PROFESSOli K. PEARSON ON THE INFLUENCE OF NATURAL 
2fi 
These equations admit of easy interpretation by spherical geometry. 
P 
Let P ]je the })ole of the great circle DEFG. Take DG = 6 ^^, DE = GF = 
Join P to E and F ; let the small circle of radius round P meet PD, PE, PF, 
and PG in d, e, f, g resj)ectively. Draw the arcs De and G/i Let the small circles 
with centres D and G and radii Dc and Gf respectively meet in Q. Join DQ and GQ. 
Then the quantities required are : 
= sin DQ, GQ. 
cos DcE, I'ls = cos Jy’G, U3 = cos DQG. 
For DE = 0 ,,, Ee = - Xi, L DEe = y ; 
hence ; cos Dc = cos 6 ^^ cos Ec = sin Xi cos = cos cqo, or cq^ = De ; 
TT 
similarly tqg = Gf. 
Next, cos DeE —cot De tan eE = cot cot Xi, or l_^ DeE = 023; similarly < F/G = 0i3. 
Lastly, from the triangle DQG : DQ = cq,, QG = eqg, and DG = ^03) 
cos DG = cos DQ cos QG + sin DQ sin QG cos DQG, 
or. 
cus ^03 - cus cj. 
cos rt. 
cos 023 ; or i__ DQG = 003. 
Thus all the relations can be expressed in terms of the sides and angles of a simple 
system of s})herical triangles. For the degree of accuracy generally possible in 
biol(,»gical and sociological investigations these triangles can be solved by a spherical 
trigononietei', such as that sold by Kreidl, of Prague."^' The changes, however, 
■whicli r23 undergoes for various values of are, indeed, lar more difficult 
to appreciate as a whole than those of Vjo or iqg. In order that they may be follov'ed 
easily, and in order to solve directly to a degree of approximation sufficient for many 
practical purposes problems in the influence of selection on ccu'relation, my assistant, 
Dr. L. N. G. Filon, has kindly drawn u|» the tables which accompany this memoir. 
* It will suffice fairly well for all 1)Ut a few special ^•alues of Vi^, r-ss, /'si. 
