Formulæ for the determination of the correlations of size and of growth increments in the developing organism

In the analysis of the growth of the higher organism it is essential to obtain definite measures of the interrelationship between certain measured magnitudes. Those which require consideration are the following:

(1) The correlations between the actual size of the organism at the various stages 1 of growth. (2) The correlations between growth increments of the organism during the several growth periods. (3) The correlations between the size of the organism at any stage and any or all subsequent growth increments.

The labor of determining these correlations by ordinary methods is excessive. If the first set of correlations (1) be determined by taking all moments about of as origin, 2 we may solve problems (2)-(3) as follows.

Problem 2.—To determine the correlations between growth increments from the moments and product moments of size at the several growth stages.

Let w, x, y, z be the dimensions of the organism at growth stages p, g, r, s. The growth increment during the intervals q-p, r-q, s-r will then be i_pq = x-w, i_qr = y-x, i_rs = z-y.

The moments Z(x), Z(x₂), Z(y), E(y₂),…, and the product moments Z(wx), Z(wy),…, Z(yz) are available for the correlations between size, which are required on their own account (Problem I).

The constants for growth increments are given by well-known formulæ

and similarly for

The product moment for any two growth increments, say i_pq and i_rs, is

In the special case in which three consecutive stages, say W, X, y, are involved we write

Problem 3.-To determine the correlation between the size of the organism at any stage and any growth increment.

The notation of problem (2) may be used. The physical constants for the growth stages and growth increments have been given.