Six equal heavy rods each of weight \(W\) are freely hinged at their ends and form a regular hexagon \(ABCDEF\) which when hung up by the point \(A\) is kept from altering its shape by two light rods \(BF\) and \(CE\). Prove that the thrusts of these rods are \[ \frac{5\sqrt{3}W}{2} \quad \text{and} \quad \frac{\sqrt{3}W}{2}. \] What would be the thrusts in the rods if a weight \(W\) were attached at \(D\)?
\(O\) is the middle point of a straight line \(AB\) of length \(2a\). \(P\) moves so that \(AP.BP = c^2\). Shew that the radius of curvature at \(P\) of the locus is \[ 2c^2r^3/(3r^4 + a^4 - c^4), \] where \(r=OP\).
Sum, for any positive integer \(n\),
\(P, Q, R\) are points on a conic with focus \(S\) which vary in such a way that the angles \(PSQ, QSR\) remain constant. Shew that the locus of the point in which the tangent at \(P\) meets \(QR\) is a conic whose focus is also \(S\).
Two blocks \(A\) and \(B\) of weight \(W_1\) and \(W_2\) respectively are connected by a string and placed on a rough table. The coefficient of friction between blocks and table is \(\mu\). A force \(P\), less than \(\mu(W_1+W_2)\), is applied to \(A\) in the direction \(BA\), and is gradually turned round in the horizontal plane. Show that the least value that \(P\) may have in order that \(A\) and \(B\) begin to slip simultaneously is \(\mu(W_1^2 + W_2^2)^{1/2}\), and that the slipping begins when \(P\) has turned through an angle \[ \cos^{-1}\frac{\mu^2(W_2^2-W_1^2)+P^2}{2\mu W_2 P}. \] Describe exactly what happens if \(\mu W_1 < P < \mu(W_1^2+W_2^2)^{1/2}\).
Three points \(A, B, C\) being chosen at random on a circle of radius \(a\), shew that the mean value of the area of the triangle \(ABC\) is \[ \frac{3a^2}{2\pi}. \]
Shew that, if \(\alpha, \beta, \theta, \phi\) lie between \(0\) and \(\pi\), and if \(\alpha+\beta=\theta+\phi\), and \(0 < \alpha - \beta < \theta - \phi\), then \[ \sin\alpha\sin\beta > \sin\theta\sin\phi. \] Deduce that, in any triangle \(ABC\), \[ 8 \sin A \sin B \sin C \le 3\sqrt{3}, \] except when \(ABC\) is equilateral; and hence, or otherwise, prove that of all the triangles which can be inscribed in a given circle, the equilateral has the largest area.
Describe briefly the process of reciprocation with respect to (a) a general conic, (b) a circle. A conic has a given focus \(S\), passes through a given point \(P\), and touches a given line \(l\). Shew that its directrix envelopes a conic which passes through \(S\).
A weight \(W\) is suspended from a fixed point \(A\) by a uniform string of length \(l\) and weight \(wl\). The weight is drawn aside by a horizontal force \(P\). Show that in the equilibrium position the distance of \(W\) from the vertical through \(A\) is \[ \frac{P}{w}\left\{\sinh^{-1}\left(\frac{W+lw}{P}\right) - \sinh^{-1}\frac{W}{P}\right\}. \]
Prove that the chords \(PQ\) of the rectangular hyperbola \(H \equiv xy-c^2=0\) which subtend a right angle at a fixed point \(A(h,k)\) envelop a conic \(\Sigma\) whose tangential equation is \[ c^2(l^2+m^2)+(h^2+k^2)lm+kl+hm=0. \] Shew that \(\Sigma\) is a parabola whose focus is \(A\) and whose directrix is the polar of \(A\) with respect to \(H\), and find the equation of the locus of the foot of the perpendicular from \(A\) to \(PQ\).