Two cylinders, similar in all respects, of radius 15 in.\ lie symmetrically in contact in a cylindrical trough of radius 54 in., and a third cylinder of radius 10 in.\ lies on the two equal cylinders. The axes of all the cylinders are horizontal and parallel. Show that if the surfaces are smooth there will be equilibrium unless the weight of the upper cylinder is greater than \(\frac{5}{8}\) the weight of one of the others. Show also that however great the weight of the upper cylinder may be there will still be equilibrium if all the surfaces are rough and no coefficient of friction is less than \(\frac{1}{3}\).
Having given an equation of the second degree in homogeneous (areal or trilinear) coordinates, determine the coordinates of the centre of the curve it represents, the equation of its asymptotes, and its species (i.e.\ ellipse, parabola, etc.). Find the equation of a conic having the triangle of reference \(ABC\) for a self-polar triangle, and a given point \(D\) for centre. Determine its species for any position of \(D\). Prove that the conic with centre \(A\) which has \(BCD\) as a self-polar triangle is similar and similarly situated.
Prove that \[ \log_e \frac{p}{q} = 2 \left\{ \frac{p-q}{p+q} + \frac{1}{3} \left( \frac{p-q}{p+q} \right)^3 + \frac{1}{5} \left( \frac{p-q}{p+q} \right)^5 + \dots \right\}, \] where \(p\) and \(q\) are both positive. Show that the error involved in stopping the series at the third term is certainly less than \[ \frac{(p-q)^7}{14pq(p+q)^6}. \]
Through a variable point on a hyperbola and through a fixed point \(A\) not on the curve pairs of straight lines are drawn parallel to the asymptotes. Prove that one of the diagonals of the parallelogram so formed touches a fixed conic. Prove also that this conic is a hyperbola if the given hyperbola is convex towards \(A\) and is otherwise an ellipse.
The ends \(A, B\) of a uniform rigid rod of length \(2l\) are constrained to move on two fixed smooth wires \(OA, OB\) at right angles, each of which is inclined at \(45^\circ\) to the vertical. \(O\) is the highest point. Prove that when the rod is horizontal equilibrium is stable.
Explain the meaning of the statement that \(\log x\) tends to infinity with \(x\) but more slowly than any positive power of \(x\). Starting from any definition of \(\log x\) that you please, prove the statement. Arrange the functions \[ x^{e^{\log x}}, \quad e^{x^{\log x}}, \quad (\log x)^x, \quad x/\log x, \] in the order of the rapidity with which they tend to infinity with \(x\). [It is assumed throughout that \(x\) is positive.]
Prove that from a point \(h\) feet above the surface of the sea the distance to the horizon is \(1\cdot23 \sqrt{h}\) miles, and that the dip of the horizon is \(1\cdot06 \sqrt{h}\) minutes, approximately. The radius of the earth is taken as 4000 miles.
A straight line meets the sides \(BC, CA, AB\) of a triangle in \(L, M, N\). The parallelograms \(MANP, NBLQ, LCMR\) are completed. Prove that \(P, Q, R\) are collinear.
A picture is hung on a vertical wall by parallel cords of length \(l\) attached to points on the back of the picture; the centre of gravity of the picture (measured on the picture) is at a distance \(a\) below and a small distance \(d\) in front of the points of attachment. The distance of the points of attachment from the bottom edge of the picture is \(b\). Show that the picture makes an angle with the vertical equal to \[ \frac{ld}{b^2+al} \] approximately, if this angle and the angle between the cords and the vertical are both small, whatever be the coefficient of friction between the wall and the edge of the picture.
Prove that, if \[ f_n(x) = \frac{1}{2^n.n!} \frac{d^n}{dx^n} \{ (x^2-1)^n \}, \] then \[ f_n(1) = 1, \quad f_n(-1) = (-1)^n. \] Prove also, by integration by parts or otherwise, that, if \(\phi(x)\) is a polynomial of degree less than \(n\), \[ \int_{-1}^1 \phi(x) f_n(x) dx = 0, \] and that \[ \int_{-1}^1 f_n(x) f_m(x) dx = 0, \text{ or } 2/(2n+1), \] according as \(m \neq n\), or \(m=n\).