State Kepler's three laws concerning the orbits of planets and shew how they are related to the theory of gravitation.
Trace the steps by which the equations of motion of a system of particles are derived from the Newtonian equations applicable to the individual particles of which the system is composed. It may be assumed that the reaction between two particles is in the line joining them. Hence develop equations connecting the external forces with the kinetic energy of rigid and non-rigid bodies.
The first and second laws of thermodynamics.
Prove the theorem that the circulation round a given circuit of particles in a non-viscous fluid is constant during the motion of the fluid. Hence develop the theory of irrotational motion.
Prove that three straight lines in space, parallel to the same plane but not to one another, can be cut by an infinity of straight lines and that the intercepts made by the three lines on any one of the cutting lines are in constant ratio.
Deduce from the focus and directrix definition of an ellipse the existence of a centre and a second focus and directrix. A variable point \(P\) is taken in a line \(AB\) between \(A\) and \(B\): prove that the ellipses of the same fixed eccentricity with \(A, P\) and \(P, B\) respectively as foci intersect on a fixed ellipse of which \(A, B\) are the foci.
Prove that, if \(T\) be a point on a diameter of an ellipse, centre \(C\), and \(V\) be the point in which the polar of \(T\) cuts the diameter, then the rectangle \(CT \cdot CV\) is equal to the square of the semi-diameter. Prove that, as \(T\) varies along the fixed diameter, the locus of the foot of the perpendicular from \(T\) on its polar is a rectangular hyperbola.
Prove that the section of a circular cone by a plane parallel to a tangent plane is a parabola. Deduce that, given a parabola in position, the locus of the vertex of a circular cone, of which the parabola is a section, is another parabola of which the focus and vertex are the vertex and focus respectively of the given parabola.
Prove that the pencil of lines formed by joining any point \(P\) on a circle to four fixed points on the circle has a constant cross ratio: prove also that \(PA, PB\) are harmonic with \(PC, PD\), provided the chords \(AB, CD\) pass each through the pole of the other. Three points \(A, B, C\) lie on a circle and \(A', B', C'\) are chosen on the circle so that \(AA', BC\); \(BB', CA\); and \(CC', AB\) are all harmonic pairs: prove that the lines \(AA', BB', CC'\) are concurrent.
Prove that, if a tangent to a parabola makes an angle \(\theta\) with the axis, the angle \(\phi\) at which the corresponding normal chord cuts the parabola again is given by \[ 2 \tan\theta \tan\phi = 1. \]