If we assume t as continuous, and carry out the addition between the limits,

μ2 =μo2 [C + 11⁄2 (b + b1 C + aa ̧) + 3% bb1] + 11⁄2 (C b2+aa)+¿1⁄2 bb.

When a, b; a,, b1; a, b; are computed from the values of the year under consideration, and the preceding and following years, which may be designated by the marks -1, 0, -1, we find

From these data the final corrected values of average statures and of their variabilities have been computed (see also pp. 1555, 1556.)

Average statures and variabilities.'

Age.

5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5 16.5 17.5 18.5

105.90 111.58 116. 83 122.04 126.91 131. 78 136.20 140. 74 146.00 152. 39 159. 72 164. 90 168.91171.07 4.62 4.93 5.34 5.49 5.75 6.19 6.66 7.54 8.49 8.78 7.73 7.22 (6.74)

Variability (4.40)|

Girls:

Average stature

Variability

104.88,110.08 116.08 121.21 126. 14 131. 27 136. 62 142. 52 148. 69 153. 50 156.50 158. 03 159. 14

It might seem that this correction could be better made by adding the proportionate amount of growth to the measurement of each individual, i. e., for those of 6 years 0 months, for instance, the amount of 6 months' growth if the measurements are to be reduced to the period of 6 years 6 months. This, however, must not be done, as small children grow differently from tall children, and therefore the amount of growth to be added differs for the various values of the measurement. That this is the case has been proved by Dr. Henry G. Beyer. I collected some statistics on this subject in Worcester, Mass., the results of which are briefly given here. I am indebted to Dr. G. M. West for many of the measurements, while others were taken by myself. The first series was taken in May, 1891. The second series was repeated in May, 1892. I give first the series of annual increases which were obtained in Worcester.

1 Figures in parentheses denote approximate values.

The Growth of United States Naval Cadets" (Proc. U. S. Naval Institute. Vol. XXI, No. 2, whole No. 74).

Increase in stature of boys.

Number of boys whose increase in stature was observed between the ages of—

Increase in centimeters.

Increase in stature of girls.

Number of girls whose increase in stature was observed between the ages of

5 and 6. 6 and 7. 7 and 8. 8 and 9. 9 and 10 and 11 and 12 and 13 and 14 and 15 and

10.

11.

12.

13.

14.

15.

16.

88888

3

1985

12227

774

10

6

11

12

98182

2

I next divided the series into two equal parts, the first embracing the short, the second the tall, individuals. The following amounts of growth were found for these two groups:

Average annual increase (d+4) in stature of short and tall children between the following years:

BOYS.

Years.

Class of children and differences.

11.

6 and 7.7 and 8.8 and 9. 9 and 10 and 11 and 12 and 13 and 14 and 15 and

10.

5.75 5.49 5.34 5.52 5.81 7.01 7.38 6.55 4.45 6.06 5.90 5.67 6.41 6.52 6.95 6.03 4.38 2.23 Difference (2A). +0.31 +0.41 +0.33 +0.89 +0.71 -0.06 -1.35 -2.17 -2.22

That there must be an interdependence between the rate of growth and the actual size attained at a certain period can be shown to be a theoretical necessity. If the variability of a series at the age t is, and if the variability of the annual increment d is m, then, according to the theory of probabilities, the variability at the age t+1 must be vμm if the amount of annual growth does not depend upon the size attained at the period t. Observations show that m is small as compared to μ. Observations also show that first increases quite rapidly from year to year, and that at the period of adolescence it suddenly decreases very rapidly. It is clear that these phenomena do not agree with the assumption made. We must conclude, therefore, that the amount of annual growth depends upon the size attained at a certain period.

It is possible to give an approximate value of this relation. If the average of all measurements for the period t is A, that for the period t, is A+d, where d is the average amount of growth for the period t1 t. We will consider in what manner a value A+d+v in the series of the period t, develops from the series of the period t.

We will suppose that the relation between the actual size of an individual and the average amount of his annual growth is expressed by the simple relation

dx=d+ax, where a is a constant.

Furthermore, we will assume that the variability of d is the same for all values of x. The annual growth of a single individual of the size A+ will be, according to these assumptions, d+ax+y, where y expresses the accidental variation of the annual increment. The size of the individual at the period t, will there

By observation we find the variability at the period t,- that is, that of -equals Therefore

As a must be a small value, the positive root only is available, and we have

It follows from this equation that as long as u, is considerably larger than u, a is positive; when, is smaller than u, it is always negative. As during the early years increases with age, among young children the small ones are in a period of retarded growth, while the tall ones are in a period of accelerated growth, while among older children when u begins to decrease again the tall ones cease growing, while the smaller ones grow rapidly.

The values given on page 1549 for the amount of growth of short and tall children may be considered as equaling

It is therefore possible to calculate a from the data contained in the table on page 1549. The two series of values show a fairly close agreement, considering the small number of repeated measurements.

is a very rough approximation to actual conditions, and that, particularly during the period preceding puberty, the distribution of annual increase will differ considerably from this law.

Dr. H. P. Bowditch, in a paper published in the Twenty-second Annual Report of the State Board of Health of Massachusetts, assumes that the growth of children is such that they always remain in the same percentile grade-that is to say, if the variability at the period t is μ, and at the period t, is μ,, then the average child which has at the period t the measurement A+x=A+μ will have at the period t, the measurement A,+. Its growth during the intervening period will therefore be

1

μ

μ