At an election the majority was 1184, which was one-fifth of the total number of votes; how many votes did each side poll?
Find the sum of 24 terms of the series \(4\frac{1}{2}+3\frac{3}{4}+3+\dots\). The sum of eight terms of an arithmetical progression is 24 and its fifth term is 1; find the first term and the common difference.
Three masses, each of 2 lbs. weight, are attached to different points on a string which hangs from a fixed point. Horizontal forces, of 1 lb. weight to the left, 2 lbs. weight to the right, and 3 lbs. weight to the left, are then applied to the different masses in the order named, beginning from the bottom, all the forces being in the same vertical plane. Find the tension of each portion of the string when in equilibrium and the angle which each portion makes with the vertical.
Shew that, if P and Q are inverse points with respect to a circle C, and P' and Q' their inverses with respect to another circle \(\Gamma\), then P' and Q' are inverse points with respect to C', the inverse of C with respect to \(\Gamma\). Shew that the locus of the inverse points of a given point with respect to a system of coaxal circles is a circle, which cuts the system orthogonally.
Two equal uniform rods \(AB\), \(BC\), each of length \(2a\), are smoothly jointed at \(B\), and are supported in the same horizontal line by a smooth support under each rod. Shew that, if the joint can only sustain a stress not exceeding \((1/n)\)th of the weight of each rod, the distance of either support from \(B\) must lie between \(\frac{na}{n+1}\) and \(\frac{na}{n-1}\), or between \(\frac{na}{n+1}\) and \(2a\), according as \(n >\) or \(< 2\).
\(A, P, Q\) are any three points on a circle such that the angle \(PAQ\) is given, find the envelope of \(PQ\); (1) when \(A\) is fixed, (2) when \(A\) takes all positions on the circle.
By assuming the properties of a complete quadrangle, or otherwise, prove that two conics have, in general, one common self-conjugate triangle; and determine the cases of (i) failure, (ii) indeterminateness. Shew that, if two conics \(S\) and \(S'\) have a common self-conjugate triangle, there are four conics (\(R\)) with respect to which each conic is the polar reciprocal of the other: but that, if \(S\) and \(S'\) have simple contact, there are only two such conics (\(R\)).
A cage weighing 3000 lbs. is being hoisted up a mine shaft at a steady speed of 4 ft. per sec., when the hoisting gear fails and the upper end of the rope is suddenly held fast at a moment when the free length of the rope is such that a load of 5000 lbs. would stretch it 1 ft. Neglecting inertia of the rope, find the period and amplitude of the oscillations of the cage, and the greatest tension in the rope.
Shew that the anharmonic ratio of the range intercepted on a variable tangent to a conic by four fixed tangents is constant. Shew that the theorem `four tangents to a conic, such that the intersection of one pair lies on the line joining the points of contact of the other pair, intercept a harmonic range on a variable tangent to a conic' may, by projection and the consideration of an appropriate special tangent, be reduced to the form `if A, B, C are three points on a circle, and B is equidistant from A and C, then the tangents at A and C cut the tangent at B in points equidistant from B': and hence prove the theorem.
Two coplanar forces are represented in magnitude and position by \(m . AA'\) and \(n . BB'\). Shew that, if \(G\) and \(G'\) are the mean centres of the points \(A\) and \(B\) and of \(A'\) and \(B'\) respectively for multiples \(m\) and \(n\), the resultant is represented in magnitude and direction by \((m+n) GG'\); but in position only if either \(AB\) and \(A'B'\) are parallel or \(AA'\) and \(BB'\) are parallel.