572 



SCIENCE. 



[N. S. Vol. V. Ko. 119, 



Age. Boys. 



12 0.10 



13 0.08 



14 —0.03 



Girls. 



— 0.11 



— 0.17 



— 0.20 



15 



-0.22 



The values cannot claim any great weight, 

 since the series of observations is very 

 small. Only about fifty individuals for 

 each year and sex are available. They 

 prove, however, that the values of a first 

 decrease until about the eighth year. Then 

 they increase and decrease again very 

 rapidly after the thirteenth year in boys 

 and after the eleventh year in girls. 



According to the assumptions made be- 

 fore, the average individual which meas- 

 ured A + X Sut the period t will measure 



(A + x) + {d + ax)=A + d + x ~\ \l~—„ 



!■>■ \ /J.J- 



at the period t^. 



If it measured 



A + d + x^J^, 



it would remain in the same percentile grade 

 while according to the above formula its 

 percentile standing will be nearer the aver- 

 age than at the initial period t. Only when 

 all the children of the initial measurement 

 A + X grow equally, i. e., if m = could 

 they remain in the same percentile grade. 

 This conclusion agrees with Dr. Henry G. 

 Beyer's observations.* 



The above approximation is fairly satis- 

 factory during the early years of childhood. 

 During the period of adolescence it is not 

 satisfactory, because the values of a are too 

 large. More extended observations will 

 enable us to include terms of higher order 

 in the considerations and to obtain more 

 accurate knowledge of the laws of growth. 



The results of this investigation suggest 

 that the differences of growth observed in 

 children of different nationalities and of 

 parents of different occupations may also 

 be partly due to retardation or acceleration 



*Proc. U. S. Naval Institute, Vol. XXI., No. 2. 



of growth, partly to differences of develop- 

 ment in the adult stage. 



In order to decide this question we may 

 assume that in the averages obtained for all 

 the series representing various social groups 

 accidental deviation from the general aver- 

 age only occurred. Then it is possible to 

 calculate the average deviation which would 

 result under these conditions. When the 

 actual differences that have been found by 

 observation are taken into consideration 

 another average deviation results. If the 

 latter nearly equals the former, then the 

 constant causes that affect each social group 

 are few and of slight importance. If it ia 

 much larger than the former, then the 

 causes are many and powerful. The pro- 

 portion between the theoretical value of the 

 deviation and the one obtained by observa- 

 tion is therefore a measure of the number 

 and value of the causes influencing each 

 series. 



I have applied these considerations to the 

 measurements of Boston school children ob- 

 tained by Dr. H. P. Bowditch. I have used 

 thirteen different classes in my calculations, 

 namely, five nationalities : American, Irish, 

 American and Irish mixed, German and 

 English ; and eight classes grouped accord- 

 ing to nationalities and occupations : Amer- 

 ican professional, mercantile, skilled labor 

 and unskilled labor, and the same classes 

 among the Irish. 



The results are as follows : 



