Meaning of capital $S$ in Student's "The Probable Error of a Mean" I'm reading through Student's "The Probable Error of a Mean" from 1908 and I'm not sure what notation he is using. He says that if $s$ is the standard deviation from a sample $x_1 x_2 \ldots x_n$ then $s^2 = \frac{S(x_1^2)}{n}-\left( \frac{S(x_1)}{n} \right)^2 = \frac{S(x_1^2)}{n} - \frac{S(x_1^2)}{n^2} - \frac{2S(x_1x_2)}{n^2}$. I'm not sure what capital $S$ represents here. I'd suspect it has something to do with sample standard deviation, but then I don't know why the equation only references $x_1$ and $x_2$ when there are $n$ variables.

$S$ is meant here as summation. $S(x_1^2) $ means the sum of squares of the sample observations initiating from $x_1^2.$ Similarly, in the same vein, with $S(x_1x_2). $
Now, is this universally used?
Well, I am not the authority here but one thing I can assert is that this is an unusual usage and quite arcane.
However, with that being said, usage of $S$ for summation is not even that of a rarity. For example, in $\rm [I], $ the author in order to distinguish the sample from population used $\sf S$ reserving $\sum$ for population:
\begin{align}\textrm{summation over the units}&~~~\mathsf S_{i=1}^n\\\textrm{summation over cross-products}&~~~\mathsf S_{i=1}^n\mathsf S_{j\ne i}^n.\end{align}
I'm not sure when the usage started (that's for history sleuths to check) but it is certainly not in tune today.

Reference:
$\rm [I]$ Sampling Theory and Methods, M. N. Murthy, Statistical Publishing Society, $1967,$ sec. $2.14, $ p. $51.$
