2G8 
Peccavimiis f 
and accordingly 
rip + Bup rip + Bup 
where x,j is measured from the sampled population array mean. 
Now we desire to obtain the various moment-coefficients of BrUp, or mean (Bmpy, 
which for convenience may be written 
There are two ways at first sight of doing this : 
(i) We may expand {Tip + BiipY in terms of 8np/np and then take the mean 
values of products such as : 
(Bnp)- {8n,jpf {Bn,.^f {Sn^-pf" 
This was the process adopted in the original memoir. It is very laborious and 
the algebra so lengthy as to lead easily to slips. Still on the present occasion we 
went to terms of a high t)rder (0») and some of the results obtained will be so use- 
ful in other investigations on probable errors of frequency constants that it seems 
worth while placing them on record here. The fourth-order mean products in Sup 
and Suiji, may be added to those given on p. 245 of the original memoir. 
They are : 
I S {8np8,r,p) = (l - I) {l + 3 (l - |) (l - '^ff)]^ , ■ 
I t{S.pBn,,M,r,W.) = - 3 (l - I) '^A^ _ I) , 
I I (SV^'^V.) = ".P (l - I) {l + (l - I) [^n,p + ,1, (l - '^^)]| , 
^ i: (S)ip'8n,jp8n,yp) = ^1 - -^j n,jpn,j.j^ (^1 _ '-^^ (2 --^^ 
I S iSnpM^p) = n„p (1 - I) |l + 3 (1 - I) np (l - |)j 
(A). 
For the Jift/i-oix\er mean products, Miss Pairman also provided the following 
values* : 
^^{on ,,i,on,jp)-np,jnp,j'^L ^ ^ 
)]}■ 
_ 6 \ Vp^ r ripg Hp^' f 4>npg , 
NJ I ~N N [ N 
* These results are of course perfectly general, that is to say we can suppress p and suppose them 
the mean variation values of elements /i^, 11^1, «,^'/, Hq,', and 7iqi, of any frequency distribution. 
