Find the mean value of the distance of a point on the circumference of a circle of radius \(a\) from \(2n\) points arranged at equal distances along the circumference. Shew that when \(n \to \infty\) the mean is \(\dfrac{4a}{\pi}\).
Prove that the locus of the centres of a family of conics through four given points is a conic \(S\), that the asymptotes of \(S\) are parallel to the axes of the two parabolas which belong to the family, and that the axes of \(S\) are parallel to the asymptotes of the rectangular hyperbola belonging to the family.
A set of \(n\) trucks with \(s\) feet clear between them are inelastic and are set in motion by starting the end one with velocity \(V\) towards the next. Find how long it takes for the last truck to start and the value of the final velocity.
Investigate the equilibrium of a beam, not necessarily uniform, acted upon by any coplanar system of forces, explaining the conceptions of shearing force and bending moment at any point of the beam: and in the case of a horizontal beam subject to any vertical loading, find relations between the shearing force \(S\) and the bending moment \(M\) at any point and the distance \(x\) from one end. A light beam, loaded with a number of concentrated loads, is supported at its ends; shew that the funicular polygon for the loads is a diagram of bending moments for the beam.
The straight line \[ lx + my + n = 0 \] intersects the conic \[ ax^2 + by^2 = 1 \] in \(P\) and \(Q\). Shew that the area of the triangle \(OPQ\) (\(O\) being the origin) is \[ \frac{n (am^2 + bl^2 - abn^2)^{\frac{1}{2}}}{am^2 + bl^2}. \]
Prove that the locus of a point, such that the tangents from it to a given conic \(S\) are harmonic conjugates of the tangents from it to a second given conic \(S'\), is a conic (the harmonic locus); and that the envelope of a straight line, such that its intersections with \(S\) are harmonic conjugates of its intersections with \(S'\), is also a conic (the harmonic envelope). Prove further that, if \(S\) and \(S'\) have contact of the third order (four-point contact), the harmonic locus and the harmonic envelope coincide.
Particles are projected simultaneously from a point under gravity in various directions with velocity \(V\). Prove that at any subsequent time \(t\) they will all lie on a sphere of radius \(Vt\), and determine the motion of the centre of this sphere.
Find the intrinsic and Cartesian equations of the curve in which a uniform heavy chain hangs when suspended from two fixed points. One end of a rough uniform chain of length \(l\) is fastened to a point on a vertical wall at a height \(h\) above the ground. Shew that the greatest distance from the wall at which the free end of the chain will rest on level ground is given by the expression \[ u\left(1+\mu \log \left(\frac{h+l+1-u}{\mu u}\right)\right), \] where \[ u = l+\mu h - \{(\mu^2+1)h^2 + 2\mu lh\}^{\frac{1}{2}}, \] and \(\mu\) is the coefficient of friction.
From \(H\) a fixed point on a parabola chords \(HP\), \(HQ\) are drawn perpendicular to each other. Shew that the locus of the intersection of tangents to the parabola at \(P\) and \(Q\) is a straight line.
\(O\) is the centre of a regular polygon of \(n\) sides and \(a\) is its distance from each side; \(P\) is a point (inside the polygon) whose polar co-ordinates referred to \(O\) as pole and any initial line are \((b, \alpha)\); \(S_m^{(n)}\) is the sum of the \(m\)th powers of the distances of \(P\) from the sides of the polygon. Shew that \[ S_1^{(n)} = na, \quad S_2^{(n)} = n \left(a^2 + \frac{1}{2}b^2\right). \] It is a known theorem that \(S_3^{(n)}\) is not (like \(S_1^{(n)}, S_2^{(n)}\)) independent of \(\alpha\); verify this when \(n=3\).