for them. But, as these sympathies produce force in the bodies, and FORCE is aniversally the product of multer and motion, Sir Richard Phillips, in his new System of Physics, contends that they are necessarily so many results of imparted motion, and that, instead of giving names to these sympathies, we ought to trace the specific motions which produce the phenomena. Upon this principle he has investigated many phenomena hitherto ascribed to innate powers of mat. ter; and he maintains, that bodies fall towards the centre of the earth, or possess the momentum of weight, owing to the two-fold motions of the earth, of which all the parts of the terrestrial mass are the patients; he also maintains, that the rotation, or motion, of the son, transferred or diffused through the gazeous medium which fills space, carries round the planets in their orbits; and he then descends to the phenomena of Atoms, showing that atomic motion is heat, and gas is atoms in motion; and hence explains the phenomena of Combustion, Animal Heat, Electricity, &c. But, as the general laws of diffused motion, by this system, accord with those of gravitation, as developed by Hooke and Newton, the mathematical resnlts are the same on either hypothesis, though the philosophy and reasoning are very different. In the preceding pages, the general principle of centrifugal and ceutripetal forces are adopted, whatever be their cause; whether, according to Newton, they consist of an original projectile force impressed at the creation, and a gravitating force arising from innate properties of matter, or whether, according to the new doctrine, the effect is occasioned by the mutual transfer of motions, wbach create action and reaction between the sun and planets, directly as their Vaantities, and inversely as the squares of their distances. The results, as go- . vered by the same law, mathematically considered, will be the same. But, for a work of general use, the received language, for many obvious reasons, has been preserved. N.B.-The mean diameter of the Earth being 7960 miles, and its mcan distance from the Sun 95,000,000, the diameters and distances of the other Planets may readily be found. As this rolume is not intended to supersede a system of Natural Philosophy, the reader who wishes to pursue these subjects further may consult ENFIELD'S Institutes, PLAYFAIR's or Robinson's Elements ; Newton's Principia (of which there is an English edition by MOTTE); VINCE's or SQUIRE'S Astronomy, Wood's, Kipling's, or SMITH's Optics ; Hutton's Course on Projectiles und Gunnery; Simson's Essays; and GREGOR Y's Mechanics. Hutton's or BARLOW'S Mathematical Dictionaries, and Young's Lectures on Natural Philosophy, wili - also supply many desideratá. Readers of the French Language will also find an inexhuustible mine of mathematical research in the Mechanique Celeste of the MARQUIS DE LA Place, and in the works of La Croix and LEGENDRE, of which & more extended notice appears in the Introduction. |