the difference of the forces in the fixed and moveable orbits, and apply it to determine the whole force on p, when the fixed orbit is an ellipse, and the force in the focus.

128. If P and S attract each other, the curve which P describes relatively to S, may be described by P round S fixed. (Newton, Book I. Prop. 58.)

129. A body revolving in a given ellipse, force in focus, leaves the higher apse; and when it arrives at the lower apse A, the absolute force is suddenly altered, so that the body describes a similar ellipse, of which A is the higher apse; and a like alteration takes place when the body arrives at the lower apse of the new ellipse, and so on at each successive apse. Find the time

longitude of moon, a = longitude of perigee,

mean longitude when = 0.

0 =

Explain fully the effects of the

several terms in the above expression on the moon's orbit.

131. A body acted on by a centripetal force varying partly 1

1

D3

D59

as and partly as is projected with the velocity which would be acquired in falling from infinity, at an angle with the distance whose tangent = 2, the forces being equal at the point of projection. Required the orbit described, and the time of descent to the centre.

132. In Newton, Book I. Prop. 66, explain the effect of the disturbing force of S in producing a motion of the nodes of P's orbit, and a variation of the inclination.

133. Find the horary motion of the nodes in an elliptic orbit. (Newton, Book III. Prop. 31.)

134. Find the horary increment of the area described by the moon, and compare its values at quadrature and syzygy. (Newton, Book III. Prop. 26.)

135. Determine by the principles of Newton's seventh section the spaces due to the velocity externally and internally in an ellipse, the force being in the centre.

136. Find the effect of the disturbing force of the sun on the apsides of the lunar orbit during one revolution of the (Newton, Book I. Prop. 66, Cor. 7.)

moon.

137. Find the horary variation of the inclination of the lunar orbit to the plane of the ecliptic, and thence by Newton's construction determine its mean monthly value.

138. If the velocities of two bodies at any equal distances from the centre of force be the same, and if one body move in a straight line to or from the centre and the other in a curve, their velocities will be the same at all other equal distances. (Newton, Book I. Prop. 40.)

139. If S and P revolve round their common centre of gravity by their mutual attraction, shew that each will move in the same manner as if a body were placed in the centre of gravity exerting a force varying according to the same law. What is the magnitude of this body for each of the two S and P, the force varying inversely as the square of the distance?

140. A body is projected round a centre of force varying as 1830 the distance, with a given velocity, in a given direction; find the magnitudes and positions of the axes of the orbit described, and also the periodic time. (Newton, Book I. Prop. 10, Cors. 1 and 2.)

141. A body revolves in a parabola, find the law of the force tending to the focus. (Newton, Book I. Prop. 13.)

142. In different conic sections described round the same centre of force, situated in the focus, the latera recta are as the squares of the areas described in a given time. (Newton, Book I. Prop. 14.)

143. When a body descends from a point A towards a centre of force S, the force varying as the distance, shew that the space described, the velocity, and the time of motion, are respectively proportional to the versed sine, sine and arc of the circle of radius SA; and find the time to the centre.

144. Explain Newton's method of finding the angle between the apsides in orbits nearly circular, and apply it when the force

145. Newton, Section XI. Prop. 66.

146. In the case of the sun, moon, and earth, find the whole force on the moon in the direction of the radius vector, the orbits being considered in the same plane.

147. Determine the orbit described and the time of describing any angle when a body is projected round a centre of force

1

3

varying as at an angle whose tangent = and with a 232

velocity which is to the velocity in a circle at the same distance :: √2: 3.

148. A body describes a circle of given radius uniformly, acted upon by two forces each varying as the distance and without the plane of the circle; find the velocity of the body and the position of the plane of its orbit.

149. If a body revolve in an ellipse round the focus, prove that a progressive motion of the apse will be the effect of any continual addition of force in the direction of the radius vector during the progress of the body from the higher to the lower apse, and point out the effect on the eccentricity.

150. A body is acted on by two forces, the one repulsive and varying as the distance from a given point, the other constant and acting in parallel lines. Determine the motion of the body.

151. Prove that the centre of gravity of the earth and moon describes about the sun very nearly an ellipse in one plane, and that the area described by its radius vector is very nearly proportional to the time.

152. Find the horary variation of the inclination of the moon's orbit. (Newton, Book III. Prop. 34.)

153. As the line of nodes of the moon's orbit moves from syzygy to quadrature, the inclination of the orbit to the ecliptic is diminished; and as the line of nodes moves from quadrature

to syzygy, the inclination is increased. (Newton, Book I. Prop. 66, Cor. 10.)

154. Find the horary increment of the area which the moon describes about the earth in a circular orbit. (Newton, Book III.)

β

=

3m

8

155. s = k {sin (g0− y) + sin (2—2m-g)0+2B+y}, where s = tangent of moon's latitude, y = γ longitude of node, -ẞ the sun's mean longitude when = 0. Explain the effect of these terms, and thence shew that the inclination of the orbit is greatest when the line of nodes is in syzygies, and least when it is in quadratures.

156. Find the ratio of the diameters of the lunar orbit, supposing it to have been originally without eccentricity. (Newton, Book III. Prop. 28.)

157. If two bodies S and P attract each other mutually, the orbit which P appears to describe about S in motion may be described about S fixed, by the action of the same force. (Newton, Book 1. Prop. 58.)

158. Find the mean horary motion of the nodes of the lunar orbit supposed to be elliptical. (Newton, Book III. Prop. 31.) 159. Eliminate t from the differential equations :

160. The difference of the forces at corresponding points in the fixed and revolving orbits varies inversely as the cube of the distance. (Newton, Book I. Prop. 44.)

161. Compare the axis-major of the ellipse apparently described by P round T, with that of the ellipse described by P round T fixed in the same periodic time. (Newton, Book I. Prop. 60.)

162. A body being acted upon by a force tending to a centre, 1831 the areas described are in one plane, and proportional to the times of description. (Newton, Prop. 1.)

163. Investigate an expression for the force by which a body may be made to describe any orbit whatever round a fixed centre in the same plane with it; and apply it to find the law of force tending to the focus of an ellipse. (Newton, Props. 6 and 11.)

164. If a body by the action of any centripetal force move in any manner, and another body ascend or descend in a straight line, and their velocities in any case of equal distances be equal, their velocities at all equal distances will be equal. (Newton, Prop. 40.)

165. Prove that the difference of the forces in the fixed and moveable orbits varies inversely as the cube of the distance. (Newton, Prop. 44.)

166. Required the part of the sun's disturbing force perpendicular to the plane of the moon's orbit.

167. Find the quantities of matter of planets which have satellites. And these being given, shew how to determine the quantity of matter of those which have not satellites.

168. If x, y, z, r, and x', y', z', r' be the coordinates and distances of two planets m and m' from the centre of the sun supposed at rest, and if the distance of m from m1, xx + yy + 22 1

and Q

=

p3

prove that the axis-major of the ellipse of curvature at the point (x, y, z,) of m's orbit is

169. How does it appear that the mean motions of the planets are subject to no secular variations?

170. If a be the mean distance of a planet from the sun, and the length of the line of nodes, then the time of the planet's passage from node to node through the perihelion is

where p = periodic time of the earth about the sun, and 1 its mean distance from it.

=