32 



OF PROJECTIL'ES. 



becomes equal to twice that distance, or to the diameter of 

 thccircle(200); therefore thevelodties-sreequal (239). But 

 since the perpendicular height of the triangular element of 

 the area, of which tlie base is the element of the orbit at 

 the mean distance, is equal to the lesser axis, this element 

 is to the contemporaneous element in the circle as the 

 lesser to the greater axis, or as the whole ellipsis to the 

 whole circle (203), consequently both areas being uni- 

 formly described, the times of revolution are equal. 



248. Theorbm. If a body describes an 

 equiangular spirdl round a given point, the 

 force must be inversely as tlie cube of the 

 distance, and the velocity equal to that with 

 which a circle might be described at the 

 same distance. 



For the orbit of a body projected in any direction with a 

 velocity equal to that with which a circle may be described 

 at the same distance, will initially coincide with an elliptic 

 orbit as its mean distance ; and the inclination of the orbit 

 to the revolving radius is constant at the mean distance ; 

 for if it were eitherincreasing or diminishing, the two halves 

 of the ellipsis co\M not be equal and similar, since the 

 angles contained between the tangent and the lines drawn 

 to the foci (igs) would be different at equal distances on 

 each side of the lesser axis. It foUows therefore that the 

 velocity must always be equal to the velocity in a circle, 

 in order that the equiangular spiral may be described ; but 

 in this curve, the perpendicular on the tangent is by its 

 fundamental property always proportional to the radius : 

 the velocity must therefore be always inversely as the 

 radius; and the velocities of bodies revolving in circles must 

 be inversely as the radii, and the forces inversely as the 

 squares of the radii and the radii conjointly (24o), or in- 

 versely as the cubes of the radii. 



249. Theorem. When a body revolves 

 round a centre by means of a force varying 

 more or less rapidl}' than in the inverse ratio 

 of the squares of the distances, the apsides 

 of the orbit, or the points of greatest and least 

 elongation, will advance or recede respec- 

 tively. 



In an elliptic orbit, when the body descends from the 

 mean distance, the velocity continually prevails over the 

 central force, so as to deflect the orbit more and more 

 .from the revolving radius, until, at a certain point, it be- 



comes perpendicularto it : but, if the central force increase 

 in agreater proportion than in the ellipsis, the point where 

 the velocity prevails over it will be more remote than in 

 the ellipsis, and the apsis will move forwards. This be- 

 comes more evident by considering the extreme cases : 

 supposing the central force to vanish, the lower apsis would 

 recede to the point where a perpendicular falls from the 

 centre on the tangent ; but, supposing the force to increase 

 as the cube of the distance decreases, the curve would be 

 an equiangular spiral, and the lower apsis would be infi- 

 nitely distant. 



Scholium. The action of a second force, varying in 

 the inverse ratio oT the squares of the distances, and directed 

 to a second centre, tends in some parts of the orbit to de- 

 duct a portion of the first force which increases with the 

 distance of the body, and in other parts to increase the first 

 force in a similar manner: but the former effect is consi- 

 derably greater than the latter, so that on the whole, the 

 joint force decreases more rapidly than the square of Ihe 

 distance increases, and the apsides advance. Thus the 

 apsides of the planetary orbits have direct motions, in coa- 

 seguence of their mutual perturbations. 



SECTION IV. OF PROJECTIIES. 



250. Definition. The force of gravi- 

 tation, as far as it concerns the motions of 

 projectiles, is considered as a uniformly acce- 

 lerating force, acting in parallel lines, per- 

 pendicular to the horizon. 



251. Theorem. The velocity of a pro« 

 jectile may be resolved into two parts, its 

 horizontal and vertical velocity: the hori- 

 zontal motion will not be affected by the 

 action of gravitation perpendicular to it, and 

 will therefore continue uniform ; and the ver- 

 tical motion will be the same as if it had no 

 horizontal motion. 



For a uniformly accelerating force is supposed to act 

 equally on a body in motion and at rest, so that the vertical 

 motion will not be affected by the horizontal motion ; and 

 the diagonal motion resulting from the combination will 



