1. Consider the equations \begin{eqnarray*} x_1 + x_2 + x_3 + & \cdots & +x_n = a_1, \\ x_1 y_1 + x_2 y_2 + x_3 y_3 + & \cdots & +x_n y_n = a_2,\\ x_1 y_1^2 + x_2y_2^2 + x_3 y^2_3 + & \cdots & +x_n y_n^2 = a_3, \\ x_1 y_1^3 + x_2 y_2^3 + x_3 y_3^3 + & \cdots & +x_n y_n^3 = a_4,\\ \ldots\quad\ldots \quad \ldots\quad\ldots\quad & \ldots & \quad\ldots \quad\ldots \\ x_1 y_1^{2n-1} + x_2 y_2^{2n-1} + x_3 y_3^{2n-1} + & \cdots & +x_n y_n^{2n-1} = a_{2n}, \end{eqnarray*} where $x_1, x_2, x_3, \ldots , x_n$ and $y_1, y_2, y_3, \ldots, y_n$ are $2n$ unknown quantities.

Now, let us take the expression

and expand it in ascending powers of $\theta$. Then we see that the expression is equal to But (1), when simplified, will have for its numerator an expression of the $(n-1)$th degree in $\theta$, and for its denominator an expression of the $n$th degree in $\theta$.

Thus we may suppose that

and so $$(1 + B_1 \theta + \cdots) (a_1 + a_2 \theta + \cdots) = A_1 + A_2 \theta + \cdots.$$

Equating the coefficients of like powers of $\theta$, we have \begin{eqnarray*} A_1 &=& a_1, \\ A_2 &=& a_2 + a_1 B_1, \\ A_3 &=& a_3 + a_2 B_1 + a_1 B_2, \\ A_n &=& a_n + a_{n-1} B_1 + a_{n-2} B_2 + \cdots + a_1 B_{n-1},\\ 0 &=& a_{n+1} + a_n B_1 + \cdots + a_1 B_n,\\ 0 &=& a_{n+2} + a_{n+1} B_1 + \cdots + a_2 B_n,\\ 0 &=& a_{n+3} + a_{n+2} B_1 + \cdots + a_3 B_n,\\ \vdots&&\quad\quad \quad \quad \quad\vdots \\ 0 &=& a_{2n} + a_{2n-1} B_1 + \cdots + a_n B_n. \end{eqnarray*} From these $B_1, B_2, \ldots B_n$ can easily be found, and since $A_1, A_2, \ldots, A_n$ depend upon these values they can also be found.

Now, splitting (3) into partial fractions in the form $$\frac{p_1}{1-q_1 \theta} + \frac{p_2}{1 - q_2 \theta} + \frac{p_3}{1 - q_3 \theta} + \cdots + \frac{p_n}{1-q_n \theta},$$ and comparing with (1), we see that \begin{eqnarray*} x_1 = p_1, & y_1 = q_1;\\ x_2 = p_2, & y_2 = q_2;\\ x_3 = p_3, & y_3 = q_3;\\ \ldots\quad &\ldots \end{eqnarray*}

As an example we may solve the equations: \begin{eqnarray*} x + y +z + u + v &=& 2, \\ px + qy + rz + su + tv &=& 3, \\ p^2x + q^2y + r^2z + s^2u + t^2v &=& 16,\\ p^3x + q^3y + r^3z + s^3u + t^3v &=& 31,\\ p^4x + q^4y + r^4z + s^4u + t^4v &=& 103,\\ p^5x + q^5y + r^5z + s^5u + t^5v &=& 235,\\ p^6x + q^6y + r^6z + s^6u + t^6v &=& 674, \\ p^7x + q^7y + r^7z + s^7u + t^7v &=& 1669, \\ p^8x + q^8y + r^8z + s^8u + t^8v &=& 4526,\\ p^9x + q^9y + r^9z + s^9u + t^9v &=& 11595, \end{eqnarray*} where $x, y, z, u, v, p, q, r, s, t$ are the unknowns. Proceeding as before, we have $$\frac{x}{1 - \theta p} + \frac{y}{1 - \theta q} + \frac{z}{1 - \theta r} + \frac{u}{ 1 - \theta s} + \frac{v}{1 - \theta t}$$ $$ = 2 + 3 \theta + 16\theta^2 + 31 \theta^3 + 103 \theta^4 + 235 \theta^5 + 674 \theta^6 + 1669 \theta^7 + 4526 \theta^8 + 11595 \theta^9 + \cdots $$ By the method of indeterminate coefficients, this can be shewn to be equal to $$ \frac{2 + \theta + 3 \theta^2 + 2 \theta^3 + \theta^4}{1 - \theta - 5 \theta^2 + \theta^3 + 3 \theta^4 - \theta^5}.$$ Splitting this into partial fractions, we get the values of the unknowns, as follows :

\begin{array}{l|l} x = - \frac{3}{5} , & p = -1\\ y = \frac{18 + \sqrt{5}}{10}, & q = \frac{3 + \sqrt{5}}{2}\\ z = \frac{18 - \sqrt{5}}{10}, & r = \frac{3-\sqrt{5}}{2}\\ u = - \frac{8 + \sqrt{5}}{2 \sqrt{5}}, & s= \frac{\sqrt{5}-1}{2},\\ v= \frac{8-\sqrt{5}}{2 \sqrt{5}}, & t= - \frac{\sqrt{5}+1}{2}. \end{array}