Karl Pearson 
155 
We must therefore in order to find P enter the Tables of "Goodness of Fit" 
with = Xo^ ~ with n' = 7i — q + 1*. 
The reader who is famiHar with the theory of the multiple total and partial 
correlation coefficients will note how closely analogous the formulae (xxxiii) to 
(xxxv) are to results in that theory. The fundamental determinant of the 
correlation may be made to agree with A, if we merely write r^s' = cos {ss'). 
In fact both theories really reduce to the discussion of the formulae of spherical 
trigonometry in multiple space. 
(7) A simple illustration of the above formulae may be taken from the case 
of the distribution of mortality in two districts, where the problem is to ascertain 
the probability that the difference of mortality observed allowing for the frequency 
of the age groups could be due to random sampling. 
Let the population sampled be represented by 
(a) 
^3 
A 
L, 
L. 
Lu 
A 
A, 
^3 
As 
Au 
P 
where Dg are the dead, the living and As the exposed to risk in the sth age group, 
there being u such groups in the total population P, of whom in the given period 
A die and A survive. 
Let the districts be represented by 
d^ 
h 
h 
h 
H 
and 
(y) 
dl 
d,' 
h' 
W 
a,' 
respectively. 
Then the 
problem i 
p 
d; 
8' 
A' 
(Is 

««' 
1^' 
distributions could both be samples of the same first population. 
The general formula has been given by me| ; we have to evaluate 
'ffj_f±\'] 
V v' 
p^p'JP) 
* The tables are constructed for n' - 1 independent variables ; in our case there are n - q such 
variables, hence n' -\=n - q. 
t Biometrika, Vol. vni. p. 252. 
