Shew that the series \[ \frac{1}{1^p}+\frac{1}{2^p}+\frac{1}{3^p}+\dots \] is convergent only when \(p>1\). Discuss the convergency of the series \[ \frac{1^p}{2^q}+\frac{2^p}{3^q}+\frac{3^p}{4^q}+\dots \] where \(p\) and \(q\) are positive numbers.
Prove that any number of coplanar forces not in equilibrium can be reduced to a single force or a couple. If the forces \(P_1, P_2, P_3\), acting at points whose coordinates are \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) in directions making angles \(\theta_1, \theta_2, \theta_3\) with the axes of \(x\), reduce to a single force, find the equation of its line of action.
Explain the meaning of limiting friction and total resistance, and find the least force which will just pull a heavy body up an inclined plane. Shew that the greatest inclination to the horizon at which a uniform rod can rest in a rough sphere of radius \(a\), and angle of friction \(\lambda\), is \(\tan^{-1}\frac{a^2\sin\lambda\cos\lambda}{c^2-a^2\sin^2\lambda}\), where \(c\) is the distance of the rod from the centre of the sphere.
Five equal uniform rods AB, BC, CD, DE, EA are hinged together and the framework is supported with AB and BC in a horizontal line resting on two smooth pegs, and DE also horizontal. Shew that the distance between the pegs is \(1\frac{3}{5}\) times the length of a rod.
Prove that if the ends of each of two diagonals of a complete quadrilateral are conjugate points with respect to a given conic, the ends of the third diagonal will also be conjugate points.
Given two tangents to a conic with their points of contact and one other point of the conic, give a construction for the centre of the conic.
Prove that the locus of points whose tangents to the two conics \[ S = ax^2+by^2+cz^2=0, \quad S' = a'x^2+b'y^2+c'z^2=0, \] form a harmonic pencil, is \[ F = \Sigma aa'(bc'+b'c)x^2 = 0. \] Deduce that, for non-degenerate S, S', the invariant condition that F should degenerate into two straight lines is \(\Delta\Delta' = \Theta\Theta'\).
Prove that the problem of drawing through a given point P a quadric cone intersecting a given conicoid in two conics whose planes intersect in a given line \(l\), has an infinity of solutions or none according as P does or does not lie on the polar line of \(l\).
Prove that if a quadric cone has one set of three mutually perpendicular generators it has an infinite number of such sets. What does this theorem become when, by a projective transformation, the circle at infinity common to all spheres is transformed into an arbitrary conic?
Prove that as a point P describes, from end to end, a generator of a ruled but non-developable surface, the tangent plane at P rotates through a total angle \(\pi\).