Raymond Pearl 
243 
and with breadth for both conjugants and non-conjugants. These index cori'e- 
lations were all calculated by formula, and not from tables directly. That the 
formula gives very close results for such correlations has been pointed out by 
several workers, notably C. D. Fawcett* and Macdonnellf. Pearson:}: has shown 
that in terms of the organic correlations Vy, ... r^^, and the coefficients of variation 
V1...V4 of four variable characters, ...op^, the coefficients of correlation p between 
the two indices , and — has the value 
In the present instance it is desired to correlate the length-breadth index x-^jx. 
with first length x.^, and then breadth x^. For the index-length correlation substi- 
tuting the proper value in (i) we get 
In the same wa}^ for the index-breadth correlation we have 
.(ii). 
.(111). 
In the values of p in each instance there are clearly two factors, («) the 
true organic correlation arising from the existence of an organic correlation 
and (b) the spui'ious correlation between the index and the characters concerned. 
The expression for the spurious correlation in the case of the index-length corre- 
lation is 
= -7==-- (1\ ), 
V ?.']'- + V.,: 
and that for the index-breadth correlation is 
/Oo= /-=-^" (\)- 
V?',- -t- v.? 
The latter differs from (iv) only in being positive where that is negative. In 
the following Table X. there are given in the colunni headed "Gross" the values 
calculated from formulae (ii) and (iii), i.e. the values for p. In the column 
headed "Spurious" are given the vahies of p„ calculated from (iv) and (v), and 
finally in the column headed " Net " we have the portion of the gross correlation 
due to true organic correlation between the index and the character, or in other 
words, the value tabulated in this column is p — po- 
The results from the index correlations are rather remarkable. In spite of the 
fact that the index is formed by taking 100 times the breadth divided by the 
length, the net organic correlation of index with length is in every case positive, 
* Biometrika, Vol. i. p. 461. t Ihid. Vol. iii. p. 238. 
t Proc. Roy. Soc. Vol. GO, p. 493. 
