H. E. SoPER, A. W. Young, B. M. Cave, A. Lee, K. Pearson 357 
Thus the divergence begins as early as the term in — 1)^ and (Ixvii) can 
only be trusted for rough approximations to p. 
Illustralion. Suppose r -6, what is the "most likely" value of p, on the 
assumption of equal distribution of ignorance ? Let n = 25, then we find from (Ixviii) 
P = -59194, 
while (Ixvii) gives -59182, an agreement adequate for most statistical purposes. 
If n = 5, and r = -6, we find 
^ = -55058 from (Ixviii), 
= -56695 from (Ixvii). 
There is now considerable divergence in the two methods and another approxi- 
mation is desirable. Let us take p^^ = pr = -33035, then to find we have 
n-2 
(w - 1) X •890,8689£^„ = (2n - 3) x -33035 + 
En-l 
will be given by (xlvii) and equals -885,5939, whence we determine 
E., = 1-189,9819, E^ = 1-246,8946. E^ = 1-324,1082, 
and accordingly from (Ixv) 
e = -001,3081, 
rp = + e = -331,8581, 
and p = -553,097, a value not far removed from that found by the first approxi- 
mation. We conclude that even when n is small, quite good results will be found 
from (Ixviii) and that it is probably better to use this rather than (Ixvii) in such 
cases as the starting point for a second approximation. 
Case (ii). Close d priori Knowledge of p. 
We will now suppose m small, so that the first approximation to p may be taken 
as p. We substitute in (Ixiii) p = p + ifi and we find, neglecting squares of ijj, 
0 _ 
ni {n - 1) ^ + V ^^"-^ + ~ ''"^^"^ = (1 - r - 2p^) (/•/„ + nri/-/„+i) 
where 
This leads us to 
dz 
.' 0 (cosh z — pr)"-! ' 
(1 -p'^) rl„ - pl„_^ 
^"-^ + 1)) + ^ ('^ + 1) - (1 - P') nrH,^, 
whence remembering that 
n{l - {rpf) - (2n - 1) rp/„ + {n - 1) /„_i. 
