Prove that the series \(\sum_0^\infty x^n \sinh(n+1)\alpha\) is convergent if \(x\) is numerically less than \(e^{-\alpha}\), \(\alpha\) being assumed positive and the sum is \(\sinh\alpha / (1 - 2x \cosh\alpha + x^2)\); but that the series \(\sum_0^\infty x^n \sin(n+1)\alpha\) is convergent provided \(x\) is numerically less than unity, the sum being \(\sin\alpha / (1-2x\cos\alpha+x^2)\).
Verify by the use of partial fractions or otherwise that \begin{align*} \operatorname{cosec}(x-\alpha_1) \operatorname{cosec}(x-\alpha_2) &= A_1 \cot(x-\alpha_1) + A_2 \cot(x-\alpha_2), \\ \operatorname{cosec}(x-\alpha_1) \operatorname{cosec}(x-\alpha_2) \operatorname{cosec}(x-\alpha_3) &= B_1 \operatorname{cosec}(x-\alpha_1) + B_2 \operatorname{cosec}(x-\alpha_2) + B_3 \operatorname{cosec}(x-\alpha_3) \end{align*} where \(A_1 = \operatorname{cosec}(\alpha_1-\alpha_2)\), \(B_1 = \operatorname{cosec}(\alpha_1-\alpha_2) \operatorname{cosec}(\alpha_1-\alpha_3)\) with corresponding forms for the other constants. Extend the process to the product of \(n\) distinct cosecants, explaining why the sums contain the \(n\) cotangents or the \(n\) cosecants according as \(n\) is even or odd.
From a point \(O\) a normal \(OP\) is drawn to a curve and \(P\) is not a singular point on the curve: shew that the radial distance from \(O\) to the curve has \(OP\) as a maximum or a minimum according as \(O\) lies on one side or other of the centre of curvature at \(P\). Shew also that the length of the perpendicular drawn from \(O\) on a tangent of the curve has \(OP\) as a maximum or minimum according to the position of \(O\) relative to \(P\) and the centre of curvature: mark the result on a diagram.
A uniform circular cylindrical log of radius \(a\) and weight \(W\) lies with its axis horizontal between two rough horizontal parallel rails at the same level and at a distance \(2a \sin\alpha\) apart: shew that, if a gradually increasing couple be applied to the log in a plane perpendicular to the rails and axis, the log will turn over one of the rails when the couple is of magnitude \(Wa \sin\alpha\), provided the angle of friction \(\epsilon\) is greater than \(\alpha\); but otherwise the log will turn about the axis when the couple is \(Wa \sin\epsilon \cos\epsilon \sec\alpha\).
A heavy spherical ball of given resilience is to be projected with given initial speed from one given point so as to reach another given point after impact on a smooth vertical wall (the points being on one side of the wall but not in the same vertical plane perpendicular to the wall): shew that there are two directions of projection both in the same vertical plane (provided the given speed be great enough) and that the lower trajectory takes less time although the speed at arrival is less.
Prove that in the steady circular motion of the bob of a simple conical pendulum, the circular path is at a depth \(g/\omega^2\) below the point of suspension, where \(\omega\) is the angular velocity. Shew also that, if the resistance of the air be not neglected and the string be attached to the end \(A\) of an arm \(OA\) of length \(a\) which rotates in a horizontal plane about the fixed point \(O\), then the angular velocity may be so adjusted that the bob in steady motion describes a horizontal circle with the same angular velocity as \(A\) but so that its position in its circle is always ninety degrees behind that of \(A\). Prove that in this case the resistance \(R\) of the air is given by \(R = ma\omega^2\), where \(m\) is the mass of the bob, and that the velocity \(v\) of the bob corresponding to this resistance \(R\) is given by \[ v^2 = (l^2-a^2)\omega^2 - g^2/\omega^2, \] where \(l\) is the length of the string.
Greek views of a future life.
Athleticism in Greece.
The place of ceremonial in Roman life.
War and Literature.