Prove the following construction for solving graphically the quadratic equation \(x^2 - px + q = 0\). Describe a circle on a diameter whose extremities are the points \((0, 1)\) and \((p, q)\): then the roots of the equation are the abscissae of the points in which this circle cuts the axis of \(x\).
Assuming the logarithmic series, obtain superior and inferior limits for the remainder after \(n\) terms in the expansions in ascending powers of \(x\) of (i) \(\log_e (1+x)\), (ii) \(\log_e \{1/(1-x)\}\), (iii) \(\log_e \{(1+x)/(1-x)\}\). Prove that if these series are used to calculate \(\log_e(128/125)\) correct to ten places of decimals, six terms must be taken in each of the first two series, while two are sufficient in the third case; and, using the tables provided, obtain in each case the remainders correct to two significant figures.
An island consists of two conical peaks, 420 ft. and 330 ft. respectively above sea level, connected by a narrow neck of land along which runs a sharp ridge which rises to 140 ft. above sea level. Draw an imaginary set of contour lines for every 50 ft. above sea level, to fit in with these data.
Prove the formulae
A framework of six equal light rods, smoothly jointed, forms a hexagon \(ABCDEF\) which is stiffened in the form of a regular hexagon by light rods \(BF\), \(CE\) and \(CF\). It is hung from \(A\) and \(B\) with \(AB\) horizontal, and masses of 30 lbs. and 50 lbs. are hung at \(D\) and \(E\). Draw the force diagram, and find the stress in the rod \(CF\); also determine which of the nine rods are in compression.
Solve the equations \begin{align*} x+y+z &= 4, \\ yz+zx+xy &= 1, \\ x^4+y^4+z^4 &= 98. \end{align*}
The circle of curvature of a curve, at a point \(P\), may be defined (1) as a circle which passes through \(P\) and has its centre at the limiting position of the intersection of the normals at \(P\) and a neighbouring point \(Q\); (2) as the limit of a circle which touches the curve at \(P\) and passes through a neighbouring point \(Q\); (3) as the limit of a circle passing through \(P\) and two neighbouring points \(Q\) and \(R\); (4) as the circle which has the closest possible contact with the curve at \(P\). Give a careful proof that these four definitions are equivalent. Obtain formulae for the radius of curvature of curves given in the forms (1) \(s=f(\psi)\), (2) \(f(x,y)=0\), (3) \(x=\phi(t), y=\psi(t)\); and apply one or other of them to the parabola, ellipse, four-cusped hypocycloid, and catenary.
A bicycle cyclometer mechanism consists of a fixed wheel A which has 22 internal teeth: rotating freely alongside it on the same axis is a wheel B with 23 internal teeth. A loose arm, also on the same axis, is rotated by the striker once for 5 revolutions of the bicycle wheel, and this arm carries on an excentric pin two wheels C and D fixed to one another, of which C has 19 external teeth and meshes with the inside of A, whilst D has 20 external teeth and meshes with the inside of B. Find the diameter of bicycle wheel for which B will make one revolution per mile.
Prove that \[ \cos \theta \cos \theta + \cos^2 \theta \cos 2\theta + \dots + \cos^n \theta \cos n\theta = \cos^{n+1} \theta \cot \theta \sin n\theta. \]
A tripod of three equal rods \(DA\), \(DB\), \(DC\), each of weight \(W\), and smoothly jointed together at \(D\), carries a weight \(W'\) at \(D\), and stands on a smooth horizontal plane. Equilibrium is preserved by strings joining the foot of each rod to the middle points of the other rods. Apply the principle of virtual work to shew that, if \(l\) is the length of each string and \(h\) the height of \(D\) above the plane, the tension in each string is \(\frac{(3W+2W')l}{18h}\).