A. RiTcniE-ScoTT 97 
(mean in h) -^^ = ^6^ + ^r^ J^' r - ^r^ + ^t^ 2^3' - 
(mean in c) = .^t„ + / - .j^' 1^2 + 2^3' i^s - 
(mean in c?) = + ^0^ r + ^t.^ ^9^ + 1T3 1^3 + .... 
= + Uri - ^r^) (A' ' A) r + (.r,' + ,t.,) (,9/ + A) 
+ (2T3' - 1T3) (2^3' - A) r' + etc (7). 
lb will be noted that when one set of categories is symmetrical about the mean, 
i.e. when say 2^1' = i"^! tlie terms of odd degree in r vanish. This corresponds 
to the fact that symmetrical categories may be reversed without altering the 
numerical value of the marginal totals and their relation to the central frequency ; 
but such reversal will change the sign of r. 
§ 4. Standard Deviation of r by Enneachoric Method. 
We have now to determine the probable error of r found in this manner. 
Throughout what follows differentials will be used to indicate random sample 
variations, i.e. it is always supposed that the variations are small as compared to 
any quantity varied so that all the dn's are small, or all the n's are large quantities. 
m,t = N z (x, //, r) dx . dy =f{h,, kt, r) (8). 
J —GO -00 
Since the variations of the means and the standard deviations are, in this form 
of , involved in the variations of li^ and , we have 
of „ df df , 
dnist dn^ + ^ dkt + ^ dr (9). 
Evaluating the differential coefficients, 
This is the area of that portion of the dichotomic plane x = h;. which bounds 
the quadrant mjj. But the area of the whole dichotomic plane is 
r + « N 
N z ih„ y, r) dy = ^ - e -^-^V2 ^ j^h, (11), 
.' -00 \/ ^TT 
SO that if we write ^ = NH,A,t (12), 
where A., = -^r 1^^' « (13), 
the factor A^t will be that fraction of the whole dichotomic plane section, which 
bounds the quadrant m,t and will have no dimensions. The value of A^t may be 
taken directly from Sheppard's tables of the probability integral entering with 
Biometrika xii 7 
