The line \(AB\) is equal in length to \(A'B'\) and in the same plane: shew that \(AB\) can always be moved to coincidence with \(A'B'\) by a rotation about a point in the plane except in the case when, \(AA'\) being equal, parallel and in the same sense as \(BB'\), translation is alone necessary. Shew also that a line can be chosen in the plane so that the image of \(AB\) by reflexion in the line can be moved by translation parallel to the line to coincide with \(A'B'\). Shew that two successive reflexions of a plane figure in lines in the plane are equivalent to a rotation and that any odd number of reflexions is equivalent to a single reflexion and translation.
Shew that, if two points at distance \(a\) apart be inverted with regard to an origin distant \(e\) and \(f\) from them respectively, the distance between the inverse points is \(\frac{a}{ef}\) if the radius of inversion be of unit length. Shew that the problem of inverting three points \(A, B, C\) with regard to a point in their plane so that the inverse points are the vertices of an equilateral triangle admits of two real solutions and that the two centres of inversion are inverse points with regard to the circle \(ABC\).
The coefficients \(a, b, c, a', b', c'\) are real in the quadratic expressions \[ f(x) = ax^2+bx+c, \quad \phi(x) = a'x^2+b'x+c' \] and a value of \(\lambda\) is taken so that the roots \(x_1, x_2\) of \(f(x)-\lambda\phi(x)=0\) are real. Prove (1) that, if \(f(x_1), f(x_2)\) have opposite signs, the like result holds whatever other such value \(\lambda\) has and moreover the roots are real for all real values of \(\lambda\): (2) that, if \(f(x_1), f(x_2)\) have the same sign, the like result holds for all such pairs of roots but the values of \(\lambda\) giving real roots are restricted.
Prove by means of the expansions or otherwise that, when \(n\) is a positive integer and \(x\) is positive and less than \(n\), \[ \left(1+\frac{x}{n}\right)^n < e^x < \left(1-\frac{x}{n}\right)^{-n}. \] Deduce that the inequalities are also true when \(x\) is negative and numerically less than \(n\). Prove also that, when \(x\) is positive, \(\left(1+\frac{x}{n}\right)^n\) is nearer to \(e^x\) than \(\left(1-\frac{x}{n}\right)^{-n}\) is.
Shew that for the special value \(\lambda = -\frac{2a^2b^2}{a^2+b^2}\) the conics \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad \frac{x^2}{a^2+\lambda} + \frac{y^2}{b^2+\lambda}=1 \] cut orthogonally whatever be the angle between the oblique axes of reference. Shew that the four common points are then given by \(\frac{x^2}{a^2} = \frac{y^2}{b^2} = \frac{1}{a^2+b^2}\). Shew also that these results mean geometrically that, if any rectangle be described about an ellipse, there is a concentric hyperbola through the points of contact having the sides of the rectangle as normals: shew that of the family of hyperbolas obtained by varying the rectangle one is confocal to the ellipse, the diagonals of the rectangle being then along the axes of the ellipse.
Shew that, if \(c^2
Shew from the differential coefficients that the functions \[ x - \log(1+x), \quad \frac{2x}{2+x} - \log(1+x) \] are respectively positive and negative when \(x\) is positive. Shew also that, when \(a\) and \(h\) are positive, \(\log(a+\theta h) - \log a - \theta\{\log(a+h)-\log a\}\) considered as a function of \(\theta\), has a maximum for a value of \(\theta\) between 0 and \(\frac{1}{2}\).
Shew that a parallelogram of freely jointed rods is in equilibrium under forces in its plane at the joints, if the polygon of the forces is closed and has its diagonals parallel to the rods. Shew also that the form is stable, supposing one joint fixed and the forces to retain their magnitudes and directions in space, provided the sums of the tensions in opposite rods are both positive, a thrust being reckoned negative.
A particle \(P\) is moving under the law of acceleration \(n^2.OP\) towards a fixed point \(O\): initially the particle is moving away from \(O\) with velocity \(v_0\), and its distance from \(O\) is equal to \(x_0\). Shew that the particle next comes to rest at a distance \(\sqrt{(x_0^2+v_0^2/n^2)}\) from \(O\): and calculate the time which elapses. An elastic string has one end fixed at \(A\), and at the other end is fastened a particle heavy enough to stretch the string (statically) to twice its natural length \(a\). The particle is dropped from rest at \(A\); what time will elapse before the particle next comes to rest?
A bullet of mass \(m\) is fired into a block of wood of mass \(M\), which hangs by vertical cords of equal length, the other ends of the cords being fastened to fixed points on the same level. The bullet penetrates a distance \(a\) horizontally into the block and in the subsequent motion the block rises through a height \(h\). Calculate the velocity with which the bullet strikes the block, and shew that the average resistance to penetration is equal to \(gM(1+M/m)h/a\).