Prove that in polar coordinates \(r\frac{d\theta}{dr}\) is the tangent of the angle between the radius vector and tangent to a curve. In the case of the curve \(r^n=a^n\cos n\theta\), prove that \(a^n\frac{d^2r}{ds^2}+nr^{2n-1}=0\).
Evaluate \[ \int_0^\infty x^2\sin x, \quad \int_0^\infty \frac{xdx}{(1+x)(1+x^2)}, \quad \int_a^b \frac{dx}{(a+b-x)^2\sqrt{(x-a)(b-x)}} \quad (b>a). \] Find a formula of reduction connecting \[ \int \cos m\theta \cos^n\theta d\theta \quad \text{with} \quad \int \cos m\theta \cos^{n-2}\theta d\theta. \]
Trace the curve \[ x(y^2-\frac{1}{2}a^2) - y(x^2-\frac{1}{2}a^2) = a^3, \] and shew that the radius of curvature where it meets either axis is \(\frac{25}{12}a\sqrt{2}\).
Find graphically the greatest root of the equation \[x^3 - 3x + 1 = 0,\] exhibiting the third place of decimals with certainty and an estimate of the fourth, and explaining the procedure.
Prove Brianchon's Theorem. A conic is drawn to touch four tangents to a given conic and the chord of contact of one pair of tangents; prove that it also touches the chord of contact of the other pair.
Prove that a force acting in the plane of a triangle \(ABC\) can be replaced uniquely by three forces along the sides of the triangle. If each of a system of coplanar forces be replaced in this way by forces of type \(p \cdot BC, q \cdot CA\) and \(r \cdot AB\), shew that the necessary and sufficient conditions that the system reduces to a couple are that \(\Sigma p = \Sigma q = \Sigma r\).
Prove that the locus of a point which moves so that the ratio of its distances from two fixed points is constant is a circle. Having given the distance between the fixed points and the ratio, find the position of the centre and the radius of the circle.
Explain the general principles of the method of inversion in pure geometry, and state and prove what seem to you to be the simplest and most important general theorems connected with it. Supposing that \(P, Q, \dots\) are points, and \(A, B, C, \dots\) circles, and \(P', Q', \dots, A', B', C', \dots\) their inverses with respect to a circle \(S\), prove:—
A cone of semi-vertical angle \(\alpha\) is bounded by the vertex and by a plane cutting the axis at an angle \(\beta\), at a distance \(b\) from the vertex. Sketch the form of a flat metal sheet which can be bent into the form of this cone, giving some dimensions; and find a convenient method of plotting the bounding curve. Either prove that it may be plotted from the focal radii of an auxiliary ellipse of eccentricity \(\tan \alpha \cot \beta\) and latus rectum \(2b \sec \alpha\), or give some other method.
A circle passing through the foci of a hyperbola cuts one asymptote in \(Q\) and the other in \(Q'\). Shew that \(QQ'\) touches the hyperbola or is parallel to the major axis.