which we know to be the case from other and more direct principles. 508. Let a and 6 be the semi-axes of the ellipse; then the velocity at the extremity of the minor axis is b a P b pedo odo = 2s 2 x du the whole time of descent to the focus. Again, by 484, the time of the body's moving from the nearer apside to the extremity of the axis minor is T' = NH Х (-e). 509. Let R be the distance from the centre of force at which the body is projected; then since odoc Fdę = V = n 1 Let a = 00, and § = R, then fo 1 R~1 But since the velocity in any curve is that which would be ac ) quired by the bodys's descent along chord of curvature (PV) 510. If the centre of force be any where about a circle, we learn from (504) that F= 8cre (ne - a* + 39) r being the radius of the circle, and a the distance of the centre of force from that of the circle. Now when the force is in the circumference a = r, and p is positive 8cm .::F= Let q = 2r; then 162 2 = 511. Let the earth’s radius be R, and the space 1 F:9 :: : fallen through in a second at its surface; also let p be the periodic time of the Moon. Then if $ be the distance of the moon from the earth, and F the force of the earth's attraction upon the moon, we have 1 R: or F = Rpg ga Now by (440), we get 27 xe ✓ Х :: p = R'o pt and & R$g} pf 4*** the distance required in feet, g being equal to 32 2 feet, and p expressed in seconds. 512. 2 s= SP; Generally the time in the parabola is (see 484). ✓** ( – )* (+ r) go being = SA and and the time of falling from rest through any space ? - I when FESTA F=is (495). * {(sv – vyt + (r – vers.- -)} and when x = 0 2.1 513. Let x be the angle between the tangent or direction of the body's motion and S; then p = g. sin. y. dy de and dp de : (438). But us : 0 :: dp P v and v' being the velocities in a curve, and a circle at the same distance. Hence 1:09 :: sin.y: sin. : sin. y te cos. *. the .(a) Hence, since y cannot exceed 90°, if de be considered positive, dit is positive, or negative, or y is increasing or decreasing, according as v' is > or < than v. If dę be negative or be decreasing, then dy is positive or negative, that is, 4 is increasing, or decreasing, according as v is > or <v'. These results indicate a defect in the enunciation of the problem. 514. Let R be the earth's radius. Then since the velocity of a body revolving at the surface is ✓ R? R we have for the new trajectory c= P x V = 6 = x : odh a |