M. A. Lewenz and K. Pearson 
373 
the two quantities or characters v and w ; then the proper formula for Dr Beddoe 
to have taken would have been : 
'~{p. - m,) + ''^flZl^^^^^ ^ (p, - ,n.) ig). 
Dr Beddoe has assumed that : 
Such a result is contrary to all anthropometric experience*. 
There is of course no reason why a formula like (g) should not be tried for 
skull capacity, but there is only one scientific method of determining it, namely 
the discovei-y of the actual means, standard deviations and correlations of the 
quantities involved for a long series of skulls. We cannot, in the light of modern 
statistical science, assume : 
C = A + D,p,-\- D,p., 
tacitly put A = 0, and take a series of guesses at and Do, " feeling our way." 
But the labour of investigating a formula like {g) is considerable, and no wise 
person would undertake it without first considering its chances of giving a 
remunerative result. Let us first enquire into this. 
Formula (g) may be written if subscript i stands for / : 
C = 7» + (7i + 72^)/^i. 
whence : = (71 + 70 iiii) 8pi + 7., nij,^ 81, 
and ultimately : 
o-c f'ci = (71 + 7-j I'll) o"^, + 72 ^Upo Oi, 
o-c = (71 + 72 '"«■) o-^i ^cp, + J2 1'ip.i o-i r,i. 
Eliminating 71 + 72 m,; we have : 
^ _ J_ rp,i - Vci r,p ^ 
Now we know that ?v,- is usually very small, and therefore the terms r^p^ 
and TciVp^i will be small; further, r^p^ is always positive. Hence 72 will be of 
opposite sign to Vp^i. Now, so far as we know, nobody has worked out the 
correlation of p^ and /, but since the correlation of p^ and G is positive and large 
thei-e can be no reasonable doubt that that of and / will have the same sign 
as that of G and /. But the correlation of capacity and cephalic index is negative 
for dolichocephalic races and positive for brachycephalic races. This conclusion 
* See, for example, the multiple regression formulae given for the long hones by K. Pearson, 
Phil. Trans. Vol. 192, A, pp. 180-7, for cranial measurements by A. Lee, Phil. Tram. Vol. I'.lfi, A, 
pp. 235-6, and for the like measurements by S. -Jacobs, A. Lee, and K. Pearson, Bioiiietrika, Vol. ii. 
pp. 349—355. The list might be indefinitely extended. 
