44 
Choice in the Distribution of Observations 
For this value Tr-^^r^ — ^ . which is the ratio between the number of observations 
2 (1 + a) 
at one end of the range and the total number of observations, is -2202562. 
As oo-; = -ydl + S.2, we find, from (59) and (54), 
- ^71 + 3(1 + ") ^2 , 5 [^±^ + a)_{^W\ . 
2<^.-^(^l+ 1+3^ ^ +4 1 + 0^ ; (bU), 
for a = -7873500 the curve is 
= ^ {3-46837 - 6-27862x2 + 6-27862,t*}, 
which has minima at a; = zt ^77:- 
The extreme values in the range of observations are therefore 
a, = ^.1-8624 fora.= |0^^ 
and a„ = ^. 1-3779 for x = ± -70711. 
(4) For a f unction of the third degree we have, from (46). 
■^=0_ct2 9(l + a)(l + 5a) 
andfrom (50). "^^^^2(1 +a) + y^-} (61)- 
Hence the condition that they are equal is 
9(1 + 5a) (1 + 10a) = 32 (2 + 15a) 
or 90a2 - 69a - 11 = 0, 
with one positive root a = -9021461. 
From (60) and (55) we find 
_ ^7 1 . 3_(1 + a ) ^2 , 5 [2 + 3(l+a) jx^ - 1)^ 
^''""iVV l+3a "^4 l + 6a 
7 (1 + a) [2 + 5(l + 3a)(a;2_ 1)]2 
.(62), 
4 (1 + 3a) (1 + 10a) 
which for a = -9021461 becomes 
3(tI==~ {3-67775 + 17-78799x2 - 48-56651x4 + 30-77852x«}. 
Besides the minimum for x = 0 this curve has other minima for x2 = -815820 
and maxima for x^ = -2361366. 
The maxima and minima are as follows : 
Forx^l^^ = 1.9177, 
„ X = ± -48594 Gy = ~„ . 2-3612, 
„ x=± -90323 (T,, = ^. 1-6055. 
