Interpret in projective geometry the projections of (i) a circle, (ii) a right angle, (iii) a pair of equal angles, (iv) the middle point of a line, (v) the foci of a conic. Illustrate by stating in a form true for all conics the property that the angle at the centre of a circle is double that at the circumference.
Lines are drawn through the vertices \(A, B, C\) of a triangle making angles \(\pi/3\) in the same sense with the sides \(BC, CA, AB\) respectively and forming a triangle whose sides are \(B'C', C'A', A'B'\) respectively. Show that the triangles \(ABC, A'B'C'\) are equal in all respects, and show also that \(AA'^2+BB'^2+CC'^2=9R^2\), where \(R\) is the radius of the circumcircle of \(ABC\).
Illustrate the methods of expressing the ratio of two rational functions of \(x\) as a sum of partial fractions by considering the cases \[ \text{(i) } \frac{x^4+x^2}{x^6-1}, \quad \text{(ii) } \frac{x^6}{(x-1)^3(x^2+1)^2}. \]
The equation of a conic is \(ax^2+2hxy+by^2+2gx+2fy+c=0\). Show how to determine the lengths of its axes and the coordinates of its centre. Find the tangential equation of the conic and show that the foci are given by \(X^2-Y^2=\Delta(a-b), XY=\Delta h\), where \(X=Cx-G, Y=Cy-F, \Delta = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}\), \(C, F, G\dots\) are minors of \(c,f,g\dots\) in \(\Delta\).
Investigate the tangential equation of the circular points at infinity and show that conics confocal with \(\Sigma=0\) are represented tangentially by \[ \Sigma + \kappa(l^2+m^2+n^2-2mn\cos A-2nl\cos B-2lm\cos C)=0, \] using trilinear coordinates. Find the \(x, y, z\) equations of conics inscribed in the triangle of reference and confocal with \(x^2+y^2+z^2=0\).
A curve touches the axis of \(x\) at \(x=0\) and \(P\) is a point on it at a distance \(s\) from \(O\) measured along the arc. Prove that neglecting \(s^4\) and higher powers of \(s\), the coordinates of \(P\) are \(x=s-\frac{s^3}{6\rho^2}, y=\frac{s^2}{2\rho}-\frac{s^3}{6\rho^2}\frac{d\rho}{ds}\), where \(\rho\) and \(\frac{d\rho}{ds}\) are the values at \(O\) of the radius of curvature and its differential coefficient with respect to \(s\). The normals at \(O\) and \(P\) meet in \(Q\). Prove that to the same degree of approximation the difference between \(QO\) and \(QP\) is \(\frac{s^3}{12\rho^2}\frac{d\rho}{ds}\).
Find the equations of the tangent and normal at any point of the curve whose coordinates are given by the equations \[ x=c(m\cos n\theta+n\cos m\theta), \quad y=c(m\sin n\theta+n\sin m\theta), \] in terms of the parameter \(\theta\), and show that the evolute is a curve of the same species. Show that the \((p,r)\) equation of the curve is \[ r^2=(m-n)^2c^2+\frac{4mn}{(m+n)^2}p^2. \]
Show that a couple is equivalent to another couple of equal moment in the same or any parallel plane. Prove that the resultant of two couples in different planes is another couple; obtain the law of composition of couples. Two couples in the faces \(ABC, ABD\) of a tetrahedron are represented by the areas of these faces in magnitude also; find the direction of the axis of the resultant couple in each of the two possible cases.
Prove the isochronism of the cycloid under gravity; show that the projection of the particle on any vertical line moves with simple harmonic motion; and prove that when the particle starts from rest at the cusp, its acceleration is constant in magnitude and directed towards a point moving uniformly in a horizontal line.
Prove the formula \(F = \frac{h^2}{p^3}\frac{dp}{dr}\), for a particle describing a plane orbit under a force \(F\) to the pole, \(p\) being the perpendicular on the tangent from the pole, \(r\) the radius vector, and \(h\) a constant. If the orbit is an ellipse with the centre as pole, show that \(F=\mu r\), and find \(\mu\) in terms of \(h\) and the constants of the ellipse. If the particle is projected with velocity \(V\) at a distance \(c\) from the centre of force and at right angles to the line joining it to the centre of force, calculate the area of the orbit.