Prove that \[ x^{2n} - 2x^n y^n \cos n\theta + y^{2n} = \prod_{r=0}^{n-1} \left\{x^2 - 2xy \cos\left(\theta + \frac{2r\pi}{n}\right) + y^2\right\}. \] A regular polygon of \(n\) sides is inscribed in a circle. The projection on the plane of the circle of the line joining the centre to any point \(P\) in space makes an angle \(\alpha\) with the radius to one of the vertices, and \(r_1\) and \(r_2\) are the greatest and least distances of \(P\) from the circumference of the circle. Prove that the product of the distances of \(P\) from the vertices of the polygon is \((s^{2n} - 2s^n d^n \cos n\alpha + d^{2n})\), where \(r_1+r_2=2s, r_1-r_2=2d\).
A uniform rod rests with its ends on a smooth parabolic wire, whose axis is vertical and vertex downwards. Shew that, if the length of the rod exceeds the latus rectum, a position of equilibrium is possible in which the rod passes through the focus, and prove that it is stable.
The following readings connect the candle power and voltage of an incandescent lamp.
Find the intrinsic and Cartesian equations of a heavy uniform chain suspended from two fixed points. A uniform chain of length \(l\) rests in a straight line on a rough horizontal table. One end is raised to height \(h\) above the table and the chain is on the point of motion. Shew that the length of the straight part on the table is \[ l+\mu h - \{(\mu^2+1)h^2+2\mu lh\}^{\frac{1}{2}}. \]
Distinguish between the latent heat of saturated steam and the excess of energy of a unit mass of saturated steam over that of a unit mass of water at the same temperature. Calculate the numerical difference between them for a pressure of 226 lbs. per square inch, having given that the volume of a pound of saturated steam at this pressure is 2.06 cubic feet, and the volume of a pound of water 0.017 cubic feet. The unit in which the result is expressed must be stated.
Obtain the equation of a tangent to the circle \((x-a)^2 + (y-b)^2 = c^2\) in the form \((x-a)\cos\theta + (y-b)\sin\theta = c\). Hence shew that the equation of the tangents to this circle from the point \((h,k)\) may be written as \[ \{(x-h)^2+(y-k)^2\}\{(h-a)^2+(k-b)^2-c^2\} = \{(x-a)(h-a)+(y-k)(k-b)\}^2. \]
The tractive effort of an electric train is uniform and equal to the weight of 4 tons. The road resistance is 40 lbs. wt. per ton of the train, and the brake resistance is an additional 200 lbs. wt. per ton. The train is taken from one station to the next, distant half a mile, in \(1\frac{1}{4}\) minutes, full power being kept on until the speed reaches 30 miles an hour, when the train "coasts" at a uniform speed until power is shut off and the brakes are put on. Shew that the mass of the train is approximately 85 tons.
If \(A, B, C\) are angles such that \(A+B+C=0\) shew that \[ \frac{1+\tan A \tan B \tan(C+D)\tan D}{1-\tan A \tan B \tan(C-D)\tan D} = \frac{\cos(A-D)\cos(B-D)\cos(C-D)}{\cos(A+D)\cos(B+D)\cos(C+D)}. \]
Prove that the path of a projectile under gravity is a parabola. The velocity of projection being given, find (i) the envelope of all possible paths, (ii) the locus of their vertices. Hence shew that all points lying between a certain spheroid and a certain paraboloid can only be reached by plunging fire. Determine (i) the least possible velocity of projection with which a given point can be reached, (ii) the area above the horizontal commanded on a vertical plane at a given distance from the point of projection.
A steel bar with rectangular faces has diagonal lines drawn on one of its faces, dimensions 9" by 2", and is subjected to a tension of 4 tons per square inch of section. Find the change of angle between the lines, having given that Young's modulus is 13000 tons per square inch, and that Poisson's ratio is 0.25. (Poisson's ratio is the ratio of the numerical magnitude of the lateral strain to that of the longitudinal strain.)