H. E. SoPER, A. W. YouNa, B. M. Cavk, A. Lee, K. Pearson 355 
an equation identical with what we obtain by the "equal distribution of our 
ignorance." The same resvdt is also reached if m be only moderately large and 
n very big. In other words "the equal distribution of our ignorance," even if we 
really have some knowledge of the frequency distribution of p, will not lead us 
badly astray in the case of big samples. The matter is quite otherwise, however, 
in the case of small samples ; unless our knowledge is very limited {tn very large) 
we have no right whatever to take (Ixiv) as applying to such small samples. Indeed 
when m is fairly small p will not differ substantially from p, and the solution of 
(Ixiii) will differ widely from that of (Ixiv). We may consider these cases in 
succession. 
Case (i). Venj slight knowledge of p, or on the other hand a large sample. 
Here we are justified in using (Ixiv). We can attempt its solution in two 
different ways as in the case of the mode. 
Let p be the most likely value of p and let us write pr = p^^, then 
{I - p^)rl„ = pl^_.^, 
or {r"^ - Pi) L = Pi I n-1' 
Now let ^ first approximation to p^ and suppose p-^^ = p^^ + e, where 
e is small. Then 
(,2 _ _ 2p„2,) (7^' + = (^^2 + ,) (/'^_^ + ,7^;), 
where IJ ^ 
] 0 (cosh Z - po2)« • 
Hence remembering that 
n{l- Po') I'n+x = (2w - 1) po^7„' + (n - 1) 
we find e= 4' jr^ - Po^) - Po^I'n- ^ 
I'n-i + Po' in + 1) /„' - n {r^ - p,^) ' 
or 
.= (l-,o^) (.-po*)^„-Po'- 
(l-Po") - (»^-l)(^'-po*) +PoH('" + l)(l-Po*) - (2'H- l)(r2-po'')}^„ 
(Ixv), 
M — 2 
where (1 - p^") (n - 1) E,, = (2n - 3) + (Ixvi), 
and E, = I,'II\,_,. 
Now (Ixv) and (Ixvi) may be treated exactly like the corresponding equations 
for the determination of the mode. If n be moderately large, we may put 
E„ = En_-^ = E' 
in (Ixv), and if we know obtain the value of E', which value substituted for 
E„ in (Ixvi) gives us e and thus a new approximation. If we cannot guess a good 
value for p^^ (although p^^ = p^ is in this case usually sufficient) we can treat the 
