H. E. SoPER, A. W. Young, B. M. Cave, A. Lee, K. Pearson 351 
Starting from the equation (xxxix) 
{n - 4) pH,_^ = (p2 - p^) - 1) 7„, 
dz 
where Z„ = ^ j^rr — - and p = or, r being the modal value of r, we may 
" j 0 (cosh Z - |02)«-1 1^ 1^ ' b 'J 
expand /„ and 7„„j^ in terms of powers of p^, the coefficients involving 
TT 
We find at once 
[n — l)n ^ 
{n - 4) p [q„_^ +{n-l) p^q,, + - ^ P^ln+i 
{n — 1) w (n + 1 
= ip^ - P*) {n - 1) U„ + ^^P^^n+l + ^ ^ 
(n -I- 21 ^ 
P^qn+:i + 
1.2.3 + ) 
n {n +l){n + 2) „g 
1.2.3 
Rearranging and substituting pr for and noting that 
m — 1 
= ^^^^ ?m-l > 
we have 
p{n-l)q„^ rqn-, (n - i - p^n - l)^) + l{n - 1) r^pq, (2 (n - 3) - p^.^s) 
+ i^Ll^ll^ r3p2^„^^, (3 (^^ - 2) - _,_ i)2) + etc (Iviii), 
where the form of the successive terms is sufficiently obvious, and the series con- 
verges rapidly if p be small. 
For the particular case in which we have chiefly used this equation to determine 
r, namely samples of three, r corresponds to an antimode and the ecj^uation is for 
w = 3 : 
2p% = - rq^ (1 + 4p2) _ Pq,9p' + Pq,p^ (3 - IQp^) 
+ r'q.p^ (8 - 25yc2) + Pq^p* (15 - 36p2) 
+ 7-^q^p^ (24 - 49/32) + Pqsp'' (35 - 64p2) 
+ etc (hx). 
An equation which led to r with singular accuracy and comparative ease for small 
values of p by aid of Table X for q,^ . 
(7) Tables and Models. 
Table A (p. 379) gives the values of the mean, of the mode, of the standard devia- 
tion, of and and thus of the skewness of the frequency distributions of r. It 
will be seen that long after we have reached the Hmit of what are usually treated 
as small samples, the skewness of the distribution of r is very considerable. The 
% 
