TO THE THEORY OF EVOLUTION. 
35 
and (up to the above degree of approximation) the volume of a “ tetrahedron ” on a 
££ sphere ” in hyperspace of four dimensions. In fact, the whole theory of hyperspace 
“ spherical trigonometry ” needs investigation in relation to the properties of multiple 
correlation. 
In our illustrations (viii.) and (ix.) will be found examples of the above formulae 
applied to important cases in triple and quadruple correlation in the theory 
of heredity. I consider that the formulae above given will cover numerous novel 
applications, for many of which greater simplicity will be introduced owing to the 
choice of special values for the h’s or for the correlation coefficients. 
(8.) Illustrations of the. New Methods. 
Illustration I. Inheritance of Coat-colour in Ilorses. —The following represents 
the distribution of sires and fillies in 1050 cases of thoroughbred racehorses, the 
grouping being made into all coat-colour classed as ££ bay and darker,” ££ chesnut and 
lighter ” :— 
Sires. 
Colour. 
Bay and 
darker. 
Chesnut and 
lighter. 
g 
Bay and darker . . . 
631 
125 
756 
pH 
Chesnut and lighter 
, 147 
147 
294 
778 
272 
1050 
a 
b 
CC -f- I) 
c 
cl 
c -f- d 
a + c 
b + d 
N 
i 
Then we require the correlation between sire and filly in the matter of coat-colour, 
and also the probable error of its determination. 
We have from (iv.) and (v.) 
«i = 
«2 - 
_ (ci + c) — (b + cl) 
N 
(a + b) — (c -{- d ) 
N 
= \f % j'f’dx = -481,905. 
= V 4 j N’dy = -440,000. 
Hence from the probability integral tables 
h = '64630, k = '58284. 
We have then: log HK = 1*037,3514 by (xvii.), 
F 2 
