Solve the equations \begin{align*} (y-c)(z-b) &= a^2, \\ (z-a)(x-c) &= b^2, \\ (x-b)(y-a) &= c^2. \end{align*}
Express as partial fractions \(\displaystyle\frac{ay}{(y+a)^2(y-a)}\) and deduce the partial fractions for \(\displaystyle\frac{x(x^2-1)}{(x^2+x-1)^2(x^2-x-1)}\).
Solution: \begin{align*} && \frac{ay}{(y+a)^2(y-a)} &= \frac{A}{y+a} + \frac{B}{(y+a)^2}+\frac{C}{y-a} \\ \Rightarrow && ay &= A(y+a)(y-a)+B(y-a)+C(y+a)^2 \\ y=a: && a^2 &= 4Ca^2 \\ \Rightarrow && C &= \frac14 \\ y =-a: && -a^2&=-2Ba \\ \Rightarrow && B &= \frac{a}{2} \\ y = 0: && 0 &= -a^2A-aB+Ca^2 \\ && 0 &= -a^2A-\frac{a^2}{2}+\frac{a^2}{4} \\ \Rightarrow && A &= -\frac14 \\ \Rightarrow && \frac{ay}{(y+a)^2(y-a)} &= -\frac{1}{4(y+a)} + \frac{a}{2(y+a)^2}+\frac{1}{4(y-a)} \end{align*} If \(a = x\), and \(y = x^2-1\) then \begin{align*} && \frac{x(x^2-1)}{(x^2+x-1)^2(x^2-x-1)} &= \frac{x}{2(x^2+x-1)^2} + \frac{1}{4(x^2-x-1)} - \frac{1}{4(x^2+x-1)} \end{align*}
Two figures \(ABC..., A'B'C'...\) in the same plane are related in such a way that points correspond to points and straight lines to straight lines. If \(AA', BB', CC'\)... all pass through a fixed point \(O\) show that the meets of \(AB, A'B'; AC, A'C'; BC, B'C';...\) all lie on a fixed straight line. A circle is given in the plane of the paper. Show how to obtain from it, by means of a ruler and pencil only, a conic section and establish criteria to determine whether this conic section is an ellipse, parabola or hyperbola.
Obtain the formulae of transformation from trilinear co-ordinates \(\alpha,\beta,\gamma\) referred to a given triangle to cartesian co-ordinates \(x,y\) referred to given rectangular axes in its plane. By this means find the condition that the lines \[ l\alpha+m\beta+n\gamma=0, \quad \lambda\alpha+\mu\beta+\nu\gamma=0 \] may be perpendicular. Deduce the equation of the director circle of the conic \[ p\alpha^2+q\beta^2+r\gamma^2=0. \]
The equations of two circles in space are \begin{align*} 2x+2y-z=0, &\quad 5x^2+5y^2+8z^2-12yz+12zx-8xy=9, \\ 2x-y+2z=0, &\quad 5x^2+8y^2+5z^2+4yz-4zx-4xy-6x+12y+12z=0. \end{align*} Find whether these circles interlace or not.
Prove that the curvature \(\kappa\) of a twisted curve is given by \[ \kappa^2ds^4 = A^2+B^2+C^2, \] where \[ A = d^2ydz-d^2zdy, \quad B=d^2zdx-d^2xdz, \quad C=d^2xdy-d^2ydx. \] In what sense is this equation invariant? Find the curvature of the curve whose equations are \[ x=a\cos\theta, \quad y=b\sin\theta, \quad z=c\theta. \]
Explain what is meant by the differential \(du\) of a function \[ u=f(x,y,z). \] Account for the identities, with the usual notation, \[ dx=\Delta x, \quad d\Delta x=0, \] the variables \(x,y,z\) being independent. Prove that, if \(f\) is differentiable and \(x,y,z\) are themselves differentiable functions of any number of given variables, then \[ du=Adx+Bdy+Cdz, \] the coefficients \(A,B,C\) being identical with those which occur in the expression for \(du\) when the variables \(x,y,z\) are the original independent variables.
The function \(F(x,y)\) is continuous in \((x,y)\) in a neighbourhood of a certain point \((a,b)\) and \[ F(a,b)=0. \] Investigate conditions under which the equation \[ F(x,y)=0 \] determines, in some neighbourhood of \(a\), a function \(y=\phi(x)\) which reduces to \(b\) when \(x=a\). Find also conditions for \(\phi(x)\) to be
Explain what is meant by saying that the series \[ u_1+u_2+\dots+u_n+\dots \] is convergent. Prove that if this is so, then, as \(n\to\infty\), \[ u_n\to 0. \] Prove further that, if, for each \(n\), \[ u_{n-1}\ge u_n > 0, \] then the series cannot be convergent unless \[ nu_n\to 0. \] Construct a series of positive terms to show that, if the condition \(u_{n-1}\ge u_n\) does not hold, then this result may not be true. State what you consider to be the sum of the series \[ 0+0+\dots+0+\dots \] and give carefully the reasons which support your statement.
Explain what is meant by the uniform convergence of a series and give an example of a series which converges in \(0 < x\le 1\) and yet is not uniformly convergent in any interval \(0\le x \le \delta\), however small the positive number \(\delta\) may be. It is given that \[ f(x)=f_1(x)+f_2(x)+\dots+f_n(x)+\dots \] for all values of \(x\) in some neighbourhood of \(a\), and that \(f_1(x),f_2(x),\dots\) are differentiable in this neighbourhood. Prove that, if the series \[ f'_1(x)+f'_2(x)+\dots+f'_n(x)+\dots \] converges uniformly in some neighbourhood of \(a\), then \(f(x)\) is differentiable at \(a\) and \[ f'(a)=f'_1(a)+f'_2(a)+\dots+f'_n(a)+\dots. \]