117 



(it seems unnecessary to demonstrate this hère) and I hâve 

 accordingly taken them so. 



The curves ail begin at the same point of the axis of 

 the abscissae marked in the figure. The end-points lie 

 at a, b, c, d, e. In reality, therefore, the curves are 

 dissymmetrical; they always extend much further on the 

 left side of the maximum than they do on the right-hand 

 side; but the tail of the curve on the left-hand side is so 

 close to the axis of the abscissae, that is to say, the 

 frequencies of the smaller abscissae are so small, that 

 even for moderate values of n they become quite insensible. 

 For n = 20 I hâve drawn in the figure the normal curve 

 having its maximum coincident with that of the point- 

 binomial. As we see it is already ail but wholly coincident 

 with the dissymmetrical curve. For still larger values of n 

 the dissymmetry would very soon disappear even in the 

 most accurate drawings. 



In fact, and hère lies the explanation of the seeming 

 paradox, for large values of n, the only part of the curves 

 of any importance, is that on both sides of the maximum 

 and this part becomes rapidly normal. 



8. How do skew curves orginate? If therefore any 

 causes whatever always produce normal curves, how do the 

 skew curves originate? Though we may not at first sight see 

 this, we may see at once the necessity of their existence. 



Suppose, as before, that we find the diameters of certain 

 ripe berries to be distributed in a normal curve. 



Let us suppose further, what in most cases must be 

 quite near the truth, that thèse berries are perfectly 

 similar, and let the question be put: What will be the 

 frequency-curve of the volumes of thèse berries? 



It must be évident at once, that the form of this curve 

 must be wholly determined by that of the diameters and 

 a little reflection will easily prove that it cannot be a 

 normal curve. 



