116 



the efFect of which follows any arbitrary law, we will 

 still find the normal curve as the Hmit of the frequency- 

 curve ^). 



The simplest proof of this of which I know, is that 

 given by Crofton Phil. Trans. vol. 160 p. 175. By such 

 a proof the truth of B e s s e l's resuit is extended to causes, 

 the efFect of which is dissymmetrical. 



The only conditions of its validity are the conditions 

 a, b, c enunciated above. 



7, Examplc of the way in which dissymmetrical 

 Point'Binomials tend to become normal. It will be well 

 to illustrate the way in which such a point-binômial as (4) 

 tends to become a normal curve. For, if we consider that 

 in such a binomial as for instance: 



(i + l)" (5) 



the first term is (})" and the last (f )", so that, whereas a 

 déviation of 



— nA will occur ({)" times 

 the déviation 



+ nA will occur (|j)" times, 

 that is 3" times as often, it seems at first sight difficult to 

 imagine, how it is that the point-binomial corresponding 

 to (5) still tends to become symmetrical. The seeming 

 paradox is easily explained however. 



In fig. 6 is given a représentation of the point-binomials 

 corresponding to (5) for the values n = 4, 8, 12, 16, 20. 

 In order that the continuons curves, which we draw 

 through the points of the point-binomial be quite com- 

 parable, it is necessary to plot the coefficients of (5) with 



intervais in the abscissae which are proportional to 



Yn 



\) In nearly every conceivable case the approximation will even be 

 much more rapid than in the case of the point-binomial; see note of 

 Bravais in Quetelet's Théor. des Prob. p. 421. 



