Godfrey H. Thomson 
227 
It can now be shown that the methods which Professor Urban employed in 
comparing these two hypotheses are incorrect and inadequate. What these in- 
adequate methods are can best be shown by continuing the above example, which 
is taken at random from among Urban's material. 
We have already found the squares x"^ of the differences between theory and 
observation in the case of the normal integral, or as Ui'ban calls it the (/> (7) hypo- 
thesis. They are given in the table just above, and 
6' (,7;=) = -00455189. 
We now proceed to form the analogous quantity in the case of the arctan. hypothesis. 
Grams 
Observed 
P' 
X 
x^ 
84 
•0233 
•0795 
•0562 
•00315844 
88 
•0267 
•1086 
•0819 
•00G70761 
92 
•1167 
•1682 
•0515 
•00265225 
96 
•3567 
■3259 
+ 
•0308 
•00094864 
100 
■6100 
•6464 
•0364 
•00132496 
104 
•8833 
•8222 
+ 
•0611 
•00373321 
108 
•9300 
•8872 
•0428 
•00183184 
•02035695 = ,S' (.1-2) 
Urban now compares the ^ (7) hypothesis with the arctan. hypothesis by comparing 
•00455189 with •02035695, 
and deciding that as the former is smaller, therefore the ^ (7) hypothesis is superior. 
This procedure is firstly inaccurate and secondly inadequate. It is inaccurate 
because not S{x-) but S{x-[p'f/) should be compared, and it is inadequate because 
no idea is given whether the observed difference is significant or not. 
The former point deserves a little more examination, because it is another form 
of an error which Urban was the first to correct, in this same article. In the form 
of the Constant Process as it left the hands of G. E. Miiller, certain weights are to 
be used on the observation equations. These weights may be called Mtiller's 
weights. Urban pointed out, however, that they needed amendment, and published 
{loc. cit.) a table of weights to replace them. These weights differ from MiiUer's by 
the factor lj4<pq, which arises in Urban's treatment from an application of what he 
calls Bernoulli's Theorem. It is these very Bernoulli weights, Ijpq, which Urban 
himself has omitted in his above comparison of the (f> (7) and arctan. hypotheses. 
In order to discuss the inadequacy of his comparison we need a measure of the 
probable error of the quantity P used above. 
