Prove that \[ \frac{1}{1!(2n)!} + \frac{1}{2!(2n-1)!} + \frac{1}{3!(2n-2)!} + \dots + \frac{1}{n!(n+1)!} = \frac{2^{2n}-1}{(2n+1)!}. \]
It is given that \[ k_1/(x-a_1) + k_2/(x-a_2) + \dots + k_n/(x-a_n) = 0, \] where \(k_1+k_2+\dots+k_n = 0\). Prove that, if \(x=(py+q)/(ry+s)\), \(a_1 = (pb_1+q)/(rb_1+s), \dots, a_n = (pb_n+q)/(rb_n+s)\), where \(ps-qr \neq 0\), then \[ k_1/(y-b_1) + \dots + k_n/(y-b_n) = 0. \]
Prove that a rigid body possesses a centre of gravity such that if it be freely suspended at that point and allowed to hang under the influence of gravity it will be in equilibrium in any position. A tripod consists of three equal uniform rods \(AO, BO, CO\) rigidly connected at \(O\) so that they are at right angles to one another. If the tripod be hung from the point \(A\) show that the plane \(ABC\) makes an angle \(\tan^{-1} 2\sqrt{2}\) with the horizontal.
Discuss the connection between Newton's laws of motion and the fundamental statical postulates, such as the composition, independence and transmissibility of forces. Forces are represented in magnitude and line of action by coplanar lines \(AA', BB', \dots\). Prove that the resultant is represented in direction by \(GG'\) and in magnitude by \(n.GG'\), where \(n\) is the number of forces in the given system, and \(G\) is the centroid of the points \(A, B, \dots\), \(G'\) is the centroid of the points \(A', B', \dots\). Also, give a logical statement of the development from the fundamental postulates of the statical theorems on which you base your proof.
Prove that, if \(\alpha, \beta, \gamma\) are the distances of the corners of an equilateral triangle of side \(c\) from any point in its plane, \[ \alpha^4 + \beta^4 + \gamma^4 - \beta^2\gamma^2 - \gamma^2\alpha^2 - \alpha^2\beta^2 - c^2(\alpha^2 + \beta^2 + \gamma^2) + c^4 = 0. \]
Find all the solutions of the simultaneous equations \[ \sin x = \sin 2y, \quad \sin y = \sin 2z, \quad \sin z = \sin 2x, \] each angle being restricted to be positive and less than \(\pi\).
Two stopping points of an electric tramcar are 440 yards apart. The maximum speed of the car is 20 miles per hour and it covers the distance between stops in 75 seconds. If both acceleration and retardation are uniform and the latter is twice as great as the former, find the value of each of them, and also how far the car runs at its maximum speed.
Define the Potential Energy of a connected system of bodies under the action of given external forces. Give an outline of the theory by which, when the Potential Energy is known for all possible positions of the system, the positions of equilibrium and their stability can be investigated, stating the principles that are assumed in the investigation. Two equal smooth circular cylinders of radius \(c\) are fixed with their axes parallel and in the same horizontal plane at a distance \(b\) apart. A cube of side \(2a\) rests with two adjacent faces touching the cylinders. Shew that, if \(a+c<\sqrt{2}b\), and \(a^2+c^2>b^2\), there are two positions of equilibrium in which the plane through the highest and lowest edges of the cube makes an angle \(\cos^{-1}\{(a+c)/\sqrt{2}b\}\) with the vertical. Also shew that these positions are unstable.
Shew that, if the perimeter of a regular polygon differs from the circumference of the circumscribing circle by less than 1 per cent., the least possible number of sides of the polygon is 13.
Assuming the formula \[ \sin n\theta / \sin\theta = (2\cos\theta)^{n-1} - (n-2)(2\cos\theta)^{n-3} + \dots + (-1)^{\frac{1}{2}(n-1)}, \] for \(n\) an odd integer, prove that, if \(m\) and \(n\) are odd integers having no common factor, \[ \frac{\sin mn\theta \sin\theta}{\sin m\theta \sin n\theta} \] is a polynomial in \(\cos\theta\) of degree \(mn-m-n+1\), the factors of which are \(\cos\theta - \cos(r\pi/mn)\), where \(r\) is any integer from 1 to \(mn-1\), which is not a multiple of \(m\) or \(n\). Hence, or otherwise, prove that \[ 16 \cos\alpha \cos 2\alpha \cos 4\alpha \cos 7\alpha = 1, \] where \(\alpha=\pi/15\).