Old expected value or summation notation from the early 20th century

Question

I've tried looking everywhere online for this but I have turned up absolutely nothing.

In William Gosset's original paper "The Probable Error of a Mean" where he constructs his now famous t-distribution, there's this notation for what seems to be a summation. The pattern of the usage of the notation indicates that he may be writing the identity that in modern terms can be written $Var[X] = E[X^2] - E[X]^2$. Past that point, the notation leaves me grasping at straws. You can see it below.

All right, so what's up with this? I'm guessing this is some very archaic math form for summation - I mean it just has to be. Past the second equal sign, I'm just confused, again mainly because I don't recognize his notation. Why doesn't he index his variables of summation with a dummy variable (if that's even what he's trying to do?) And what's with the lack of comma separating the sample values, should that be expected from these older papers?

Here's a link to the paper in its entirety.

Answer 1

Try reading $S(x_1)$ as $\sum\limits_i x_i$, then $S(x_1^2)$ as $\sum\limits_i x_i^2$ and $S(x_1 x_2)$ as $\sum\limits_{i,j} x_ix_j$

Answer 2

In more familiar terms, by the formula for the square of a sum,

$$\frac1n\sum_{i=1}^n x_i^2-\left(\frac1n\sum_{i=1}^n x_i\right)^2=\frac1n\sum_{i=1}^n x_i^2-\frac1{n^2}\sum_{i=1}^n x_i^2-\frac2{n^2}\sum_{i=1}^n\sum_{j=i+1}^n x_ix_j.$$

The notation $S(x_1x_2)$ is questionable, because it doesn't clearly shows that the summation indexes do not cover the whole range $[1,n]\times[1,n]$.