Three infinite parallel wires cut a plane perpendicular to them in the angular points \(X,Y,O\) of an equilateral triangle, and have charges \(e,e,e_0\) per unit length respectively. Prove that the limiting lines of force which pass from \(X\) to \(O\) make at starting angles \((2e-5e_0)\pi/6e\), and \((2e+e_0)\pi/6e\) with \(XO\), provided that \(e>2e_0\). Sketch the lines of force, and determine the distance of the point of equilibrium from \(XY\).
Prove, by inversion or by the method of images, that if a small sphere, of radius \(a\), be made to touch a large one, of radius \(b\), the ratio of the mean density on the small to that on the large sphere tends to \(\pi^2/6\), as \(a/b \to 0\).
Two spheres, radii \(a,b\), have their centres at a distance \(c\) apart. Prove the approximate formula \(p_{12}=1/c\), showing that the error is of order \((a/c)^7\) or \((b/c)^7\).
Construct a triangle of which the sides are bisected at three given points. Prove that it is a definite problem to construct a closed polygon the sides of which are bisected at \(n\) given points in assigned order, provided \(n\) be an odd number; and that the solution may be got by constructing any unclosed polygon which has its sides bisected at the points, and then starting again from the point midway between the two free ends of the polygon. Extend the problem by projection to the case of each side being divided harmonically by one of the points and a fixed straight line, and give a diagram of the construction necessary in the case of the triangle and three points.
Prove that it is always possible to draw a straight line to cut two given non-intersecting lines in space at right angles. Prove also that, if the acute angle between the lines is \(\alpha\), then four lines, two lines or none can be drawn to cut the given lines so that the acute angles of the intersections have an assigned value \(\theta\), according as the assigned \(\theta\) is greater than both angles \((\pi-\alpha)/2\) and \(\alpha/2\), lies between them or is less than both.
Discuss the family of conics \(x^2/\lambda + y^2/(r^2-\lambda) = 1\) in a manner analogous to the case of confocal conics, shewing that the family consists of a set of ellipses and a set of hyperbolas of two types. Shew also, with a diagram, that the four lines \(x \pm y \pm r = 0\) divide the plane into nine regions of which five only are entered by the conics; and that through any point in each of the five regions two conics pass, either both ellipses or both hyperbolas of the same type. Prove that the tangents to the two conics at a common point meet the axes in concyclic points.
Prove the algebraic theorem that, if the product of \(n\) positive factors has an assigned value \(C\), the sum of the factors is least when they are all equal. Discuss by the differential calculus the problem of determining \(n\) so that the sum of the factors is least of all; and shew that in the special cases of \(C\) having any of the values \(2^2, (3/2)^3, (4/3)^4, \dots, (20/19)^{20}, \dots\) the value of \(n\) is not definite but either of two consecutive integers.
Prove that the series \(\sum_0^\infty x^n \sinh(n+1)\alpha\) is convergent if \(x\) is numerically less than \(e^{-\alpha}\), \(\alpha\) being assumed positive and the sum is \(\sinh\alpha / (1 - 2x \cosh\alpha + x^2)\); but that the series \(\sum_0^\infty x^n \sin(n+1)\alpha\) is convergent provided \(x\) is numerically less than unity, the sum being \(\sin\alpha / (1-2x\cos\alpha+x^2)\).
Verify by the use of partial fractions or otherwise that \begin{align*} \operatorname{cosec}(x-\alpha_1) \operatorname{cosec}(x-\alpha_2) &= A_1 \cot(x-\alpha_1) + A_2 \cot(x-\alpha_2), \\ \operatorname{cosec}(x-\alpha_1) \operatorname{cosec}(x-\alpha_2) \operatorname{cosec}(x-\alpha_3) &= B_1 \operatorname{cosec}(x-\alpha_1) + B_2 \operatorname{cosec}(x-\alpha_2) + B_3 \operatorname{cosec}(x-\alpha_3) \end{align*} where \(A_1 = \operatorname{cosec}(\alpha_1-\alpha_2)\), \(B_1 = \operatorname{cosec}(\alpha_1-\alpha_2) \operatorname{cosec}(\alpha_1-\alpha_3)\) with corresponding forms for the other constants. Extend the process to the product of \(n\) distinct cosecants, explaining why the sums contain the \(n\) cotangents or the \(n\) cosecants according as \(n\) is even or odd.
From a point \(O\) a normal \(OP\) is drawn to a curve and \(P\) is not a singular point on the curve: shew that the radial distance from \(O\) to the curve has \(OP\) as a maximum or a minimum according as \(O\) lies on one side or other of the centre of curvature at \(P\). Shew also that the length of the perpendicular drawn from \(O\) on a tangent of the curve has \(OP\) as a maximum or minimum according to the position of \(O\) relative to \(P\) and the centre of curvature: mark the result on a diagram.