Prove that the envelope of the line \(L\cos\theta+M\sin\theta=N\), where \(\theta\) is a parameter, is the conic \(L^2+M^2=N^2\). Two straight lines are drawn at right angles to each other, one touching \((x-a)^2+y^2=b^2\) and the other touching \((x+a)^2+y^2=c^2\). Prove that the envelope of the line joining their points of contact is the conic \[ (b^2+c^2)x^2+(b^2+c^2-4a^2)y^2+2a(b^2-c^2)x+a^2b^2+a^2c^2-b^2c^2=0, \] and that the tangents to this conic from any point on either circle meet the other circle at the ends of a diameter.
The perimeter and area of a convex pentagon \(ABCDE\) which is inscribed in a circle are denoted by \(2s\) and \(S\) respectively; and the sums of the angles at \(E\) and \(B\), \(A\) and \(C\), \dots are denoted by \(\alpha, \beta, \dots\). Shew that \[ s^2(\sin 2\alpha+\sin 2\beta+\sin 2\gamma+\sin 2\delta+\sin 2\epsilon)+2S(\sin\alpha+\sin\beta+\sin\gamma+\sin\delta+\sin\epsilon)^2=0. \]
Prove that if an algebraic equation \[ f(x) = x^n + a_1x^{n-1} + \dots + a_n = 0, \] has all its roots real, the derived equation \[ f'(x) = 0, \] has all its roots real. Prove that if \((n-1)a_1^2-2na_2\) is negative, the roots of the original equations are not all real.
If \[ f(p,q) = \int_0^{\pi/2} \cos^p x \cos qx dx, \quad (p>0), \] shew that \[ \left(1-\frac{q}{p}\right)f(p-1,q+1) = f(p,q) = \left(1+\frac{q}{p}\right)f(p-1,q-1). \]
Prove that the circle of curvature at a point \((x,y)\) will have contact of the third order with the curve if \[ (1+y_1^2)y_3 = 3y_1y_2^2 \] where \[ y_1 = \frac{dy}{dx}, \quad y_2 = \frac{d^2y}{dx^2}, \dots. \] Find the points on the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \] which have this property.
A gate of weight \(W\) is hung by means of two circular-headed staples driven into the gate at \(C, D\), and placed over two L-shaped staples driven into the gate post at \(A, B\). Prove that the pressure on the upper hinge will be \(W\sqrt{a^2+b^2}/b\) or \(Wa/b\) according as \(CD\) is just a little less or greater than \(AB\). \(2a\) is the horizontal length of the gate and \(b=CD\).
Two equal cards rest against one another on a perfectly rough horizontal table with their lowest edges parallel and at a distance apart equal to the length of either card. Shew that there is equilibrium only in the symmetrical position if the angle of friction \(\epsilon\) between the cards is less than \(\frac{1}{4}\pi\), that every position is one of equilibrium if \(\epsilon\) exceeds \(\tan^{-1}2\), and that if \(\epsilon\) lies between these limits the greatest angle \(\alpha\) which either card can make with the horizontal is given by the equation \[ \cos(\alpha-\epsilon)+2\cos\epsilon\cos 2\alpha = 0. \]
A train of mass 300 tons is originally at rest on a level track. It is acted on by a horizontal force \(F\) which increases uniformly with the time in such a way that \(F=0\) when \(t=0, F=5\) when \(t=15\); \(F\) being measured in tons weight, \(t\) in seconds. When in motion, the train may be assumed to be acted on by a frictional force of 3 tons, independent of the speed of the train. Find the instant of starting and shew that at \(t=15\) the speed of the train is 0.64 feet per second, and that the Horse-Power required at this instant is about 13.
A homogeneous sphere of mass \(M\) is placed on an imperfectly rough table, the coefficient of friction being \(\mu\). A particle of mass \(m\) is attached to the extremity of a horizontal diameter. Shew that the sphere will begin to roll or slide according as \(\mu\) is greater or less than \[ \frac{5(M+m)m}{7M^2+17Mm+5m^2}. \]