28 Choice in the Distribution of Observations 
(11) In Section I under (7) it was found that 
/ "'^''^'^ = a2 (^^ + 1) 
when the integration was taken over the places of observation. For the present 
distribution f {x) is \, ^ {x) constant and jifj {x) dx = N , hence the mean of „o-| in 
the range of observations is for a uniform continuous distribution 
^ {n + 1). 
For the grouped observations in Section IT we find by integration of the 
formulae for functions from the first to the sixth degree that 
.2 / 1 
1 a' 
^^aldx = ^ (n + 1) 
2n + 1 
IV. Uniform continuous distribution of obsermtions with constant standard 
deviation. Special formulae. 
(1) Let no'l — „_-i^al be indicated by S„, then the formulae (32) and (33) 
give us 
Sr. 
a" 
N 
^2 r; 
3a 
5 
(1 - 3a-2)2 
N'4: 
ct2 7 
N ■ 64 
a2 lla^ 
N ■ "64" 
(3 - 30x2 35,^4)2 
15 - 70,r2 + 63.«4)2 
Sr = 
13 
iV ■ 9 X 256 
from which we form „ct;, beginning with 
a" 
rr2 
(15 - 315x2 j_ 945.^4 _ 693a;G)2 
.(36), 
N 
-r2 
fl + 3x2) 
,a; = ^ (1 + 3x2 + 5 (1 _ 3,^2)2) = ^ . _ (i _ 2x2 + 5^4)^ 
and further in the same way 
., 1 
...(37). 
t2 25 
^al ^ • II (9 - 36x2 + 294x4 - 644x« + 441x8) 
= "^-^ . — (25 + 175x2 _ 1750x4 + 6510x«- 9555x8+ 4851x": 
iv 64 
= — -^(175 - 10.50.^2+ 17325x4 - 93660x« + 225225x8 + 
" N ' 256 
- 245322x" + 99099x12) ; 
