48 
PROFESSOR K. PEARSON OX THE INFLUENCE OF NATURAL 
X., — k.,, . . . X,, — k„) = expt. — ^ {x^ — k^)~ -4- {x., — 7:^)- + . . 
4- 2a (.-fj — Jc{){x. — 7-.) — . . . + {x„_^ — Jc„_y)(x„ — J:„)} . 
Thus, to determine the probability of survival, we require to kno-v the values of 
the as and Jcs in terms of the h’s, Iis, and c’s. The shortest method of finding is 
to put Xy = 7’j, X., = /q, . . . x^i — Ii\„ and then note that : 
T\> 
^ ('0 — rq — — 1 ,,) 
A ('q = /q, = /i'o, . . . o'n — />',;) 
(xci.). 
Since 
p = zj'Z, and 
we see that the relations for p, given s and Z, and for 2 , given p> and Z, are cyclicly 
interchangeable if at the same time we change to — <?jj, to — c.^.z ... Cu to 
— Ci 2 . . . to — If o-j, (To, . . . rr„ be the standard deviations of the 
unselected population, this amounts to changing cr^, cto, . . . o-,^ to \/ — lo-^, v'^— Icr.o, 
. . . Icr,; respectively. Thus the results which give the probability of survival 
in terms of unselected and selected populations can always by an easy interchange 
be used to obtaiii the selected population from a knowledge of the unselected 
po])ulation and of the probability of survival. 
Let the unselected population be defined by nr-, . . . c^, (t_-, . . . cr,„ and >qo, 
^ 10) ^ 23) • . • ^ .'( — I, «• 
Let the selected population be defined by ?)q + H^, nq H.), . . . + H,, 
. . • —)() and ijj, 1 ^ 3 , V 03 , . . . 
Let the constants of the probability of survival function, or rq^, . . . a„i„ 
. . . «;;_!,«) be expressed by 6 q, ^.n, . . . q,, p^.,, p^g, . . . p„-i,n as if it were a normal 
correlation surface. 
Then the problem will be solved, if we know the Jcs, s’s, and p’s in terms of the cr’s, 
r’s, H’s, N’s, and r’s. 
Equating the squares, joroducts, and linear terms in the :r’s in the equation 
•p = z/Z, we have at once the system : 
= h 
(I,, 
UU ^ UW) 
I — c 
nr*, 
for all values of u and v from 1 to n. 
(xcii.). 
— o,i/q — o,../,-., — ... — a,,k, — ... — ajxu 
— — h,iHi ~ — ... — 7>,,H(, — ... — h,,TL, . . (xciii.), 
for all values of v from I to n. 
* These must not be confused with the .sj, So, . . . s,j, piz, pis, p^s, . . . pq,q-i constants of the q selected 
organs of the jirevions discussion, The new cjuantities may be in part iinaginarii. 
