46 Choice in the Distribution of Observations 
The maxima and minima are : 
For X = -0000 = ^ • 1 -728, 
„ x=± 4843 a„=°^,-. 2-116, 
„ x=± -8585 cr^ = ^^^ . 1-655, 
„ x=± 1-0000 CT, = ^^. 2-161, 
and this distribution which has -1708 x N observations at each end of the range 
may be considered satisfactory. 
(7) From (46) and (50) we find, for a, function of the fourth degree, 
'^=!)_o^ 225 (1 + a) (1 + 14a) 
~ iV^ • "64 1 + 15a 
x'=i Q.2 (2 3 
and 44=,^5(l+a) 
N ' ' ' (l + 10a^l + 15aj ' 
which are equal when 
D (1 + 14a) (1 + 10a) = 64 (1 + 12a) 
or 1260a2 - 552a - 55 = 0, 
that is when a = -5217564. 
The formula for ^ct'I, found from (62) and (56), is 
„ _(t2 I 3_(l + a) ^ 5 [2 + 3(l + a)(x'^-l)p 7 ( 1 + a)a:^[2 + 5(1+ 3a)(x^-l)]2 
iV^ l+3a "^4 l + 6a ^4 (l + 3a)(l + 10a) 
9 (l + a)[8 + 20(2 + 9a)(x2-l) + 35(H-6a)(a;2-l)2]2| 
64 (1 + 6a) (1 + 15a) 
For a = -5217564 it becomes 
a" 
.(63). 
al = ^{5-03367 - 19-72772x2 + 133-0171 lx« - 235-96817x« + 122-67868s8}. 
The maxima and minima are as follows 
For x = \ ^ a = 2-244 
„ x=±-3130 a, - . 2-041, 
cc=±-6844 a., = --°,,.2-575, 
„ x=±-9361 CT, = -^,. 1-856. 
a 
VN 
a 
We have again as for the function of the third degree brought a,j down below 
one of the maxima of 4a,,, although since has a maximum at a; = 0 the demand 
x=0 .r-=l 
that (jy = Uy is not so exacting as for ga^ which has a minimum at x = 0. 
