How to calculate the Doolittle Skill Score? A statistic called the Doolittle Skill Score (DSS) is sometimes calculated in relation to the deterministic forecasting of binary events. Hogan and Mason (2012) p35-36 provides the following instruction on two different ways of calculating the DSS:

I call one method the 'abcd' method and the other the 'hfs' method. They're both meant to be equivalent. The methods must be understood with respect to the following 2x2 contingency table:

I've written a MATLAB script with example data here. It seems to me that the two methods do not give the same results.
Are these two calculation methods inconsistent with one another? If so, is there some way I can easily tell which is wrong? If the hfs version is wrong, how can I calculate the DSS using the hfs method?
One possibility to find the real DSS would be to look at the original article. Unfortunately it is from 1885 and I can't find it. In any case in this research area the terminology often diverges from what was intended by the originator.
Doolittle, M. H. (1885). The verification of predictions. Bulletin of the Philosophical Society of Washington, 7, 122-127. Chicago
Hogan, R. J., & Mason, I. B. (2012). Deterministic forecasts of binary events. Forecast Verification: A Practitioner's Guide in Atmospheric Science, Second Edition, 31-59.
 A: This is unhelpful, but I used Mathematica (https://sourceforge.net/p/bcapps/bcapps/ci/master/tree/STACK/bc-doolittle.m) to simplify DSS_hfs and got:
$\frac{a d-b c}{\sqrt{(a+b) (c+d) (a+b+c+d)^2}}$
which is different from the denominator in DSS_abcd.
I'm not sure why the definitions are different, but they do appear to be.
A: Armistead (2016) provides the solution, which is that the score - which he calls Doolittle’s $i_1$ - should in fact be rendered as
$\frac{(ad-bc)^2}{(a+b)(a+c)(b+d)(c+d)}$
Armistead explains on p.65 that

The equations from Doolittle’s era display an obsolete notation, and
are occasionally misstated. Hogan and Mason (2012, table 3.3, p. 36,
and p. 52) represent Doolittle’s $i_1$ as
$\frac{(ad-bc)}{\sqrt{(a+b)(a+c)(b+d)(c+d)}}$, which yields the square
root of Doolittle’s $i_1$ instead of the true value. This appears to
be a substantive misunderstanding of Doolittle’s measure; the authors
also incorrectly represent $i_1$ as being the equivalent of
$\sqrt{\frac{\chi^2}{n}}$ instead of $\frac{\chi^2}{n}$.

Armistead, T. W. (2016). Misunderstood and Unattributed: Revisiting MH Doolittle's Measures of Association, With a Note on Bayes’ Theorem. The American Statistician, 70(1), 63-73.
