Any point \(S\) on a sphere is displaced on the great circle through a fixed point \(O\) on the sphere to a point \(S'\) by the aberration law \(\sin SS' = k\sin SO\), where \(k\) is any finite number less than unity. Prove that, if \(S\) lie on a great circle having any fixed point \(P\) as pole, \(S'\) lies on a small circle with \(P\) as pole.
Shew that the necessary and sufficient condition that the three pairs of points \(A, A'; B, B'; C, C'\) on a straight line should belong to an involution is \[ BC' \cdot CA' \cdot AB' + B'C \cdot C'A \cdot A'B = 0. \] State the corresponding condition that three pairs of lines through a point should belong to an involution pencil, and prove that if two of the pairs are at right angles, the third pair are also at right angles.
A tetrahedron has each edge perpendicular to the opposite edge. Prove that the four perpendiculars from the vertices on the opposite faces are concurrent, the perpendicular from each vertex on the opposite face passes through the orthocentre of that face, and the sum of the squares of opposite edges is the same for each of the three pairs.
Discuss the number of conics which pass through \(m\) given points and touch \(n\) given lines in the several cases when \(m\) and \(n\) are zeros or positive integers such that \(m+n=5\).
Shew that the equations \[ x:y:1 = a_1 t^2 + 2b_1 t + c_1 : a_2 t^2 + 2b_2 t + c_2 : a_3 t^2 + 2b_3 t + c_3 \] are equivalent to equations of the form \[ t^2:2t:1 = A_1 x + A_2 y + A_3 : B_1 x + B_2 y + B_3 : C_1 x + C_2 y + C_3. \] Shew also that, if the equations be taken as the definition of a conic where \(x\) and \(y\) are rectangular coordinates and \(t\) a variable parameter, the existence of foci and directrices is established by the fact that \(\alpha\) and \(\beta\) can be chosen to make both the expressions \(a_1t^2+2b_1t+c_1-\alpha(a_3t^2+2b_3t+c_3)+i\{a_2t^2+2b_2t+c_2-\beta(a_3t^2+2b_3t+c_3)\}\) perfect squares. Find the focus of the parabola \(x=t^2-2t+2, y=t^2+1\).
Shew that, if \(m,n,a,b\) be real and \(m \neq n, a \neq b\), the expression \(\dfrac{m^2}{x-a}-\dfrac{n^2}{x-b}\) is such that (1) there are two real values between which it cannot lie for real values of \(x\), and (2) the imaginary values of \(x\) which make the expression real and intermediate between these two real values are all included in \(\dfrac{bm^2-an^2}{m^2-n^2} + \dfrac{(a-b)mn}{m^2-n^2}(\cos\theta+i\sin\theta)\) where \(\theta\) is any angle not zero or a multiple of \(\pi\).
Discuss the expression of a rational function of \(x\) as the sum of a polynomial and of partial fractions whose denominators are powers of linear or quadratic functions of \(x\) and numerators respectively constants or linear functions of \(x\). Illustrate by the cases \[ \text{(i) } \frac{x^5}{(x+1)(x+2)}, \quad \text{(ii) } \frac{x^5}{(x^2+x+1)(x+1)}, \quad \text{(iii) } \frac{x^4}{(x^2+1)^2(x+1)}. \]
Solution:
Find the asymptotes of \[ x^2(y+a)+y^2(x+a)+a^2(x+y)=0, \] and trace the curve.
Prove the existence of an 'instantaneous centre' for the motion of a flat body in its own plane. [A diagram shows a linkage with fixed points A, D and joints B, C. AB=BC=4, CD=2. Angles BCD and CDA are right angles.] In the figure the rods \(AB, BC, CD\) are smoothly jointed at \(B\) and \(C\), and \(AB\) and \(CD\) can turn about the fixed points \(A\) and \(D\) respectively. The angles \(BCD\) and \(CDA\) are right angles and \(AB=BC=4\) feet and \(CD=2\) feet. If the angular velocity of \(CD\) is given find the angular velocity of \(AB\) and of \(BC\) when the rods are in the position shewn. If each rod is uniform and weighs 2 lbs. per foot of its length, find by Virtual Work or otherwise the couple acting on \(CD\) which will keep the rods at rest in the position shewn with \(AD\) horizontal.
Prove that, when a particle describes a path under the action of a force directed to a fixed point, the radius vector drawn from the point to the particle describes equal areas in equal times. A particle of mass \(m\) is held on a smooth table. A string attached to this particle passes through a hole in the table and supports a particle of mass \(3m\). Motion is started by the particle on the table being projected with velocity \(V\) at right angles to the string. If \(a\) is the original length of the string on the table, prove that when the hanging weight has descended a distance \(a/2\) (assuming this possible) its velocity will be \[ \frac{\sqrt{3}}{2}\sqrt{(ga-V^2)}. \]