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

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?

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 $$[1,n]\times[1,n]$$.

• couldn't you correct this by changing the upper limit of summation? i've seen it done before by other people when squaring sums. Oct 11 '19 at 8:03
• @sqrtpapi2001: ooops, my bad, when the factor $2$ is pulled out, the triangular domain must be used. Now fixed.
– user65203
Oct 11 '19 at 8:07
• what is the "triangular domain"? That sounds interesting. Oct 11 '19 at 8:16
• @sqrtpapi2001: come on, check the shape of the domains.
– user65203
Oct 11 '19 at 8:21
• when you say "domain" i imagine you're talking about that cartesian product you have there that you're saying is the (ambiguous) range of all index of summation values for i and j. what makes it triangular? also what area of math is this coming from, i find that quite interesting. Oct 11 '19 at 8:27

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 $$\sum\limits_{i,j} x_ix_j$$

• And what about $S(x_1x_2)$ ?
– user65203
Oct 11 '19 at 7:30
• @YvesDaoust - edited Oct 11 '19 at 7:32
• Why do you think he opted to skip writing the indices? Is there a relevant technical reason? Oct 11 '19 at 7:34
• @sqrtpapi2001 It was the style of the time, and probably easier to typeset. I came across it when commenting on the preface of Fisher's 1925 Theory of Statistical Estimation, where he said "Given any series of proper fractions $P_1,P_2,\ldots,P_s$, such that $S(P_k) = 1$" to mean that they added up to $1$. Oct 11 '19 at 7:41
• I don't quite agree with your retranscription of $S(x_1x_2)$.
– user65203
Oct 11 '19 at 7:45