A rectangular tank 6~ft. long, 5~ft. wide and 4~ft. deep stands on a slope with the two corners at the ends of one short side of the bottom face 3~ft. and 2~ft. above the lowest corner. How much water will the tank hold?
Two smoothly jointed uniform beams \(AB\), \(BC\), lengths \(l\), \(3l\) and weights \(W\), \(3W\), rest in a horizontal line on three smooth supports at \(A, D, C\), where \(AD = DC\). Find the reactions on the supports. If the beam \(BC\) is cut at \(E\), between \(D\) and \(C\), at a distance \(x\) from \(D\), show that the couple which, applied to the rod \(BE\), maintains the rods \(AB, BE\) in their horizontal position is \(W(l - x^2/2l)\).
Prove that if tangents are drawn from a point to all the conics touching four given straight lines the pairs of tangents so obtained form a pencil in involution, of which the double rays are the tangents to the conics of the system which pass through the given point. Hence prove that the two confocal conics through a given point cut at right angles.
Give an account of the methods of conical and orthogonal projection. State in each method the figure corresponding to (i) a circle, (ii) two sets of straight lines cutting at right angles, (iii) a bisected straight line. Illustrate your theory by extending to conics the theorems
One of the internal common tangents of two circles touches the circles at \(P\) and \(Q\), and meets the external common tangents in \(A\) and \(B\). Prove that \(AP = QB\).
\(ABCDE\) is a pin-jointed framework in a vertical plane. It is free to turn about a fixed horizontal axis through \(E\). To \(C\) is attached a load of 100~lbs. and a horizontal force \(P\) just maintains the frame with the side \(AD\) vertical. Find the force \(P\), the reaction at \(E\) and the forces in all the bars, preferably by means of a reciprocal force diagram. % The following is a description of the diagram provided in the paper. % A framework ABCDE is shown. % AD is a vertical member. A is above D. % The angle at A is 90 degrees, formed by members AB and a horizontal line extending from A where force P is applied. % B is to the left of AD. C is below B and D. % E is a point between B and D, connected to both, and also to C. % Members are AB, BC, CD, DA, BE, EC, ED. % The structure is hinged at E. % A load of 100 lbs hangs from C. % A horizontal force P acts at A, pointing to the right. % Dimensions are given: % AB = AD = 50" % BC = DC = EC = 30" % BE = DE = 25"
A uniform rod is placed over a rough horizontal rail and rests with one end against a rough vertical wall, the rail being parallel to the wall and perpendicular to the rod. If the rod is on the point of slipping up the wall, show that it will make an angle \(\theta\) with the horizontal, where \[ a \cos^2 \theta \cos (\theta + \lambda + \lambda') = c \cos \lambda \cos \lambda', \] \(\lambda, \lambda'\) being the angles of friction at the points of contact of the rod with the rail and wall. It is assumed that the rod is sufficiently long to make such a position of equilibrium possible. The length of the rod is \(2a\), and the rail is at a horizontal distance \(c\) from the wall.
Prove that, if \(A, B, C, D\) are the angular points of a rhombus, taken in order, the locus of a point \(P\), which moves so that \(PA, PC\) are harmonic conjugates of \(PB, PD\), is the ellipse whose axes are \(AC\) and \(BD\).
Prove that, if \(P\) and \(Q\) are two given polynomials in \(x\), with no common factor, it is possible to find two other polynomials \(A\) and \(B\) such that \[ AP + BQ = 1. \] Prove further that, if \((A_1, B_1)\) is one solution of the problem, the most general solution is \((A_1 + CQ, B_1 - CP)\), where \(C\) is any polynomial, and that, if \(A\) is restricted to be of lower order than \(Q\), there is one and only one solution. Hence (or otherwise) shew that, if \(f(x), \phi(x)\) are two polynomials of order \(m, n\) respectively, with no common factor, and \(\phi = (x-a)^p (x-b)^q \dots\), where \(a, b, \dots\) are different, and \(p, q, \dots\) are positive integers, then \(f(x)/\phi(x)\) can be expressed in the form \[ R(x) + A_1/(x-a) + \dots + A_p/(x-a)^p + B_1/(x-b) + \dots + B_q/(x-b)^q + \dots, \] where \(A_1, \dots B_1, \dots\) are constants, and \(R\) is a polynomial of order \(m-n\) if \(m \ge n\) but otherwise zero.
Prove that the centroid of a sector of an ellipse bounded by two conjugate semi-diameters lies on a diagonal of the parallelogram of which the semi-diameters are two sides.