194 
Miscellanea 
Lastly : 
= 2^,.(<fe^</>,^^)+f + 1 (xxxii) 
Then substituting from (xxix), (xxx), (xxxi) and (xxxii) in (xxvi) we have : 
+ 2iV^,,(^</,p2,^,^l^^J (xxxiii) 
When the contingencies, mean and mean squared, approach zero, the terms of the third, 
fourth and fifth orders may be neglected as compared with that of the second order and we find 
V=^' %=7l (^^^^^^ 
But if C be the coefiicient of mean squared contingency : 
(T. , 
and accordingly ac = — , = ( 1 - C^) <t, (xxxv) 
Hence the probable error of C 
= -67449 (1-C2)^ cr^, 
and in the particular case of no contingency 
•67449, , . . 
= — ^ by (xxxiv). 
Hence unless a coefficient of mean squared contingency be two or three times this value, we have 
no evidence that the quantities under discussion can be considered as contingent on each 
other. 
The general expression for cr^ in (xxxiii) can be dealt with in several ways. It might be 
thought that being of changing sign, the cubic terms as well as those of the fifth order 
in -^pg would be small ; but this is not our experience in actual application. Terms will occur 
in which m^, is very large as compared with UpnJN owing to the existence of a few isolated 
units in outlying compartments, and it by no means follows that the second term is less than 
the first, or the sixth term less than the third. We have not succeeded in getting any 
