A box in the form of a cylinder of height \(b\) with its generators vertical is divided into two parts by a horizontal partition at height \(a\), (\(a
If \(U\) and \(p\) denote the energy per unit mass and the pressure of a substance, supposed expressed as functions of the temperature \(T\) and the specific volume \(v\), establish the relation \[ \frac{\partial U}{\partial v} = T\frac{\partial p}{\partial T}-p. \] A substance is such that \[ U=av+C_vT, \] where \(a\) and \(C_v\) are constants. Show that it has an equation of state of the form \[ (p+a)f(v)=T. \] Show further that the specific heat at constant pressure \(C_p\) is independent of \(T\), and that if it is also independent of \(v\) then \[ f(v) = \frac{v+k}{C_p-C_v}, \] where \(k\) is a constant.
In a spherical triangle show that \[ \sin(B+C) = \frac{\sin A(\cos b+\cos c)}{1+\cos a}. \] The Foucault siderostat consists of a mirror which is constrained to move so that a ray from some given star \(A\) is reflected due South at all hour-angles. If \(A'\) is the point (due South) where the reflected beam from \(A\) meets the celestial sphere, and \(B'\) is the corresponding point for the ray reflected by the same mirror from some other star \(B\), prove that as the stars change in hour-angle the locus of \(B'\) is in general a small circle round \(A'\). Further, if \(\phi\) is the position-angle of \(B\) with respect to \(A\), show that when the hour-angle of \(A\) is positive (i.e. after culminating) the position-angle of \(B'\) with respect to \(A'\) is \(\phi'\), given by \[ \phi' = \phi + P\hat{A}A' + P\hat{A'}A, \] \(P\) denoting the North Pole. Hence, by means of the result quoted above for a spherical triangle, show that the rotation of \(B'\) with respect to \(A'\) is counter-clockwise or clockwise according as \(A\) is or is not circum-polar; and that in the critical case in which \(A\) just sets, the orientation of \(B'\) with respect to \(A'\) remains fixed (\(\phi'\) is measured positive in a clockwise direction).
Given two intersecting straight lines and a point in a plane, shew how to draw the straight line, the intercept on which between the given lines is bisected at the point, and also the two lines for which the intercepts are trisected.
In a plane a circle is given and two points external to it. Shew how to construct the two circles which pass through the given points and touch the given circle.
In a parabola \(SY\) is the perpendicular from the focus \(S\) on the tangent at the point \(P\) and \(A\) is the vertex, prove that \(SY^2 = SA.SP\). Prove that, if \(PM, PN\) be the perpendiculars on the tangent at the vertex and on the axis respectively, \(MN\) touches a parabola.
Prove that, if \(A\) and \(B\) two points on a conic be each joined to four given points on the conic, the resulting pencils of four lines have the same cross ratio. Two points \(S\) and \(H\) are the foci of a variable conic inscribed in a triangle \(ABC\): shew that, if \(S\) describes a straight line, \(H\) describes a conic circumscribing the triangle \(ABC\).
Prove that through two circles which are plane sections of the same sphere it is possible to construct two cones and that the line joining their vertices is the polar line with regard to the sphere of the line of intersection of the two planes.
Prove that \(x = \mu^2 - \lambda^2, y=2\lambda\mu\) is a point of intersection of the two confocal parabolas of the systems obtained by making in turn the two parameters \(\lambda\) and \(\mu\) the one constant and the other variable. Shew also that the two parabolas cut at right angles.
Prove that the equation of the line of a chord of the ellipse \(x^2/a^2+y^2/b^2=1\) may be written \[ \frac{x}{a} t_1 t_2 \left(\frac{a}{x}+1\right) - \frac{y}{b}(t_1+t_2) + \left(1-\frac{x}{a}\right) = 0, \] where \(t_1, t_2\) denote the tangents of the halves of the eccentric angles of the ends of the chord. A point \(Q\) on the auxiliary circle of an ellipse is joined to the extremities \(A, A'\) of the major axis, the joining lines cutting the ellipse again in \(P\) and \(P'\) respectively: shew that the line \(PP'\) envelopes an ellipse.