376 
Miseellmiea 
It is unnecessary to form the fifth sum in detail cas the total of the fourth gives the only 
term required. The totals check the work throughout. The rough moments (not reduced 
to unit frequency) from the equations given above are 978, 2318, 6612 and 21458 which agree 
with the result we should obtain by direct calculation. 
If the rough moments about the mean are required the following is a convenient form : 
V3 = 6.S'4/^, -3vo{d+\)-d{\-\-d)(2 + d), 
i'4 = 24,Sy.S'i-2^3{l+2(c;+l)}-.'3 {6(l+t^)(2 + c^)-l}-c^(l+c^)(2 + (;)(3 + rf), 
where d is the distance between the mean and the point used for calculating the untransferred 
rough moments. Obviously d^S.jSi- 
I find that if there are few terms to be dealt with the direct method is quicker but for 
longer series the above has a distinct advantage. The summation method is really due to 
Mr G. F. Hardy, but though he has remarked that it is practically the same as the method 
of moments, I have not traced any previous note of the actual connection between the two 
methods as statistical processes, though Professor Pearson tells me he believes the summation 
method of reaching moments is given by Coradi in a paper on the Abdaidc-Abanowitz integrator, 
and J. Massau, " Mumoire sur I'integration graphique et ses applications," Paris, 1887, gives 
the connection between successive integrations and the moments. I understand that the iiitegraph 
does not give very satisfactory results for the higher moments, but though the method is 
like the one with which we are dealing, the latter is clearly not open to the same objection. 
The method can be extended to enable us to deal with correlation tables. In order to find 
the coefficient of correlation we require the means and standard deviations of the a^s and ys 
and the .ry-moment. The mean and s. d.'s can be found by treating the totals of the ^'-columns 
or ,y-rows in the same way as we did the entries in Table II. but a quicker method can be 
devised, which however will be easier to explain with the help of an example. 
TABLE III. Correlation Table. 
X 
Totals 
1 
'2 
3 
1 
2 
6 
10 
20 
3 
41 
2 
1 
5 
30 
9 
2 
47 
y 
3 
9 
28 
30 
7 
74 
k 
1 
11 
16 
10 
38 
Totals 
4 
20 
79 
75 
22 
200 
We now obtain from this table another (Table IV.) in the same form giving the y-sum 
of it by summing each column continuously, and then obtain Table V. by summing Table IV. 
across continuously. 
TABLE IV. y-sum of Table III. 
X 
Totals 
1 
2 
3 
k 
5 
1 
4 
20 
79 
75 
22 
200 
y 
2 
2 
14 
69 
55 
19 
159 
3 
1 
9 
39 
46 
17 
112 
4 
1 
11 
16 
10 
38 
Totals 
8 
43 
198 
192 
68 
509 
