Find the asymptotes of the curve \[ x^4 + 3x^2y + 2x^2y^2 + 2xy + 3x+y = 0; \] determine on which side or sides the curve approaches each asymptote, and where it cuts the asymptotes.
Three particles of equal mass are connected by light rods forming an equilateral triangle \(ABC\) with the particles at the corners and rest on a smooth horizontal plane. Shew that, if an impulse be applied to the particle at \(A\) in a direction parallel to \(BC\), the motion of the particles can be obtained by rolling the circumcircle of the triangle \(ABC\) along a certain straight line.
Find a formula of reduction for the integral \[ \int_0^{\pi/2} \sin^m x \cos^n x \,dx \] reducing one of the indices by two; and evaluate \(\int_0^{\pi/2} \sin^2 x \cos^6 x \,dx\). Find the integrals \[ \int \frac{dx}{\sqrt{1-x^2}}, \quad \int \frac{\sqrt{1-x^2}}{x} \,dx. \]
Prove that, if \[ u_n = \int_{-a}^a (a^2-x^2)^n \cos bx \,dx, \] \[ b^2 u_{n+2} - 2(n+2)(2n+3)u_{n+1} + 4(n+1)(n+2)a^2 u_n = 0, \] where \(n\) may be assumed to be positive.
Two equal light strings of length \(l\) are hung at their upper ends from two fixed points distant \(a\) apart in the same horizontal line, \(a\) being small compared with \(l\). Their lower ends are joined to one another and to a third equal string, from the lower end of which a small mass is suspended. The mass is drawn aside in the vertical plane containing the two fixed points through a distance \(x\) from the position of equilibrium. Shew that the time of a complete oscillation is \[ 4\sqrt{\frac{l}{g}} \left\{\sqrt{2}\cos^{-1}\frac{2}{3} + \sin^{-1}\frac{2}{\sqrt{7}}\right\}. \]
The two parabolas \[ y^2 = 4ax, \quad y^2=4bx, \] are drawn, where \(a\) and \(b\) are both positive; lines are then drawn through their foci perpendicular to their common axis. Find the area between the two parabolas which is contained between these lines; and shew that the volume generated by a revolution of this area round the axis is \(2\pi (a-b)^2(a+b)\).
A particle \(P\) is attracted towards each of four points \(A, B, C, D\) by forces equal to \(\mu_1 PA, \mu_2 PB, \mu_3 PC, \mu_4 PD\). Shew that it will rest in equilibrium only at the centre of gravity of four masses proportional to \(\mu_1, \mu_2, \mu_3, \mu_4\) placed at \(A, B, C, D\). Shew also that if a force \(Q\) be applied to the particle its position of equilibrium will be displaced from this position a distance \(Q/(\mu_1+\mu_2+\mu_3+\mu_4)\) in the direction of \(Q\).
A solid hemisphere rests with its base in an inclined position at an angle \(\theta\) to the horizontal, its curved surface resting on a horizontal plane (coefficient of friction \(\mu\)) and against a vertical plane (coefficient of friction \(\mu'\)). If the hemisphere is on the point of slipping shew that \[ \frac{c \sin\theta}{a} = \frac{\mu(1+\mu')}{1+\mu\mu'}, \] where \(a\) is the radius of the hemisphere and the centre of gravity is on the axis of symmetry at a distance \(c\) from the centre. If \(\mu=\mu'\) and \(c/a = 3/8\), shew that there is no position of limiting equilibrium for the hemisphere if \(5\mu > \sqrt{31}-4\).
\(OA\) is a slightly compressible vertical rod of height \(h\) and negligible mass (modulus of compressibility \(\mu\)) freely pivoted at its lowest point \(O\). \(AB\) is a slightly extensible cord of natural length \(l\) (modulus \(\lambda\)). \(B\) is a point in the horizontal plane through \(O\) distant \(a\) from \(O\) where \(a^2=l^2-h^2\). A horizontal force \(P\) is applied at \(A\) in the direction \(BO\). Shew that the horizontal and vertical components of the displacement of \(A\) are approximately (neglecting \(x^2\) and \(y^2\)) \[ x=P\left(\frac{h^3}{a^2\mu} + \frac{l^3}{a^2\lambda}\right), \quad y=\frac{Ph^2}{a\mu}. \]
A number of weights are to be hung on a light string so that the vertical lines drawn through them are at equal horizontal distances apart and so that the particles lie on a curve of the form \(a^2y=x^3\), where \(Ox\) is a horizontal axis and \(Oy\) an upward vertical axis (only positive values of \(x\) are to be considered). Shew that the weights of the particles must be in arithmetical progression.