261. Now, the areas described by the radii vectores varying with the times, the fourth term of the proportion (A) (art. 259.) becomes, on substituting this expression for r2 and P for dt 365.256384, e sin. ◊ (1 — e2)* and this is the time in which the planet, setting out from the perihelion, would describe about the sun an angle equal to 0. 262. In the elliptical theory the three laws of Kepler may be considered as facts obtained from observation: but some of the comets describe about the sun ellipses whose major axes are of such extent that, during the time in which the comets are visible, their paths differ but little from portions of parabolas; and, for the sake of facilitating the investigation of the elements, they are considered as such. Now if a body, as a comet, be supposed to move in a parabolical curve, the sectoral area described in a given time by a radius vector cannot be compared with the entire area, since the latter is infinite; neither can the time in which the body describes any arc of the curve, nor the time in which the radius vector describes any angle at the focus, be compared with the time of a revolution, since the curve line does not return into itself; it will be necessary, therefore, to show how the laws of Kepler relating to movements in ellipses, require to be modified when they are to be applied to the movement of any body in a parabola. 263. By Kepler's theorem (arts. 203. and 258.) we have, for any planet moving in an elliptical orbit whose semitransverse axis is represented by a, u being the excentric anomaly, e the excentricity of the orbit on the supposition that the semi-transverse axis is unity, and nt a2 representing the mean angular motion in the time t, the latter being reckoned in days from the instant that the planet was in the perihelion point v. Then, if D represent the perihelion distance VF (fig. to art. 260.), that is, if D = a—a e, in which case e=1 =u−(1–2) sin. u: and if the ellipse become a parabola, in which case a (in the second member) Now, developing the arc u in terms of its tangent, we have (Pl. Trigon., art. 47.) u = tan. u— tan.3 {u+ &c.: But (art. 204.), O representing the true anomaly, we have and substituting for e its equivalent above, the equation becomes the factor D and different powers of tan entering into all the terms of this second member. From this investigation it may be inferred that, in different parabolical orbits, the squares of the times in which the moving body, or its radius vector, describes about the focus equal angles, reckoned from the perihelion, are to one another as the cubes of the perihelion distances. 264. From the proportion (a) (art. 259.) we have do dt equal to (the sectoral area described in one day, in an ellipse) aa (1—e2)‡π, that is, in different a2 (1 — e2)3π or to P ellipses, the sectoral areas described in equal times vary as the square roots of the parameters: let such area be represented by A. Now, putting D for the perihelion distance in an ellipse, we have (as in the preceding article) e =1 ; whence (1-e2)t= D a we have A= P Consequently, in different parabolical orbits the values of A, the sectoral areas described in a unit of time or in equal times, will vary with the square roots of the perihelion distances or with the square roots of the parameters: and, in the same parabola, D being then constant, the sectoral areas described by the radii vectores in a unit of time, or in equal times, are equal to one another; or the areas vary with the times as in elliptical orbits. 265. Again, in a circular orbit, in which e=0, the value of απ A becomes ; also, in the parabola and circle, the values of P A represent the areas described by the radii vectores in a unit of time; therefore the areas described in any equal times, in the parabola and circle, are to one another as the values of A, or as (2D) to at: and if, in the parabola, the perihelion distance D be equal to a, the radius of the circle, the areas described in equal times in the parabola and circle are to one another as √2 to 1; or (since, as above, the sectors in each vary with the times) the times of describing equal sectors in the parabola and circle are to one another as 1 to 2. 266. The area of a sector described by the radius vector of a parabola in turning upon the focus through any angle 0, reckoned from the perihelion point, may be found thus: Let F (fig. to art. 260.) be the focus, v the vertex of the parabola, and let P be the place of a comet, from which let fall PR perpendicularly on FV. Also let the parameter of the axis (=4FV) be represented by p, and the true anomaly VFP by ; then, by conic sections, FP= = P 1 + cos. (1+cos. )cos.20* Now the area of the parabolical sector FVP = RP.RV + FR. RP, or =RP (RV + FV): but RV = p. 1 cos. = p sin.210 4 1 + cos. O' 4 cos.240' and R P = F P sin. 0, 2 F P sin. cos.: therefore sin.310 sin. 10 cos.310 cos. 0 = {p), But if FV, the perihelion distance, be represented by a (= the area of the sector will be expressed by a2 (tan.3 10+ tan. 0); and when 90°, the area of the sector FVP' is equal to a2. If the perihelion distance be unity, the area of the sector whose angle is will be tan.3 10 + tan. 10, and that of the sector FVP will be . 267. The time in which the radius vector would describe the sectoral area FVP' may be found thus:-Let the earth be supposed to describe a circular orbit about F with a radius equal to unity; then, since in any one orbit the areas vary as the times, =area of the circle): :: P (=365.256384) : the time in which the radius vector of the earth would describe a circular sector equal to 3); and (art. 265.) 4 2 : 1 :: P: T(=109.6154 days), the time of describ 3 п ing the equal parabolical sector FVP' in which the perihelion distance is unity. But the areas in any one parabola varying as the times, ( (area FVP) T:: tan.3 10 + tan. 10 : T (tan.30 +3 tan. 10), and the last term of the proportion expresses the time in which a radius vector of the parabola whose perihelion distance is unity describes the sector corresponding to an angle from the perihelion. Again, in different parabolical orbits, the times in which the radii describe equal angles about the focus vary (art. 263.) with D (D being the perihelion distance): therefore, in a parabola whose perihelion distance is D, the time of describing the sector corresponding to an anomaly ◊ is equal to T.D (tan.310+ 3 tan. 10). 268. Let the elementary sector described by the radius vector FP of a parabola in a unit of time, as one day, be ᏧᎾ represented by FPP, and let represent the angular velocity, dt or the angle PFp (a circular are having unity for its radius and subtending the angle PFP). Now, since by conic sections area of the parabolical sector PFp described in a unit of time; and (art. 264.), ‡ D2 being the area FVP' when the perihelion 1 D2 do cos.10 dt T: 1 ( = one day). to be inferred that in any one parabola, the angular velocity varies inversely with the square of the radius vector. 269. It sometimes happens that a comet may be seen at night on the meridian of a station, and in such a case the geocentric right ascension and declination of the comet, like those of any other star, can be observed by the transit telescope and mural circle; but frequently a comet, when visible, is too near the sun to be on the meridian when that luminary is below the horizon, and therefore its apparent place must be ascertained by other instruments. With a |