Karl Pearson 
407 
Let y be the observed size of any other organ at the same interval after the mean time, then 
its probable size at the mean time is given by 
y^y-^^if-t). 
Clearly means of x' and y' will be equal to means of x and y, and we easily find* 
<y\ = t\ (1 - r\^\ <rV = (1 - r\t) (i) 
Pxy '^xt'^yt 
Ex 
(ii) 
where B^'y' is the correlation of the x' and y' characters allowing for growth and p^y of x and y, 
or without allowance for growth. 
If X and y are the same organ for the pair, we have 
(iii) 
where r is the correlation of growth and interval of time for the organ in question. 
If we apply these formulae to the values of p given in Table VI and to the values of r^t 
given in Table VII, we deduce the values of the homotypic correlations, when allowance is 
made for growth. We find 
TABLE VIII. 
Homotyposis allowing for Growth -f". 
Lengths of 1st and 2nd Members 
•569+ -074 
Breadths of 1st and 2nd Members 
•400 + -151 
Length of 1st and Breadth of 2nd Member ... 
•491 + 1 
Indices of 1st and 2nd Menibei-s 
•043 + ? 
Length of 1st and Breadth of 1st Member 
•425 + 1 
The first two of these results are quite as satisfactory as we could expect from such data. 
They fit in fairly well with the homotypic correlation distribution clustei'ing about •o. The 
fourth is very improbable. But the di-op from ^664 to •I 14 in index correlation between Dr 
Simpson's first and second series of measurements is so inexplicable that we cannot possibly 
expect anything from the index figures. The third result clearly does not satisfy the theorem 
I have given for cross-homotyj^osisj. But the lowness of the index correlation, and the fact that 
the length of the 1st member is moi-e highly correlated with the breadth of the 2nd member 
than with that of the 1st would undoubtedly be explicable, if, as I have suggested, there be some 
principle of " compensation in division " at work. We can only hope that at some future time 
we may have available a much more extensive system of growth measurements, made on the same 
long series as the homotypic measurements. Without this correspondence it is impossible to 
distinguish how much of the degree of resemblance between members resulting from the same 
division is due to growth and how much to pure homotyposis. But the results for growth 
in Table VII combined with those for pure homotyposis in Table VIII suffice to show that 
great care must be used in not treating what are really coefficients of gross resemblance as 
in Table V— due to pure homotyposis + growth + individual environment of the pairs of mem- 
bers — as in any way a pro23er measure of the first factor, i.e. of pure homotyposis only. 
* These formulae may be obtained by straightforward algebra from the above results, but they are 
really simple cases of a theory developed at length in an unpublished memoir on "Selection allowing 
for Growth " ; (ii) is clearly the partial correlation coefficient for the time-factor constant. 
t Probable errors in first two cases calculated from unpublished formulae. 
+ Phil. Trans., Vol. 197, A, p. 287. 
