Karl Pearson 
237 
triangle, then 23P1, 31P2} 12P3 the cosines of the angles, and i?2.3iJ -^1.23; -^3.12 
are the cosines of the perpendiculars from the angles on the opposite sides. If 
a, b, c are the sides, A, B, C the opposite angles, pa, pb, pc the perpendiculars on 
the sides, then the above relation is 
sm^ pa = 
sin^fe sin^c 
sin"" a 
1, — cos C, — cos B 
cos C, 1 — cos A 
cos B, — cos A, 1 
It is greatly to be desired that the "trigonometry"' of higher dimensioned plane 
space should be fully worked out, for all our relations between multiple correlation 
and partial correlation coefficients of n variates are properties of the "angles," 
"edges" and "perpendiculars" of sphero-polyhedra in multiple space. It would 
be a fine task for an adequately equipped pure mathematician to write a treatise 
on " spherical polyhedrometry " ; he need not fear that his results would be without 
practical application for they embrace the whole range of problems from anatomy 
to medicine and from medicine to sociology and ultimately to the doctrine of 
evolution. 
Lastly we may turn to (vii), and express it also in multiple and partial correla- 
tion coefficients. We have 
/ A 
s's' s 'is" / ^s's'ss As"s".ss / 
t:at:,V a,, a,,,, v 
A^' Ajv; 
A A 
A 
^s"s"f:s' \ 
\\/A,v„A,v's. VA.v.s A,V3"," V A,v'.«A, 
or using the correlation symbols 
{l-R^.^.„){l-R^.,.„){l-R^■■.^.„) i 
s"P\-n ^ 
V(l-72V.,-..')-n)(l-i?V.3-(n-«) 
1 
Or, 
V(l - i2^.i_,s',_„) (1 - R\^-is")-nW{l - R\.r-n) (1 - R\".r-n) 
X (sV'Pl-(s)-n ss"Pl-W)-n X ss' Pl-is'l—n) • 
"Pl-n = (1 - R'^s.l-n) 
(1 - (1 - jgv.,_j 
(1 - i2^.i_,,,-„) (1 - R\.,-U;-n) (1 - i2V.l-(.')-n) (1 - ^V.l-K",-n) 
'<"Pl-(.s-l-ra " ss"Pl-is')-n X ss'Pl-(s"l-M 
This is a complicated form and unlikely to be of material service. 
If we use the symbol 'P„_3 to represent the determinant 
1, .„'pi_(,,")_„, sx"Pl-U)-n 
.■.■s'Pl-(.0-,i, 1, jV'Pl-(,s)-n 
ss"Pl-{s')-m t's'Pl- {s)-n> 1 
.(xviii). 
