which we know to be the case from other and more direct principles. 508. Let a and b be the semi-axes of the ellipse; then the velocity at the extremity of the minor axis is a × {√(ap — g3) a 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 509. Let R be the distance from the centre of force at which the body is projected; then since vdv= Fdę = dp (4-4). ? = R, then 1 N- 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) 4 510. If the centre of force be any where about a circle, 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 511. Let the earth's radius be R, and 9 the space fallen through in a second at its surface; also let p be the periodic time of the Moon. Then if ę be the distance of the moon from the earth, and F the force of the earth's attraction upon the moon, we have the distance required in feet, g being equal to 32 expressed in seconds. 512. t= 1 feet, and p Generally the time in the parabola is (see 484). r being and the time of falling from rest through any space - z when 22μ tit:: ç3 : √ 2 √ (ç − r) × (§ + 2r) VOL. II. 2√2 3 2 A 513. Let x be the angle between the tangent or direction of the body's motion and ę; then `v and v' being the velocities in a curve, and a circle at the same distance. Hence dy v2: v2 :: sin. : sin. ↓ : sin. ¥ + e cos. 4. dz Hence, since cannot exceed 90°, if de be considered positive, dis positive, or negative, or is increasing or decreasing, according as v' is> or < than v. If de be negative org be decreasing, then dy is positive or negative, that is, is increasing, or decreasing, according as v is> or <v'. These results indicate a defect in the enunciation of the problem. Then since the velocity of a body revolving at the surface is |