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

+0.5 >> -0.5, thus covering the whole year, we find

+0.5
Hi=f(C+at+bt?) [110? (1 + a,t +1, t) + a.t +0,1") at.

f

--0.5 u=h'[C+1 (1+0, C+aa) + Bobb,] +1(CD:taa)+30. When a, b; a, b,; 2, bz; 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

C=1-15b,

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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,

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Boys:

Average Variability (4.40) Girls: Average

stature Variability

stature.. 105.90 111.58 116, 83 122.04 126.91 131.78 136.20 140.74 146.00 152.39 159.72 164.90 169. 011171.07 4. 6:2 4.93 5.31 5. 191 5.75 6.19 6.66 7.51 8. 49

8.78 7.73

7.22 (6.74)

101.88 110.08 116.08 121.21 126. 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 inonths' 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. 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).

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Increase in stature of girls.

Increase

in centi. meters.

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.

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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:

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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 ll, 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 vil + m2 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 ll. Observations also show that it 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 ! is the average amount of growth for the period t, t. "We will consider in what man. ner a value A+u++ 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

de=l+-ax, where a is a constant.

Furthermore, we will assume that the variability of l, is the same for all values of x. The annual growth of a single individual of the size A + x 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 therefore be

1+2+0+ ax +y=4+0+.

Y=V-X (1+a). The probability of finding the variation x is

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By observation we find the variability at the perio:l t, - that is, that of 2-equals thy. Therefore

hl=l? (1+a).+m";

- m.

-1.

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

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It follows from this equation that as long as !l, is considerably larger than ll, a is positive; when ll, is smaller than 11, it is always negative. As during the early years he 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 it begins to decrease again the tall ones cease growing, while the smaller ones grow rapidly.

The values given on page 1519 for the amount of growth of short and tall children may be considered as equaling

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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.

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Annual amount of growth d + ac 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 ll, and at the period t, is ļly, then the average child which has at the period t the measurement A+x=A+will have at the pe

pl riod t, the measurement A, +1] Its growth during the intervening period will

Il therefore be

A,+24,-A-mu=,- A+",—x.

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