Problems

Prove Leibnitz's rule for the repeated differentiation of the product of two functions of \(x\). \par Prove that \[ \left(\frac{d}{dx}\right)^n \frac{\log x}{x} = (-1)^n\frac{n!}{x^{n+1}}\left(\log x - 1 - \frac{1}{2}-\dots-\frac{1}{n}\right). \]

Prove that for a curve, the radius of curvature \(\frac{ds}{d\psi}\) is equal to \[ \left\{1+\left(\frac{dy}{dx}\right)^2\right\}^{\frac{3}{2}} / \frac{d^2y}{dx^2}. \] Prove that the length of the radius of curvature of the curve \(y=\frac{x^3}{3}-x\) is a minimum at points for which \(x=\pm 1.07\), approximately.

Write a short account of the method of reciprocation, shewing particularly how to reciprocate a circle into a conic of any species. Give some examples shewing the power of the method. \par \(S\) is the focus of a given conic and a line \(l\) meets the corresponding directrix in \(Z\). \(l'\) is the line joining \(Z\) to the pole of \(l\). A second conic is drawn having \(S\) as one focus and touching \(l, l'\). A common tangent to the two conics touches them at \(Q, Q'\); shew that \(QQ'\) subtends a right angle at \(S\).

\(OX, OY\) are conjugate lines with respect to a fixed conic. \(A\) is any fixed point. A fixed circle through \(O\) and \(A\) cuts \(OX\) in \(P\), and \(AP\) meets \(OY\) in \(Q\). Shew that the locus of \(Q\) is a conic. \par As a particular case, shew that, if a point moves so that the line joining it to a fixed point is perpendicular to its polar with respect to a conic, the curve traced out is a rectangular hyperbola with its asymptotes parallel to the axes of the original conic. \par Deduce that four normals can be drawn from any point to an ellipse.

Shew that the coordinates of any point on a conic can be expressed in terms of a parameter \(t\) by the equations \[ \frac{x}{at^2+2bt+c} = \frac{y}{a't^2+2b't+c'} = \frac{1}{a''t^2+2b''t+c''}. \] Find the condition that \(\lambda x + \mu y + \nu = 0\) may be a tangent, and obtain (i) the foci, (ii) the director circle, (iii) the conditions for the conic to be a parabola, or a rectangular hyperbola.

Shew how to sum the series \(a_0+a_1x+\dots+a_nx^n+\dots\), whose coefficients satisfy the relation \(a_n+pa_{n-1}+qa_{n-2}+ra_{n-3}=0\), \(p, q, r\) being given numbers. \par In the case where \(3a_n-7a_{n-1}+5a_{n-2}-a_{n-3}=0\), and \(a_0=1, a_1=8, a_2=17\), shew that \(2a_n = 20n-7+3^{2-n}\).

Shew that

1. [(i)] \(1+2(\cos\alpha+\cos 2\alpha+\dots+\cos n\alpha) = \sin(n+\frac{1}{2})\alpha \operatorname{cosec} \frac{\alpha}{2}\).
2. [(ii)] \(1+2(\cos\alpha\cos\theta+\cos 2\alpha\cos 2\theta+\dots+\cos n\alpha\cos n\theta) = \frac{\cos n\alpha \cos(n+1)\theta - \cos(n+1)\alpha\cos n\theta}{\cos\theta-\cos\alpha}\).

Deduce that \begin{align*} 1^2\cos\theta+2^2\cos 2\theta+\dots+n^2\cos n\theta \\ = \tfrac{1}{2}\operatorname{cosec}^2\tfrac{1}{2}\theta\{(n+\tfrac{1}{2})^2\cos n\theta - n^2\cos(n+1)\theta - \sin(n+\tfrac{1}{2})\theta\operatorname{cosec}\tfrac{1}{2}\theta\}. \end{align*}

Prove that the following definitions of the curvature of a curve at a point \(P\) lead to the same value.

1. [(i)] The rate, in radians per unit of arc, at which the tangent turns.
2. [(ii)] The limit to which the reciprocal of the radius of a circle touching the curve at \(P\) and passing through an adjacent point \(Q\) tends, as \(Q\) tends to \(P\).

Examine the nature of the evolute of a given curve in the neighbourhood of the following points on the given curve

1. [(a)] a point of inflexion,
2. [(b)] a cusp,
3. [(c)] a point of maximum or minimum curvature.

Trace the curve \(b(ay-x^2)^2=x^5\), and shew that the evolute has a point of inflexion corresponding to the origin on the curve.

Give some account of the theory of a framework of rods, dealing with (i) the number of rods necessary for a frame with \(n\) joints to be just rigid, (ii) the graphical determination of stresses (\(\alpha\)) in a `simple' frame under given forces at the joints, (\(\beta\)) in a non-simple frame. \par Illustrate by the two cases of a pentagon \(ABCDE\) of rods jointed at the corners, (a) with connecting rods \(AC, CE\), (b) with connecting rods \(AD, CE\), in equilibrium under three given forces at the corners \(B, C, E\).

A thin wire has the form of a circle in a vertical plane with centre \(C\). \(A, B\) are pegs attached to the wire so that \(CA, CB\) make angles \(\alpha\) on opposite sides of the downward vertical through \(C\). A small ring of mass \(M\) can slide on the wire, and is attached to two strings passed over the pegs with masses \(m\) hanging from their ends. Write down the potential energy of the system when the radius to \(M\) makes an angle \(\theta\) with the vertical. \par Hence discuss the stability of equilibrium positions in the cases \(M \gtrless m\sin\frac{1}{2}\alpha\).