12 
PROFESSOR K. PEARSON ON THE INFLUENCE OF NATURAL 
Now suppose in the exjDression (xxix.)'”® we were to put Xy, .Xo, . . . equal to 
liy, ho, . . . hj respectively, then the system of ec^uations to find the means of 
Xy+i, for this array would be the "a — q equations 
X 
' (J+\ ~h ?+l ^ ?+2 fi" ■ ■ ■ X — 0. 
!7+2 ~l~ ^2, ?+2 ^^2 ~i~ ••• fi” ^V’? + 2 ^?+l!!?+2 ^ !? + l ~i~ *^y+2,!?+2 ^ !?4 2 ~h • • • ^n.q + Z^ 
Cy„ hy + Co, ho + ... + Jhj + Crj+y,, x' + y-{-c j+o^, • • • fi" x'„ = 0 . (xxxiv.), 
where x'g+y, x'g+o, . . . x',, are the “co-ordinates of the centre” of the array of the 
71 — q organs. 
If we were to solve these equations we ought to get precisely the solution for 
x', (m > < n -{- 1) that we have found for x', in (xxxii.) above, where none of the 
coefiicients involve correlation-coefficients other than those of the first q organs 
among themselves and with the oigan. This result floAvs from the pretty obAuous 
law that the mean of the organ for an array determined l3y values of the 
first q organs cannot l)e in any Avay dependent oji our considering the relation of this 
selection of q organs to any additional organs l)eside the : see p. 9 . 
Thus the solution of (xxxiw) is simply obtained l)v })utting = q 1, q 2 , 
q S, . . . 7 } successively in (xxxiii.). 
Let us now select the first q organs not with al)solute values, but varying about 
means /q, /q, . . . /q, AAuth standard deviations s., . . . ,sq, and with mutual corre¬ 
lations pyo, pi3, . . . pyg, P23> • • • P2?) • • • P'7-vj- have then to multiply C in (xxix.)^'" 
by an exponential quadratic function of the Xy-\-hy, .To+Zoi, . .. i.e., the selective 
correlation surface, and divide it by another exponential quadratic surface, i.c., the 
primary correlation surface of the q organs Xy, x., . . . x,^. This folloAvs, since the 
frequency of each complex of n organs must be reduced in the ratio of the selected 
to the primary frequency of the complex of q selected organs. But it Avill be clear 
that such a reduction must give us a result of the following form for the final 
frequency surface of the n organs : 
Z = constant X expt. — L {cn-Xf -f c^oX.j^ -f- . . . -f CggXf 
“1“ ^? + l,? + 2^?+U -f . . . 4 - 
-h 2 c,,i,„x,„Xi.„ -h . . . {v' and v'' < q -j- 1) 
-f 2 c,.,,XrT, -h . . . (i; < 7 -f 1 and u > q) 
-b + . . . (n' and ii” > q) 
-f- linear terms in x,,, {v' and v" <7+1) • (xxxv.), 
where the c’s denote the changed c’s. 
