Trace the curve \(y=e^{1/x}\). Find the inflexions and the asymptotes.
Prove the following formulae for the radius of curvature at any point of a plane curve \[ \text{(i) } r\frac{dr}{dp}, \quad \text{(ii) } \left\{u^2+\left(\frac{du}{d\theta}\right)^2\right\}^{3/2} / \left\{u^3\left(u+\frac{d^2u}{d\theta^2}\right)\right\} \quad (u=1/r). \] Prove that the distance between the origin and the centre of curvature at any point of \(r^n=a^n\cos n\theta\) is \[ \{a^{2n}+(n^2-1)r^{2n}\}^{1/2} / \{(n+1)r^{n-1}\}. \]
Prove that the area of one loop of the curve \(x^4-2xy a^2+a^2y^2=0\) is \(\frac{1}{6}a^2\).
Show that \((y-c)^2+\frac{1}{2}(x-c)^3=0\) is a family of solutions of \[ y+\frac{1}{2}p^3-(x+p^2)=0, \quad \text{where } p=\frac{dy}{dx}. \] Find the envelope of the family, and show that \(y=x\) is a cusp locus.
One of Sir Walter Scott's novels.
The Turkish Empire.
A league of Nations.
Is the study of Physical Science an essential part of a general education?
The application of Chemistry to the arts.
Shew how to reduce any number of co-planar forces to a force at a given point and a couple. Find expressions for the magnitude of the force and the moment of the couple. \(AB, A'B'\) are equal lines in the same plane, \(C\) and \(C'\) their middle points. Prove that forces represented by \(AA'\) and \(BB'\) are equivalent to a force represented by \(2CC'\) and a couple whose moment is \(2AC^2\) multiplied by the sine of the angle between \(AB\) and \(A'B'\).