172 
PROFESSOR KARL PEARSOX. ^lATHEMATTCAL 
(3.) So far we have dealt only with the reconstruction of the most probable value 
of B from one organ A, but we may propose to find the most probable value of B from 
11 organs Aj, Aa, Aj . . . A,j, Let represent the correlation coefficient of B and the 
organ A,^, the correlation coefficient of A,^ and A,^-; o-q the S.D. of B, and cr^ of the 
organ A,j, nio the mean of B, and of A,j ; let Pt be the determinant 
1 Al '^\-2 '^’03 
On 
"^'lO 1 '^\2 '^’l3 • • • 
'^20 '^*21 1 ^^23 • • • '^^ 2 n 
'^’SO ^’si '^’32 1 = . . T^;, 
^'nO '^w2 ^’nS ... 1 
and E.^ 5 , the minor corresponding to 'Ifiien the general theory of correlation 
shows us that 
B — r/in = — 
lb 
E 
(10 
/ A \ CTii 
-- (Ai — m,) — ^ - 
a I ±A;„J <J 2 
E, 
(A,_ma)... - . ''^'^(A.-m.,). . (ii.) 
-•--o;) ^-,1 
is the most probable value of B, and that there is a probable error = '67449 o-,,^(E Rqo) 
in this determination. 
Thus we reach again a formula of the character 
P “ Co + CjAi + C2A2 + C3A3 + . . . + c„A;„ 
or, B is expressible as a linear function of the organs from wdiich its value is to be 
predicted. This again supposes normal, or at least " linear ” correlation. Now there 
are several points to be noticed here. 
(i.) The linear function which will give the best value for B is unique. For 
example, some anthropologists have attempted to reconstruct stature by adding 
together the lengths of femur and tibia. The proportions in which femur and tibia 
are to be combined are given once for all by the regression formula, and they are 
not those of equality. I have succeeded in proving the following general theorem, 
which settles this point conclusively. Given any linear function of the 11 organs 
Ai, Ao, A 3 . . . A,,, say 
“b ^9-^1 "h l>-iAl + hgAs + . . . + l>n^,n 
Skeleton,” Cambridge, 18-58, p. 108. Many others have been given by French writers, in some cases 
with several values cf B/A for three ranges of stature or of long bone (Topixard, Rollet, etc.). 
Dr. Beddoe has given a rule Avhich really amounts to making B a linear function of A, but his values 
for Cl and c.^ are widely divergent from what 1 have obtained by applying the theory of correlation. 
‘ Journal of the Anthropological Institute,’ vol. 17, 1888, p. 205. 
