8 
PKOFESSOR K. PEARSON ON THE INFLUENCE OF NATURAL 
R 
s. (if 
121 ^ 
1 > 1 > 
— zm I _1_ Ul2 / , PMs / 1 I / 
— T>l'^lO^T> ■^31 i> 
Ji \ J-hi -L^n 
T » ' ’2 
+ 
11 -^'11 
P P P 2 
RR„ 
X 
+ 
1) p P 
jA^ <1 \ "> 
RR 
11 
wliere is tlie sum for all values ofyj>, and for all pairs of values of y) and y, from 
2 to n inclusive, 
K + 
P' 
B1 .tJ 3 
R' 
X 
p 
+ 
‘7-< 
where li\ = Sj 
.U, 
j 
11 
For if Pt' = Pjj, or the determinant of the correlation coefficients, omitting all 
involving the first variable, i.e., the first row and colnmn of It, the determinant Pv,'^^ 
corresponding to the minor of the constituent in Pi' or to the second minor (. Pii)y,^ 
of Pi is given by 
and for yi = q, 
= (Pv/^P^II ~ 
(xxii.). 
Hence, integrating x\ between the limits + oo and — cc, we have 
N = ^'oCriO‘3 
X 
r + 30 
expt. 
J — 00 
V 
T.) / 
J'P 3 
| ) / p 
+ 1 
OS' 
T>/ 
pq / / 
j>/ ^ p'^ q 
dtp o r/a Q. . . da 
This is of precisely the same form as before, except that we have the factor 
•v/27r x/P/lP, and the multiple integral is reduced by one integration and by the 
disappearance of all correlations involving the first variable. Now, integrate with 
regard to a'o. The sole effect will he to multiply by a factor \/ 'Itt \/R'/P", where 
Pi" is the minor of R' not involving correlations of the second variable. Thus, by 
repeating the process, we have ultimately 
N = 
ZijCTicr^ 
o-„ (v/277)" v/R/1P x/R'/R" v/R"/R' 
? 
or, 
~u 
N 
(27rp' (T^CTo . . . cr„ \/R 
(xxiii.), 
wliich gives the constant of the surface. 
The preceding investigation enables us also to deal with two further points. 
(o) Given n varial)les, what is the mean xmlue Wj + .ny of the first variable and 
its variability (r\ for definite values m.^ ff- h., 7723 + h. . . . 77?,, + /?,, of the other 
(?? — 1) variables ? 
