May 6, 1892.] 



SCIENCE. 



257 



average value of the measurement, s, may be expressed by the 

 formula — 



/< 



2/^1 = 



« + »' ~ fX, i/2 7T 



fii is the measure of variability of the series and is called the mean 

 variation, or the mean variability. A series is the more variable 

 the larger fi^. 



The value of the measurement belonging to the average of all 

 those individuals who will finally reach the value s is, at any given 

 period, a function of this period and may be called t- The aver- 

 age of all those individuals who will finally reach the stature s -\- x 

 may be expressed as a function of t and x, f \^t ; x\ 



The individuals constituting the adult series will not develop 

 quite regularly, but some will be in advance of others. We as- 

 sume that at any given time these variations in period will be dis- 

 tributed according to the law of probabilities. The relative fre- 

 quency of the variation y from the period under consideration, t, 

 will be 



P^ 



The value of the measurement belonging to a child which will 

 finally reach the value s + cc, at the period t + y, will be 



J yt + y,-^J. /^ t + y expresses therefore also the relative 



frequency of the individuals measuring / (^ + y ; ^j at the 



period t among that class which will finally reach the value s + x. 

 The relative frequency of the latter among all individuals 



is ^ s + X. Therefore, the relative frequency of the value 

 r (^t -^- y ; ^) among the whole series will be 



_ — _ 11^ 



P 



f{H^y;-)=Ps + xpt + y = ^-^e 



It remains to determine /(*J -f 2/,- a^Y The function /(** '^j 



may be obtained by observations on the same individuals taken in 

 annual intervals. The form of the function will be 



Xq <X <Xi, 



and 



By means of observations we find also 

 *^ + y=^'^ +ajy + a^y- + . . . . 



Jn\^ +y)=bo+h,y + h,y'- + ... 

 By substitution we find 



yQt+ y; '^^^H + a,y+a,y^ + 



+ x(bo' +h^'y + 1)^ y + • 

 + a;=(6o" +bi"y + bJ'y'- f . . .) J 

 For a certain series of combinations of x and y this function 

 will remain constant. Then the function may be considered a 

 new variable u — 



y(^H + y;oc^ = Sf ^u. 



The probability of finding the value ** +u is 

 -I- Urn, 

 Psf ^ ,( ^ / Pf^^ '^ ^' *) ^y ^^'■^^ ^here the lim- 



ya<y<y^ 



•* 1 2/0 < 2/ < 2/1 



its depend upon x„, x^, y^, and yi, and where i? is a certain rest 

 which is determined by the same values. 



By assuming the limits a;,,, Xi, y^, y^ sufficiently narrow and 

 neglecting terms of higher degrees, which may be done on ac- 

 count of the smallneas of their factors, the equation assumes the 

 form 



T. 



_ (1 -f e m) 



+ u ~ Jlf »/ Stt ' 



2Jlf3 



M : 



This function is asymmetrical. It is, therefore, shown that the 

 asymmetry of the curves is an effect of the irregularity of growth. 



Only for flj, a^, . . . = 



(«) <«.) 

 and bu b^ . . . . = 0, the curve will be an ordinary proba- 



bility curve, c being zero in that case. When a^, a^, . . . are zero, 

 growth is regular. 



We may also draw certain conclusions in regard to the value M. 

 ^2 is the variability of period. According to the laws of proba- 

 bility this variability must be proportional to the square-root of 

 time elapsed — 



^2 = i" ^ t- 



It is also probable that 



bo= l/ '± 

 s 



t 

 We will investigate for which points 



M > ^i, 

 a, V' t + ^t_ ^^2 > ^i2 



ai V' t > ^ If ^i2 



For small values of t, 



is large, but at a certain period, 



when ai is still large the product on the left-hand side will rapidly 

 increase over that on the right-hand side until osi begins to de- 

 crease. It may be expected that in aU cases when a^ is suffi- 

 ciently large, i.e., the growth rapid, there must be a time when 

 the variability of the growing series is greater than that of the 

 adult series. 



M and //i are known by observation. Therefore n^ may be 

 computed according to the formula 



and we have, therefore, a means of determining the variability of 

 period of the growing individuals. By means of this value we 

 can also determine how many individuals of any given age will 

 have reached the adult stage. 



This theory holds good for statistics of all kinds of development, 

 whatever the cause of the development may be: for, physical 

 measurements as well as for psychical ; for growth as well as for 

 the effects of practice. Fbahz Boas. 



Clark University, Worcester, Mass., April 25. 



G. P. Putnam's Sons will publish immediately " New Chap- 

 ters in Greek History," based upon the latest archaeological dis- 

 coveries, by Professor Percy Gardner of Oxford, and " The Test 

 Pronouncer," by W. H. P. Phyfe, a companion to the author's 

 " 7,000 Words Often Mispronounced," containing the same list of 

 words, differently arranged, for convenience in recitations. They 

 also announce new supplies of Phyfe's books on pronunciation : 

 "7,000 Words Often Mispronounced," "How Should I Pro- 

 nounce," and "The School Pronouncer." 



