N is the foot of the perpendicular from the origin, O, to the tangent at \((r, \theta)\) to the curve \[ r=a(1-e\cos\theta), \quad (0 < e < 1), \] and Q is the point on ON such that \(OQ \cdot ON = a^2\). Show that \[ ON^2 = a^2 \frac{(1-e \cos \theta)^4}{1-2e \cos \theta+e^2} \] and hence that, if \(e\) is so small that powers of \(e\) higher than the first can be neglected, the locus of Q is \[ r=a(1+e\cos\theta). \]
\(f(x), g(x)\) and \(h(x)\) are functions of \(x\) satisfying the equations \begin{align*} \frac{df}{dx} &= f+g+2h, \\ \frac{dg}{dx} &= 2g+2h, \\ \frac{dh}{dx} &= 7f+8g+24h. \end{align*} Show that \(f-g\) is of the form \(Ae^{\lambda x}\), where \(A\) is a constant. Find all the solutions of the form \[ f=f_0e^{\lambda x}, \quad g=g_0e^{\lambda x}, \quad h=h_0e^{\lambda x}, \] where \(\lambda\) is independent of \(x\), giving the possible values of \(\lambda\) and the corresponding ratios of the constants \(f_0, g_0, h_0\).
Two complex variables \(z=x+iy\), \(Z=X+iY\), are connected by the relation \[ Z = \sin(\tfrac{1}{2}\pi z). \] Show that to every point in the complex \(Z\)-plane there corresponds a point in the strip \(|x| \le \frac{1}{2}\) of the complex \(z\)-plane. Show also that the lines \(x=\text{constant}\), \(y=\text{constant}\) map into certain mutually orthogonal systems of ellipses and hyperbolae in the \(Z\)-plane.
If \[ X_n = e^{-x^2} \frac{d^n}{dx^n}(e^{x^2}), \quad (n=0, 1, 2, 3\dots) \] establish the relations
The coordinates of the points on a curve are given in terms of general homogeneous coordinates by the parametric relations \(x:y:z = \theta^3:\theta^2:1\). Prove that, if the points \(A, B, C\) with parameters \(a, b, c\) are collinear, then \(bc+ca+ab=0\); and that, if the points \(L, M, N\) with parameters \(l, m, n\) lie on a conic through the vertices \(X, Y, Z\) of the triangle of reference, then \(l+m+n=0\). State and prove the converse results. The points \(A, B, C\) of the curve are collinear, and the tangents at \(A, B, C\) meet the curve again in \(P, Q, R\) respectively. Prove that \(P, Q, R\) are collinear. The conic \(X, Y, Z, B, C\) meets the curve again in \(U\), and the tangent at \(U\) meets the curve again in \(V\). Prove that \(V\) lies on the conic through \(X, Y, Z, Q, R\).
Prove that, if two conics \(S\) and \(\Sigma\) are so related that there exists one triangle inscribed in \(S\) and circumscribed to \(\Sigma\), then such a triangle can be drawn with one of its vertices at any given point of \(S\). A circle \(S\) is drawn to pass through the focus \(F\) of a parabola \(\Sigma\). The tangents to \(\Sigma\) from any arbitrary point \(A\) of \(S\) cut \(S\) again in the points \(B, C\). Prove that \(BC\) is a tangent to \(\Sigma\). (It is assumed that the point \(A\) lies "outside" the parabola.)
A number \(n\) of equal uniform rectangular blocks are built into the form of a stairway, each block projecting the same distance \(a\) beyond the one below. The top block is supported from below at its outer edge. Show that the stairway can stand in equilibrium if, and only if, \(2l > a(n-1)\), where \(2l\) is the width of each block.
Show that a uniform chain hangs under gravity in a curve (catenary) with equation that can be written in the form \[ y=c\cosh\frac{x}{c}. \] A uniform chain of length \(2l\) hangs symmetrically over two fixed smooth circular cylinders of equal radii \(r\). The axes of the cylinders are horizontal and parallel and a distance \(2a\) apart on the same level, the chain hanging in a plane perpendicular to the axes. Show that \[ l = \frac{(a-r\sin\theta)(\sec\theta+\tan\theta)}{\log(\sec\theta+\tan\theta)} + r\left(\frac{\pi}{2}-\cos\theta+\theta\right), \] where \(\theta\) is the angle the tangents at the highest points of the catenary make with the horizontal.
The point of suspension \(A\) of a pendulum is caused to move along a horizontal straight line \(OX\). The centre of gravity of the pendulum is \(G\), and \(AG=l\). The radius of gyration about any axis through \(G\) perpendicular to \(AG\) is \(k\). The pendulum can move in the vertical plane containing \(OX\). At time \(t\), \(OA=x\), and the angle between \(AG\) and the vertical is \(\theta\), supposed positive when \(GAX\) is acute. Show that \[ l\cos\theta \frac{d^2x}{dt^2} + (l^2+k^2)\frac{d^2\theta}{dt^2} + lg\sin\theta = 0. \] What condition must \(d^2x/dt^2\) satisfy in order that the pendulum can maintain a constant angle \(\alpha\) to the vertical? Show that, if this condition is maintained, the periodic time of small oscillations about the position is \[ 2\pi\left(\frac{l^2+k^2}{lg\cos\alpha}\right)^{\frac{1}{2}}. \]
A uniform solid cube of side \(2a\) starts from rest and slides down a smooth plane inclined at an angle \(2\tan^{-1}\frac{1}{4}\) to the horizontal, the orientation of the cube being such that its front face is perpendicular to the lines of greatest slope of the plane. The cube meets a fixed horizontal bar placed perpendicular to the direction of motion and at a perpendicular distance \(a/4\) from the plane. Show that, if the cube is to have sufficient velocity to surmount the obstacle when it reaches it, the cube must be allowed first to slide down the plane through a distance greater than \(107a/60\). (The obstacle may be taken to be inelastic and so rough that the cube does not slip on it.)