5G 
PROFESSOR K. PEARSON ON THE INFLUENCE OF NATURAL 
practical pur230ses we are quite safe if we assume that an individual who occurs only 
once per thousand can produce no effect on the physical evolution of the population 
as a whole. 
Now tlie form of a correlation-surface for two organs, x and y, is 
N - ■>- ^ f ?! - - ™y + 1!L'> 
^ ~ G- /I -^ e 
lira^a^ V 1 — r~ 
Let US write k~ = 
1 
^ _ -yi + jn ; then K = a constant gives a series of 
I — r- [c7 p o-^CTj cr,~] 
similar ellipses which are the contour lines of the surface, or lines of equal frequency, 
i.e., giving individuals with equal probahility of occurrence. Let the equation to 
these contour lines referred to their princijoal axes he 
Y2 
K' 
A2 + IP 
Then we have at once : 
1 
A- 
1 
>0 
(1 - rh 
+ 
1 ^ ' 
I_ -r 0 -2 
1 — cri"cr;> 
or, 
1 1 1 
AHE “ (1 - rh' (1 - r2)- “ (1 - r^) (TfV.d 
AB = cr^cTo \/I — rq A~ + B' = cry + cr/. 
Further, if (f) he the angle tlie A principal axis makes with the axis of x, vce 
have 
tan 2(/) = 2rcriarJ{(T^^ — ax). 
These fully determine the principal axes of the frequency surface. Now consider 
the frequency Ijetween the elliptic cylinders corresponding to k and /c -f 8 k ; we 
have it 
= :•' X 27TABKdK = ,t X 277crifro \/1 — kc/k = Ne~'*'K(/K. 
Hence, if N^ l^e the frequency outside any contour k, 
N, = N I* e-'-’KcU = Nc-’--’ . 
For N^ to he ‘ximo ^ have simply 
0 
-, whence k — 3•"16,923. 
log e ’ 
(cxviil.). 
* For easy calculation put y - J(r{^ + 0 - 2 '^, tan xp = crg/a-i. Then we have at once if r = cos D : 
A = y cos X) B = y sin X) 
sin 2x = sin 21 /- sin D, tan 24 > — tan 2 \p cos D. 
where : 
