Again, in expression (d) let e = 0. Then Te . 2 Ne 2mat (9) the Periodic Time in a circle when the force ce this also follows from (f). Again in the expression (d), if 4r be the latus-rectum, we have 12 X and T = 212. r 2 -e (1 – e) – sin.-1e Nu (1 – e?) Now when the Ellipse becomes a Parabola, e = 1. Consequently the time from the vertex to the extremities of the latus rectum in the Parabola is 2 Te (1 - 1) which is the same as (c). This last case affords a striking example of the utility of the Theory of Vanishing Fractions. 1 8 e :. if v and v' denote the velocities of the earth at its mean distance a and perihelion distance a ae, ae being the eccentricity, we have v: 0 :: :: Ime: 1. Q. E. D. aa a? (1 1 1 486. The centripetal force acting upon a point placed within a sphere of g, see Newton's Prop. LXXIII. Consequently, if g be the force at the surface of the sphere, and R its radius, we have F: 9:: 5 : R 9 But the velocity (V') in a circle at the Earth's surface is that which R would be acquired down with the force considered constant. Also since pip:: gd : de from similar triangles in the elemental figure, we have Rde § V (nR n-1.6) do = 1 1 {+ ...V (nRo-n–1.5") - Run + c 2 n (nR'-n-1.8") +RWN 1 ✓n It Nn ..0 = 1+wn .1. ✓ (nRe-n-1.5") – RJ n ......(6) 2n w (nR-n-1.8")+Rvn) the polar equation of the spiral. Let <= 0. Then log. 0 = 0. 2n or the number of revolutions will be infinite before the body falls into the centre, that is, it never reaches it, although it continually approaches the centre. R Again, let g = then the body will have approached half 1 2 way to the centre. In this case by (a) R ♡ (3n + 1) 4 V=L=H x V3n + 1 R R and the velocity in a circle at the distance is If the force be the same in the circle and parabola, 488. we have |