On Page 62 of my copy of Uspensky's Theory of Equations (chapter III, section 5, "Relations between Roots and Coefficients"), I've come across some notation that I'm not familiar with--to the extent that I'm not even sure how I'd google it.
The expression looks like this:
$$\left. \begin{array}{rl} P(x + b_{n+1}) = x^{n+1} &+ s_1 \\ \quad & + b_{n+1} \end{array} \right | \left. \begin{array}{rl} x^n & + s_2 \\ &+ b_{n+1}s_1 \end{array} \right | \left. \begin{array}{rl} x^{n+1} + . . . &+ s_i \\ & +b_{n+1}s_{i-1} \end{array} \right | \begin{array}{l} x^{n+1-i} + . . . + s_{n}b_{n+1} \\ \quad \end{array}$$
The polynomial in question is $P(x) = (x + b_1)(x + b_2) . . . (x + b_n)$
$s_1$ is the sum of $b_1, b_2, . . . b_n$
$s_2$ is the sum of the products taken two at a time (e.g. $b_{1}b_2$, $b_{1}b_3$, etc.)
$s_3$ is the sum of the products taken three at a time
...
$s_i$ is the sum of the products taken $i$ at a time
...
$s_n$ is the product of all $b_n$.