a centre of force situated in the centre, and if 0 be the angle described by the body from an apse in time t, prove that sin 20 = e4√kt 1 ekt +1 the force at distance 1 being represented by k. 81. Apply the differential equations of motion to determine the density of the medium, so that a body may describe a given curve about a given centre of force; and find the law of the density, when the curve is a circle and a force is situated in its circumference varying as 1 Dn 82. If a body describe an ellipse uniformly, round two centres of force situated in the foci; prove that the forces at any point of the ellipse are equal, and inversely proportional to the square of the corresponding conjugate diameter. 83. A body projected from a given point in a plane is 1827 α B attracted by forces in the direction of x, and in the direc x3 y3 tion of y; prove that if the velocity and direction of projection be rightly assumed, it will describe a circle round the origin as a centre, and find how the velocity varies in different parts of the orbit. 84. Compare the difference of the forces in the fixed and moveable orbits, with the force in a circle at the same distance described with the same angular velocity. 85. According to what law must the centripetal force vary, that the areas dato tempore in all circles uniformly described about the centre may be equal? 86. Find the disturbing forces of Venus on the earth when their heliocentric longitudes differ by 45°. 87. If two bodies S and P attracting each other with forces varying inversely as the square of the distance, revolve about each other, S being much greater than P, the actual time of one revolution will be less than if S were immoveable in the P ratio of 1 to 1+ 25· 2S R 1828 88. Supposing the orbit of a comet to lie in one plane, if the force attracting it towards the sun vary as that power of the distance whose index is 28, (8 being a small fraction,) the heliocentric angle between two successive perihelia will be 89. Prove geometrically that (ydx — xdy) is the element of a sectorial area about the origin of the coordinates. 90. Compare the force at a given point of an ellipse described about the focus, with that in a circle at the same distance described with the velocity in the ellipse at that point. 91. Find where the space due externally to the velocity in an ellipse, force in focus, is thrice the space due internally. 92. If the force vary inversely as the 7th power of the distance, and a body be projected from an apse with a velocity which is to the velocity in a circle at the same distance :: 1 : 3; find the polar equation to the curve described, and transform it to rectangular coordinates. 93. Force varying inversely as the square of the distance, if a body be projected with n times the velocity in a circle at the same distance, and in a direction making an angle a with the distance; the angle @ between the axis-major and the distance may be determined from the equation tan (0 — a) = (1~n2) tan a. 94. If equal areas be described by a body in equal times about a given point in a given plane, the body is urged by a force tending to that point. (Newton, Book I. Prop. 2.) 95. If a body be projected from a given point in a given direction with a given velocity about a centre of force varying as the distance, shew that it will describe an ellipse having its centre in the centre of force, and find the magnitudes and positions of the axes. (Newton, Book I. Prop. 10. Cor. 1.) 96. Find the law of the force tending to the focus of an hyperbola. (Newton, Book I. Prop. 12.) 97. If the mass of the earth increase slowly and uniformly, find the resulting equation of the moon's place at any given time, the orbit being nearly circular. 98. Mention the steps of the proof by which Newton shewed that every particle of matter gravitates to every other particle with a force which is inversely as the square of the distance. 99. A body falls towards a centre of force which varies as some power of the distance, determine the cases in which we can integrate so as to find the time of descent. 1 D2' 100. Knowing the force which varies as and the velocity of projection from a given point, to find the path described. (Newton, Book I. Prop. 17.) 101. State and prove Newton's construction for the path of a body projected from an apse with a velocity less than that acquired by falling from an infinite distance, and acted upon by a force varying inversely as the cube of the distance. (Newton, Book I. Prop. 41. Cor. 3.) 103. Describe the variations which take place in the inclination of P's orbit during one revolution of the line of nodes. (Newton, Book I. Prop. 66. Cor. 10.) 104. Shew how very small secular inequalities in the mean motions of two planets may be introduced when their mean motions are nearly commensurable. 105. Find the actual velocity of the point P, (Newton, Book I. Sect. vii. Prop. 39.) the force tending to C being supposed to vary directly as the distance. m where r is the radius 106. The force in an orbit c √ a2 + vector; find the angle between the apsides when the orbit is nearly circular. 107. Shew from Newton's construction for determining the horary increment of the area described by the moon in a circular orbit round the earth at rest, that the velocity generated by the tangential ablatitious force between quadrature and syzygy is to that which would be generated in the same time by the mean addititious :: 3 : π. 108. To determine the horary motion of the moon's nodes in a circular orbit. (Newton, Book III. Prop. 30.) 109. Explain fully the principles on which Newton calculates the correction in the motion of the nodes due to the unequable description of areas, and shew that the mean decrement when the nodes are in quadratures is equal decrement in syzygy. 110. A body may revolve in the equiangular spiral in a medium of which the density is inversely as the distance from the centre by a force varying inversely as the square of the distance from the centre. (Newton, Book II. Prop. 15.) Prove this geometrically and analytically. 111. If a uniform force act upon a body tending to give it a motion of rotation about an axis always perpendicular to the axis about which it is at each instant revolving, and always in the same plane, the angular velocity is unaltered. Shew hence that the angular velocity of the earth is not affected by the action of the sun and moon. 112. If gravity act upon a system of bodies m', m", . . . and h', h", .... be the vertical spaces described, and v', v", be the actual velocities of the bodies, prove that 113. Find the place of a body in a parabolic orbit at any assigned time. (Newton, Book I. Prop. 30.) 114. Find the horary variation of the inclination of the lunar orbit to the plane of the ecliptic. (Newton, Book III. Prop. 34.) 115. The centre of gravity of the earth and moon describes an orbit round the sun much more nearly elliptical than that described by the earth or moon. 116. Compare the times of bodies oscillating in a hypocycloid, revolving about the centre, and falling to the same centre, the force varying as the distance. (Newton, Book I. Prop. 52, Cor. 3.) 117. A body descends in a straight line towards a centre of force varying as the distance, find the velocity acquired in descending through a given space, and the time of descent (Newton, Book I. Prop. 38.) 118. Find the difference of the forces in the fixed and moveable orbits. (Newton, Book I. Prop. 44.) 119. Prove analytically that the areas described by a body about any centre of force are in the same plane, and proportional to the time. 120. Find the law of force acting in parallel lines by which 1829 a body will be made to describe a portion of the circle. (Newton, Book I. Prop. 8.) 121. Find the law of force tending to the focus, by which a body may be made to describe an ellipse. (Newton, Book I. Prop. 11.) 122. Explain the nature of centrifugal force, and shew that in a body revolving about a centre it varies inversely as the cube of the distance. 123. If a body be projected from a given point with a given velocity in a given direction, and acted on by a force which varies inversely as the square of the distance; find the conic section described. 124. Prove that the motion of a body P round T, when disturbed by a body S, is determined by the equation together with two similar equations in y and z, where r = PT, and by means of them exemplify fully the method of determining the variation of the elements of P's orbit arising from the disturbance, when that disturbance is small. 125. At similar points in similar curves described round centres of force similarly situated, the forces are as the squares of the velocities directly, and the distances inversely. 126. Find the place of a body in an elliptic trajectory after a given time. (Newton, Book I. Prop. 31.) 127. State and prove the proportion which Newton has given in Book I. Prop. 44, Cor. 1, for determining the actual value of |