ON A FORMULA FOR THE PRODUCT-MOMENT COEFFICIENT 
OF ANY ORDER OF A NORMAL FREQUENCY DISTRIBUTION 
IN ANY NUMBER OF VARIABLES. 
By L. ISSEELIS, D.Sc. 
1. In Biomcfrilri, Vol. XI, Part III, I have shown that for a normal frequency 
distribution in four variables, if 
Vxvzt = SSSS {n^y,t xyz(}IN 
X V z t 
denotes the product-moment coefficient of the distribution about the means of the 
four variables and q,j.y^f is the reduced moment, i.e. 
^xyzt ~ Vxyztl^x^v^z^ti 
then q^y,t = ?v^r,, + fy^r^t + (1). 
In this result any two or more variables may be made identical leading to a 
variety of results for moment coefficients of distributions containing fewer than 
four variables but of total order four, for example identifying t with x we obtain 
Qx-yz ~ *i/z ~l~ ^'^xy'^xz ■••^ (2); 
and putting y = z = t = x we find q.^i = 3; of course q„j = r^y and q^i is merely jSg. 
I suggested that (1) was probably capable of generalisation, and I now propose 
to prove a general theorem which gives immediately the value of the mixed moment 
coefficient of any order in each variable for a normal frequency distribution in any 
niimber of variables. 
2. Consider a normal distribution, total population N. Let Ni2...n denote the 
frequency of the group in which the characters differ hj x^, X2, ... x„ from the mean 
values for the whole population and let 
Pii.2i^... ,A = ^ (^i2...«a"i'-a^2'-' ■■• or,/'^)/N (3), 
denote the moment coefficient of the most general kind about the mean values of 
the characters. The corresponding reduced moment will be 
U2l^...J''==Pli'J^...nd<^^''<^2^-'^n" (4). 
Then for normal distrihuiions, 
itn\)Qodd, 5'i2...n = 0 (5), 
and if n be even, ^12... „ = S ... r;,^) (6), 
where the summation on the right-hand side extends to every possible selection of 
n/2 pairs ah, cd, ... hk, that can be formed out of the n suffixes 1, 2, 3, ... w; equa- 
tion (1) is thus a particular case of (6). 
Equation (6) is the theorem it is proposed to prove. The value of qii,2h...Jn 
is at once found for given numerical values of the indices l^, l^, ... In writing 
down (5) for + + ••• + variables and identifying the values of ?i of them with 
that of the first and so on. 
