KiKSTiNE Smith 
49 
For we get, from (64) and (58), 
, g^f 3(l+a ) 5 [2 + 3(l+a)(x^-l)P 
6t^.-^|i+ 1^3^ * + 4 l + 6a 
7 (1 + a) a;2 [2 + 5 (1 + 3a) (*2 _ 1)]2 
+ 
4 (1 + 3a) (1 + 10a) 
+ 6-4 (1+ 6:1)11 + 15,) P + (2 + Oa) - 1 ) + 35 ( I + 6a) (x' - 1)']» 
+ M (l + lo'»)'ir+21») [« + ^8 + - 1) + ^3 (1 + 10a) - 1)^]^ 
+ 2l( l + 15<.Hl + 28, )t"^+"»'^ + """-'-" 
+ 126 (3 + 40a) (x^ - 1)2 + 231 (1 + 15a) (a;^ - 1)3]2| , 
which for a = -2048019 becomes 
^al = ~ {5-58984 - 33-14234a;2 + 504-4523x4 _ 2512-673x« + 5524-186a;8 + 
- 5452-650a;io + 1974-020a;i2}. 
The maxima and minima are : 
Atx= 0 CT,^ = . 2-364, 
„ a;= ± -2216 a„=_^. 2-216, 
„x=± -4826 = ^ - 2-515, 
„«=± -6194 C7,^=_^*^. 2-427, 
„ x=± -8445 a„ = - f,, . 3-149, 
„x^± -9615 CT,^ = -^.2-485, 
„ a; =± 1-0000 CT^=^^. 3-128. 
It thus appears that this distribution which has -08499 x N observations at 
each end of the range fulfils our demand that Uy shall be apj^roximately equal 
to the greatest of the maxima. 
(11) We bring together our final results in the following table. It gives the 
distribution of observations, the maximum of Oy within the range, the value of 
Vn + 1 or the lowest maximum of — possible, which can only be obtained 
a 
by distributing the observations of the function of the nth degree into {n + 1) 
a, VN 
groups, and the value of n + 1 which is the maximum of — — for a uniform 
a 
distribution. 
Biometfika xii 4 
