156 On the Theories of Multiple and Partial Contingency 
where /; must take the value of every frequency in the cells of (j8), // the corre- 
sponding cell frequency for (y), and Ff that for (a). We have to seek for P under 
the argument n' = 1u in our tables, if we have no restrictions on our variates. 
Actually we have such restrictions, for we are going to seek the partial when we 
suppose the age groups in each sample to be constant. Now 
+ Zj. = and d/ + 1/ = a J . 
Thus it follows that 
— ^ + ^- -^ = ^ 1 (xxxvii). 
p p p p p p 
We have accordingly u equations of condition or n' as argument will be 
reduced to m + 1. Now we take 
^d^ dj-^ 
pp \p p 
p + p' Vdjp ' 
Zo Zq ^ 
pp \p P_ 
P + p' VL.jP 
Thus x' = S,-{Xs') + S,-{Ys'). 
with u conditions of form 
or in the prepared form 
_ _ (^_^\ 
V p 
Further, all the cosines like cos (ss') are zero, for no equations of condition 
involve the same variates*. Thus 
K^=Ki'+K,^+ ..:+KJ 
pp' ^ P fa. aj\^ . .... 
■= , S.^-j- — (xxxviu). 
p + p' ^ As\p p J 
Accordingly 
p + p D,\p pj L,\p pi ^ A,\p pi) 
I I ' 
We shall now substitute from the relation (xxxvii), getting rid of - — 
/' & o p P 
* This will necessarily be true if our equations of condition refer to parallel rows or columns, not if 
they refer to certain rows and columns. 
/t2 = 
