KiRSTiNE Smith 
67 
Table II (p. 50) the maximiun . 1-862. Hence when //.2 and are the moment 
coefficient of this 0 [x) the maximum is found from 
ct; = ^ (1 + 2aju.2 + ct^ix^) . 1-862. 
We find fi^ = -6270, fi^ = -5524 and ^ = 1 + l-2540a + -5524a2. 
The actual distribution is hence 
together with the clusters 
at — 1 and 1. 
1 + l-2540a+ -5524a2 
•22026 (l + a)2 
1 + l-2540a+ -5524a2 
TABLE IV. 
■ N, 
a 
Maximum of 
s/N , 
dy - — tor 
(J 
the best 
distribution 
Maximum of cr,,^^ 
for distribution 
N 
with <p{x) = — 
JN 
Maximum of ay- — for 
(7 
distribution with 
<t)(x) = c and chisters 
at ±1 
0 
1-732 
3-000 
1-862 
1 
3-000 
4-099 
3-120 
2 
4-359 
5-310 
4-453 
3 
5-745 
6-573 
5-810 
4 
7-135 
7-861 
7-178 
5 
8-522 
9-165 
8-551 
The difJerence between the first and second maxima taken as a proportion of the 
first varies from 79 per cent, at a = 0 to 8 per cent, at a = 5, while the difference 
between the first and the third maxima varies from 8 per cent, at a = 0 to 0-4 
per cent, at a = 5. The continuous distribution with clusters is therefore 
especially useful for smaller a. 
For a = 4 we find from (82) v = -9816 and for a = 5, w = -9700, both of these 
values of v are so close to 1 that if instead of using them we take the observations 
at 1 and — 1 and let the numbers of the three groups of observations be jiroportional 
to the squared standard deviations we get the maxima 7-141 and 8-544 which only 
differ quite insignificantly from the corresponding values of Table IV. 
(7) For a function of the first degree, of which the standard deviation of the 
observations is o- (1 + ax), where 0 5 a < 1, we have, according to (66) and (67), 
1 + 2aju,j + a?ix.2, , 
o:, 
-{ju,2 — 2ju,ja; + x^\ 
.(85). 
For 1x1 = — the maximum of this function is at a; = 1 , and for = at — 1 . 
As the maximum of {^i^. — 2/XiX + x"^) has the same value in both cases it is clear 
