The convergence of series of positive terms.
Maxima and minima of functions of several variables.
Ruled surfaces.
Discuss the two methods (of harmonic analysis and of travelling waves) of representing at any time the transverse vibrations of tense strings, or the displacements in plane sound waves, after an arbitrary initial disturbance. In the case of the second method, consider reflection at a boundary.
Obtain Lagrange's equations of motion, considering, in addition to the usual case, the forms appropriate to impulsive forces and to small vibrations of an elastic system about a position of stable equilibrium.
Explain, with examples, the applications of `conformal representation' to hydrodynamics and electricity.
Describe the method of time determination by meridian observations of star transits, showing how our time system is based finally on the apparent motion of the sun. State the instrumental errors which have to be allowed for, and explain how this is done.
Discuss, with the aid of Cotes's theorem and Helmholtz's formula, the properties of a system of thin coaxial lenses, and indicate the use of the cardinal points in geometrical constructions.
The rational numbers \(\frac{p}{q}\) and \(\frac{r}{s}\) are such that \(p, q, r, s\) are positive integers and \(ps-qr=1\): shew that no rational number in similar form can be intermediate in value between them unless its denominator exceeds both \(q\) and \(s\). Shew also that the rational number intermediate between them, which is nearest to \(\frac{r}{s}\) and of which the denominator does not exceed \(N\) is \(\frac{p+nr}{q+ns}\) where \(n\) is the integral quotient of \((N-q)\) by \(s\).
The quadratic equation \(x^2+2bx+c\), where \(b^2>c\), has real roots \(x_1, x_2\): form the equation of which the roots are \(x_1^2-a^2, x_2^2-a^2\) and shew that \(x_1\) and \(x_2\) are both outside the interval \(-a\) to \(+a\) provided \((c+a^2)^2 > 4a^2b^2 > 2a^2(c+a^2)\). Determine the conditions that the roots of the biquadratic \[ x^4+1+2p(x^3+x)+qx^2=0 \] may be all real and unequal.