244 "Goodness of FiV in Statistics and Physics 
where j,j8i is the first /3-coefficient of the pth array in the sampled population. 
Hence whatever be the nature of the frequency distribution in that population, 
if the number in the array be not too small, the distribution of the means of arrays 
will approach symmetry, i.e. ^B^ will be small*. 
The following values have been used in the previous investigations, some of 
which have and some of which have not, as far as I am aware, yet been publishedf . 
The remainder will be required in investigating j^B^, or the value of S {hm^YjX. 
Third Order Mean Products for Random Samples in the case of an indefinitely large 
Sampled Pojndation. 
A N V N )' 
A 
W X -''^''[^ NJV N)' 
S (8np8wgp8wg>^) 2?\j,ri,'j, / 
A = -~^N V-N)' 
ih) S{8m (8n,)^} ^Q^ 
A 
. S (8m^8n^,8n^) _ "w' /'i _ "^V^ _ ^ 
- S{(8m,)^8n,} _ o^,-,^ / _ «_x 
^"^^ A v^V Nj' 
//N S {8w^ (8n^p)^} _ / 2n„\ 
* Even if the array diverged considerably from the Gaussian value, i.e. ,,/3i = -2, say, then pBj^ for 
an array of even 10 only would be but -02 and the asymmetry very slight. 
t (a) and (g). Phil. Travs., Vol. 186, A, p. 347 (Pearson, 1894); (6) and (c), Biometrika, Vol. ix. p. 95 
(Soper, 1913); (d)-{m) probably here for the first time. 
