Rationalise the equation \(\sqrt{x}+\sqrt{y}+\sqrt{z}+\sqrt{t}=0\), and express the result in factors each of which is linear in the square roots of \(x,y,z\) and \(t\). If \(a+b+c+d=0\) and \(x+y+z+t=0\), prove that the results of rationalising the equations \begin{align*} \sqrt{ax}+\sqrt{by}+\sqrt{cz}+\sqrt{dt}&=0, \\ \sqrt{bx}+\sqrt{ay}+\sqrt{dz}+\sqrt{ct}&=0, \end{align*} are equivalent. State the two other equations which lead to equivalent results.
If \(p_r/q_r\) is the \(r\)th convergent of the continued fraction \[ a_1 + \frac{1}{a_2 +} \frac{1}{a_3 +} \dots + \frac{1}{a_n}, \] prove that \(p_n/p_{n-1}\) is equal to \[ a_n + \frac{1}{a_{n-1} +} \frac{1}{a_{n-2} +} \dots + \frac{1}{a_1}, \] and express \(q_n/q_{n-1}\) as a continued fraction. Prove that \[ a_1 + \frac{1}{a_2 +} \frac{1}{a_3 +} \dots + \frac{1}{a_{n-1} +} \frac{1}{2a_n +} \frac{1}{a_{n-1} +} \dots + \frac{1}{a_1} \] is equal to the harmonic mean of the convergents \(p_{n-1}/q_{n-1}\) and \(p_n/q_n\), and express the arithmetic mean of the convergents as a continued fraction.
The lengths of the sides of a convex quadrilateral are \(a,b,c,d\), and the sides of lengths \(a,c\) are parallel. Find expressions for the cosines of the angles of the quadrilateral in terms of the sides, and prove that, if \(2\theta\) is the sum of a pair of opposite angles, \[ \cos^2\theta = \frac{(s-a)(s-c)(b-d)^2}{bd(a-c)^2}, \] where \(2s=a+b+c+d\).
Prove that the curve \(2x^2=ay(3x-y)\) has two tangents in the direction of the axis of \(x\) and one tangent in the direction of the axis of \(y\), giving the equation in each case. Sketch the curve, and prove that the area of its loop is \(81a^2/80\).
Prove that an ellipse can be described to touch the sides of a given triangle at their mid-points, and shew how to construct its principal axes. Deduce, or prove otherwise, that there are two sets of parallel planes on any one of which the projection of a triangle given in space is an equilateral triangle.
Prove that there are four lines in a plane the respective shortest distances of which from three fixed points in the plane are in assigned numerical ratios: also that there are eight planes in space the respective perpendiculars on which from four fixed points in space are in assigned numerical ratios. Shew also that in the corresponding problems of the distances of a variable point from three fixed points in a plane and from four fixed points in space, the number of solutions in each case is two.
Prove that two conics have three pairs of common chords, and explain under what circumstances two pairs are (i) imaginary and (ii) coincident. Prove that, by suitable choice of homogeneous coordinates \(X,Y,Z\), the conics \begin{align*} 7x^2+8y^2+18z+7 &= 0, \\ % Note: original has 18z, likely a typo for 18x or 18. Assuming z is not a coordinate. Transcribing as seen. x^2+16xy-2z+16y+1 &= 0, % Note: original has -2z, likely a typo. \end{align*} assume equations of the form \(XY+Z^2=0\), and express \(X,Y,Z\) in terms of \(x\) and \(y\). % Note: The conics given seems unusual for this type of transformation, assuming there may be typos in the original paper, possibly z should be x.
Forces act in order along the sides of a convex polygon. Prove that the system is equivalent to a couple proportional to the area of the polygon in each of the following sets of circumstances:
Two equal uniform rods are fastened at right angles to one another at a common end, and, with that end uppermost, are free to slide each in contact with one of two smooth rails in the form of right circular cylinders, of equal radius \(a\), which have their axes parallel, in the same horizontal plane, and at a distance \(c\) apart. Prove that, if the length of either rod lies between \(4(a+c)\) and \(4(a+c\sqrt{2})\), there are three configurations of equilibrium, that in which the rods are equally inclined to the vertical being stable and the other two unstable.
Two equal logs of rectangular cross section, each of mass \(M\), lie close together end to end on a rough table, and a wedge, of negligible mass, which has a vertical angle of \(2\theta\), is inserted symmetrically between them; the angle of friction between a log and the table is \(\epsilon\) and that between a log and the wedge is \(\epsilon'\). Prove that a vertical blow \(I\) applied symmetrically to the wedge will cause motion to ensue, provided that \(\theta+\epsilon+\epsilon' < \pi/2\); and shew that when the blocks come to rest they are separated by a distance \[ \frac{I^2 \cos^2(\theta+\epsilon+\epsilon')}{2M^2g \sin 2\epsilon \sin^2(\theta+\epsilon')}. \]