J. Arthur Harris 
203 
Biometrika, Vol. ix. pp. 450-452, 1913. In present notation the product summa- 
tions are for the several correlations : 
For T,,,,, {S [S {o')f - S [ (iii) 
„ r,.,,, {S[^{s')f-S[i:{s'^)])IN, (iv) 
„ ^/./.. {S[S(/')?->S[S(/'2)]}/iV, (V) 
„ r„,3„ {5[S(o')S(s')]-^[2(oV)]}/iV, (vi) 
„ r„.,„ {^[S(o')S(/')]-'5[S(o7')]}/A^, (vii 
„ r,^f„ {S [S {s') S (/')] - ^ [S (viii) 
where N is the population of individuals resulting from the n {n — l)-fold 
weighting, S denotes a summation for the individual pods of a class and S as 
summation for the classes (individual trees). For data see Table IX. \ 
In the first three of these the negative sign term of the formula is merely the 
second moment of the unweighted population and in the fourth it is the product 
moment of the correlation surface — the "organic" relationship between 
ovules and seeds of the same pod — from which the three second moments, S (o'^), 
S {s'^), S (/'^), may be calculated. The 5th and 6th cross correlations may be 
determined from the other correlations by (i) and (ii), or since f = o — s the 
minus terms in (vii) and (viii) are given by 
^[S(o7')] = ^[S(o'2)]-5[S(oV)], 
^[S (.'/')] = >S[E(.V)]-^[S(0], 
which may be most easily calculated from the published tables for r^g, since all 
classes are equally large. 
IV. Presentation op Data. 
The constants for both direct and cross homotypic correlations are shown for 
all series in Table A. 
The reader, in examining these constants, will remember that the two Meramec 
Highlands series were taken from the same habitat and in the same year. They 
are known to differ only in (i) the smaller number of pods from each individual 
in the first series, (ii) the fact that the first series contains pods from 50 individuals 
which do not occur in the second lot. The Ohio and Kansas collections on the 
other hand represent materials of the same species from localities several hundreds 
of miles distant. Unfortunately, both of these contain too few individuals to be 
fully trustworthy. 
There can be no reasonable question of the statistical trustworthiness of all 
the direct homotypic correlations. The lowest is that for ovules failing per 
pod and this is in all cases six or more times its probable error. The coefficients 
''0,0.2' **siS2 from 25 to 45 times as large as their probable errors. For the most 
part the cross correlations may also be regarded as statistically trustworthy. 
