Shew that the electrical resistance between opposite corners of a framework of twelve equal wires arranged along the edges of a cube, all having the same resistance, is \(\frac{5}{6}\) of the resistance of each wire.
Two spheres of masses \(m_1, m_2\), coefficient of elasticity \(e\), and equal radii, are at rest on a smooth horizontal table. Find the direction in which \(m_1\) must be projected so as to give \(m_2\) the maximum velocity perpendicular to this direction, and determine this maximum velocity.
Find formulae of reduction for \[ \int \frac{x^n dx}{\sqrt{(ax^2 + 2bx + c)}}, \quad \int_0^\infty \frac{dx}{(x^2 + a^2)^n}, \] and evaluate the latter integral when \(n = 3\) and when \(n=5/2\).
Show that an approximate root of \(x \log_e x + px = e\), where \(p\) is small, is \[ x = e(1 - p + 3p^2/16). \]
The diagram shows a pressure gauge used to determine the pressure of nearly perfect vacua. The vessel \(V\) is lowered until the mercury falls below \(A\), thus putting \(B\) into connection with the ``vacuum'' to be measured. \(V\) is then raised, and the gas in \(B\) is driven into the fine-bore tube \(T\) as shown in the diagram. If the volume of \(B\) above \(A\) is 3540 times the volume of \(T\) per cm. of its length, what was the pressure of gas in the vacuum, the mercury levels being as shown? % Diagram shows a McLeod gauge. The difference in mercury levels in the two parallel tubes is 2.70 cm - 1.85 cm = 0.85 cm. The length of the trapped gas column in tube T is 1.85 cm. If the tube \(T\) is 25 cm. long, and if the maximum difference of level in the two tubes which can be read is 30 cm., find the greatest pressure which can be measured with the gauge. The volume of the right hand tube may be neglected in comparison with that of the vessel to which the apparatus is connected.
A gun has a given muzzle velocity and is required to hit some point of a small vertical object, of given height \(\delta h\), on the same horizontal plane. The gun is laid so that the shell will carry a horizontal distance \(R\). Prove that the object must lie in a horizontal interval of length \(\delta h \cot\alpha\), where \(\alpha\) is the angle of elevation, and show by means of a graph the relation between \(R\) and the length of the interval for all angles of elevation. The resistance of the air is neglected.
Prove that the arc \(S\) of the evolute of a given curve satisfies in general the equation \[ S = \rho + c, \] where \(\rho\) is the radius of curvature of the given curve at the corresponding point and \(c\) is a constant. Determine the total length of the evolute of the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \]
Find the equation which gives the values of \(x\) for which \(f(x)\) is stationary, where \[ f(x) = x^3 / \{e^{x/T} - 1\}; \] and show that, if \(x_0\) is a root of this equation, then as \(T\) varies, \(x_0/T\) is constant.
Evaluate the integrals \[ \int x^2 \log x \, dx, \quad \int \frac{\sqrt{(x^2-a^2)}}{x} \, dx, \quad \int \frac{1}{5+4\sin x} \, dx, \quad \int_0^{\pi/2} \cos^2 \theta \sin^2 2\theta \, d\theta. \]
Find the volume of the surface generated by the complete revolution of a circle about a tangent.