W. Palin Elderton 
157 
addition and after adding log to the former, the resulting figures were reduced 
from ten to eight places of decimals so as to avoid the error that would arise from 
the accumulation of small differences in the eleventh place in the value of log e""i 
These tables were carefully checked by addition and by examining every tenth 
value in the continuous work. The table of colog ( = log 
' n {n — 2){n — 4),.. 
was originally calculated to ten places. The only other auxiliary table required 
was for /y^— j <^%, and these values were calculated to seven places of 
2 
decimals by second differences from a table of values of — = I e~^'dt*. It was 
VTr.'o 
quite safe to omit third differences. The values of P were then calculated from 
formulae (i) and (ii) given above. In making the table, to find )(^\ a column of 
slog;i^^ was first set up, and then by means of four moveable slips of paper (two 
for n even and two for n odd) a second column calculated giving the sum of 
s log %^ colog (2s + 1 ) and log (^/y^^ • These figures were checked by addition. 
The use of slips with colog (vi) written on them saved a very large amount of 
copying. The antilogarithms of the items of the second column were then put in 
a third column and the values of a/- [ e~''^^ having been written at the top 
of it, the figures given in Table I. were found by continuous addition. The values 
for n' even were calculated in like manner. The numbers obtained were tested 
when possible against those originally published in the Phil. Mag. and against a 
few additional values calculated by Miss M. A. Lewenz. The work was of course 
checked at every stage, but when the table was completed the second differences 
in each column were examined and found to run smoothly. The like method of 
differences was appealed to in the case of discrepancies between the short table 
and the present table, which were not due to the approximate Vidue taken for the 
probability integral. It is hoped that the table as it now stands is substantially 
free from error. 
In using the present method of testing goodness of fit it is essential to bear in 
mind a warning given in the paper in the Phil. Mag. referred to above (footnote, 
p. 164): "A theoretical probability curve without limited range will never at the 
extreme tails exactly fit observation. The difficulty is obvious where the observa- 
tions go by units and the theory by fractions. We ought to take our final theoretical 
groups to cover as much of the tail area as amounts to at least a unit of frequency 
in such cases." 
Further we ought to be careful to read the corresponding areas of the frequency 
curve and not merely the mid-ordinates, when we have not a great number of 
groups, or when, although the groups are numerous, the frequency is very skew. 
* The Table in Galloway's Treatise on Probabilities was the one actually used. 
