Prove that, if the circle of curvature at any point \(P\) on the cardioide \(r=a(1+\cos\theta)\), which has its cusp at \(O\) and \(OA\) for its axis, cuts the curve again in \(Q\), then \[ \frac{1}{OQ} - \frac{9}{OP} = -\frac{8}{OA}. \]
Three rods \(BC, CA, AB\), of which the weights are \(p,q,r\), form a triangle \(ABC\) which is suspended by a string attached to the vertex \(A\). Prove that the angle \(\theta\) at which \(BC\) is inclined to the vertical in the position of equilibrium is given by \[ (2p+q+r)\cot\theta = (p+q)\cot C \sim (p+r)\cot B. \]
\((n+1)\) bricks of the same size are piled one above another in a vertical plane so that they rest, each one overlapping the one below by as much as possible. Prove that, if \(2a\) is the length of a brick, the lowest but one overlaps the lowest by a length \(a/n\). Shew also that, if each brick overlaps the one next below by a length \(a/n\), the greatest number of bricks that may be piled up is \((2n-1)\).
A string, of which one end is attached to a mass \(m\) lying on a smooth table, passes over the edge of the table, and after passing over a smooth fixed pulley close to the table and on a level with it has its other end attached to a mass \(m'\); between the table and the pulley the string hangs in a loop and supports a smooth ring of mass \(M\). The string lies in a vertical plane perpendicular to the edge of the table. Find the motion and the tension of the string, and shew that the mass \(m'\) will remain at rest if \(M=4mm'/(2m-m')\).
A smooth sphere of mass \(M\) is suspended from a fixed point by an inelastic string, and another sphere of mass \(m\) impinges directly on it with velocity \(v\) in a direction making an acute angle \(\alpha\) with the vertical. Shew that the loss of energy due to the impact is \[ \frac{1}{2}\frac{mM(1-e^2)v^2}{M+m\sin^2\alpha}, \] where \(e\) is the coefficient of elasticity.
The invariants of a system of two conics.
Envelopes of plane curves.
Curvilinear coordinates.
Differentials.
Series of complex constants.