Equations of a relatively simple form were derived in the following way for the secondary stature growth of Japanese boys and girls in puberty period between the ages 10.16.
1. Assuming that the rate of growth (dy/dt) is proportional to both the time (t), calculated after a certain stage and the difference 1-y, where l is the terminal value of the stature, y the stature at time t,
(i)
By integration:
(ii)
which is an equation of growth as referred to the origin O'. (See Fig. 1.) Let the age and the stature at 0' be θ and l0 respectively (See Fig. 2 on p. 55), and we obtain an equation of growth
(iii)
as expressed in ordinary terms (age and height).
The constants were determined using the data of Y. Yoshida and by the method of least square:
(iv)
It fits the data satisfactorily (Table 1 on p. 55)
2. Assuming that the rate of growth is proportional to y and l-y, y and t are measured from the starting point 0' of the secondary growth (i. e. of the puberty), we obtain
(v)
or by integration,
(iv)
where φ is a constant (See Fig. 3 on p. 56) Designating the coordinates of the starting point of the secondary growth by (θ, l0), and using the stature and age as the hight y and time t, we obtain:
(vii)
Numerical constants corresponding to the actual data make the equation:
(viii)
Agreement between the observed and calculated values is quite satisfactory. Both types of these formulae have been found useful for smoothing the growth curves of observed data to obtain adjusted curves of the standard values