Andrew W. Young and Karl Pearson 219 
But this is the Mean {^'n^')^, where S'Ws' is measured from the mean of n^' in the 
case where Bfig is fixed and we must reduce to the general mean, i.e. where Sn^ is 
not given, to obtain the mean value of (Sn^)^ for constant Sn^. This is done by 
adding the square of the difference between these means of «v, namely • 
We thus obtain that the Mean {8n,-)^ for constant Sn^ 
iV - ?ls - 8«s - 1\ /AT N fi '"'s' ^ , ri/{8ns)^ 
= 1 
M - Ms - 1 / N ~n,\ N - 71 J {N - «,) 
^2 
N _ ^\ _ ^ _ 2iV\ N8n, (Sn,)^ ] MHi,. {N - - n,.) 
fi^.'^ (8m.,)^ 
when we substitute MiiJN for m^. 
Thus we have to evaluate 
/2 xo,2 .0 /s n2 ^ 2iV\ iV Mean (8n,)3 
Mean {Sn,)^Sn,r = [mV ~ m) ^^""'^ ^^''^^ ~V~m)m "IV"^ 
iV^ Mean(8n .)^| M^», {N - - n,) ^ 3 
- M'^ iN - ,1.)^ I _ ^^^^ _ + ^-'^ (^"^) • 
Substituting from (a), (c) and (e), we find 
Mean (8«,)2 (S^^,)2 = ^ _ M _ j ^ ^ 
/ 2iV\ M(iV-2»,)(iV-7l,-7l,-) ^ {N - n, - n,) 
iV2 (iV - - 1 j N i^j^iN -n,)-lj 
N - Ti., - , o "sis' I ^ 
(iV - 7/,) (^-^ {N - n,) - 1 j 
This expression must be symmetrical in s and s' and this will be the case only 
. . M _ 
if the quantities ]y (-^ ~ '>h) — 1 and N — v,. in the denominator cancel with 
factors in the numerator. By taking the two terms in ^4 together we get rid of 
the N — ri^ factor and after a laborious expansion in substituting the values of the 
x's we reduce the whole expression to the comparatively simple form 
This agrees with the value obtained by Isserlis from the differential equation to 
the hypergeometric series and thus confirms his result obtained by a totally 
different procedure*. 
* Proc. Roy. Soe. Vol. xcii. p. 28. Our notations are different. 
