A circle of radius \(b\) rolls on the outside of a circle of radius \(a\) and a point on the circumference of the rolling circle traces an epicycloid. Prove that the length of the arc from one cusp to the next is \(8b(a+b)/a\), and that the area between this arc and the circle is \(\pi b^2(3a+2b)/a\).
Give an account of the method of reciprocation with respect to a circle, and illustrate its use.
Give a general account of the resolution of a fraction (whose numerator and denominator are polynomials) into partial fractions. Resolve into partial fractions \(\dfrac{x^2}{(x+2)^2(x^2+1)}\).
A pole DE, inclined to the vertical, stands at D on a horizontal plane, and A, B, C are three collinear points in the plane, such that AB=\(a\), BC=\(c\). The angles DAC, DCA (\(\theta, \phi\)) and the elevations (\(\alpha, \beta, \gamma\)) of E as seen from A, B, C are measured. Shew how to determine from these measurements the inclination of the pole to the vertical and the direction in which it leans.
Shew that the locus of centres of a family of conics through four given points is a conic. Shew also that, in the case of a family of conics having three-point contact at P and passing through a fourth point Q, the locus of centres touches the conics at P, has curvature at P of the opposite sign and of double the magnitude of that of the conics, and has PR as a diameter, where R is the middle point of PQ.
Shew that the function \(\sin x + a \sin 3x\) for values of \(x\) from \(0\) to \(\pi\) has no zeroes except the terminal ones if \(-\frac{1}{9}
Find formulae of reduction for the integrals \[ \int \sin^n x dx, \quad \int e^{-ax}\sin^n x dx, \quad \int \frac{dx}{(x^2+a^2)^n}. \] If \[ f(p,q) = \int_0^{\frac{\pi}{2}} (\cos x)^p \cos qx dx, \] shew that \[ f(p,q) = \frac{p(p-1)}{p^2-q^2}f(p-2,q) = \frac{p}{p+q}f(p-1, q-1). \]
A, B are two equal and equally rough weights lying on a rough table and connected by a string. A string is attached to B and is pulled in a direction making an angle \(\alpha\) with AB produced until motion is about to ensue. Examine the cases where one or both of the weights are on the point of motion, shewing that they arise according as \(\alpha \lessgtr \frac{\pi}{4}\). Shew that in the latter case B is about to move in a direction making an angle \(2\alpha\) with AB produced. Find the force needed in each case.
State the general principle of virtual work and prove that when applied to the case of a single rigid body in two dimensions it leads to three independent necessary and sufficient conditions of equilibrium and deduce these conditions in one of their usual forms. Four equal uniform rods, of length \((a+b)\), are hinged at A, B, C, D, and are suspended by two strings of equal length, so that the diagonal AC is vertical and A is the highest point. The strings are attached to two pegs in the same horizontal line and to two points in AB, AD at distance \(a\) from A. Prove that if the rods rest in the form of a square the inclination of the strings to the horizontal is \(\tan^{-1}(a/b)\).
A particle of mass \(m\) is attached by a string to a point on a fixed circular cylinder of radius \(a\) whose axis is vertical. The particle is projected with velocity \(v\) at right angles to the string along a smooth horizontal plane so that the string winds itself round the cylinder. Shew (i) that the velocity of the particle is constant; (ii) that the tension of the string is inversely proportional to the length which remains straight at any instant; (iii) that if the initial length of the string is \(l\) and the breaking tension is \(T\), the string will break when it has turned through an angle \(\dfrac{l}{a}-\dfrac{mv^2}{aT}\).