346 Distrihution of Correlation Coefficient in Small Samples 
To confirm this value of r we take as ths first approximation to r the value 
given in the method of the following section, i.e. r = -91344 giving p^^ = -548,064 
and 
fVi I 
n X •699,6259^„+i = (2w - 1) x -548,0640 + -^r- • 
Using (xlvii) we find £'2 = 1-103,9149 and hence 
= 1-822,4446, = 1-828,4768, 
E^ = 1-957,1738, E^ = 1-994,3056, 
leading to e = - -008,8172, 
and Po' = -539,2468, 
or r = -89875. 
It will be seen that our two methods of approaching the true value of r still 
differ to some extent, although probably serviceable enough for practical purposes. 
Accordingly we will now make a further approximation starting from r = -8980 
or = -5388, and we have 
E^ = 1-093,5399, E. = 1-783,0638, E^ = 1-792,1629, 
£5 = 1-918,2742, ^6 = 1-954,1949. 
These give e = - -000,4924, 
and consequently po" = -538,3036, 
with r = -89717, 
a value no doubt correct to four figures. 
It is clear that the process of finding the mode for n small is much more laborious 
than for n = 25 or over, because E^^ is not nearly E.^.^-^^. Actually the value given 
for E' by the simultaneous equation process from which we started is 
E' = 1-881,8787, 
which is only a rough approximation to the value E^ = 1-911,8858. That method 
must therefore be followed by further approximations when n is much smaller 
than 25. 
(5) Delerminalion of Onlinates and Mode hy Expansions. 
Approximate Expression for the Ordinates. We may proceed to expand the 
Eqn. (xxxvii) in powers of Ijn or l/(n — 1). This will involve a knowledge of the 
expansion of 
dz 
and can be achieved by a process to which Pearson drew attention in 1902*. 
Let 1 e-K^-^ + "4'^^ + ««'^« + <^«+...) (^iviii). 
cosh z — po'' 1 — po 
* Bioiiictrika, Vol. l. p. 39:5. 
