113 



of the fruits to grow A less, than the average of the 

 whole of the fruits. 



Of the yP berries (a) which at the the end of 1 May 

 had diameters deviating — A from mean, half will now 

 get a déviation of — 2 A, the other half a déviation zéro. 

 Of the yP berries (b) half will get the déviation zéro, half 

 the déviation + 2 A, so that now we will hâve, 

 déviation — 2 A + 2A 



number of berries jp \p \p. 



If we continue in this way and if we remark that the 

 coefficients are no other than the binomial numbers obtained 

 by the development of 



(i + iP 

 we easely get to the conclusion that, in order to find the 

 distribution of the diameters after the opération of n 

 causes, we will only hâve to develop the binomial 



(i + i)" = a)"(l + " + ""^^y-- +.... + " + l) ... (1) 



and we wil hâve 



déviation-- nA in (|)"p cases 



— {n — 2)A ' „ (i)"pn 



n{n - 1) 



in — 4)A „ {^)"p 



1.2 " ...(2) 



+ n{n ~ 2)A „ (i)"pn 

 + nA M p 



Taking the déviations as abscissae and the frequencies 

 as ordinates, we get a séries of n 4- 1 points. 



This figure is called a Point Binomial. 



The continous curve which may be drawn through 

 thèse points rapidly converges to the normal curve as n 

 becomes larger and larger. 



Prof. Gai ton (Natural Inheritance p. 63) has constructed 

 an apparatus which gives an extremely instructive illustra- 

 tion of this way of génération of the normal curve. 



