115 



— A as r : s (taking for the sake of convenience the 

 numbers r and s so, that r ^ s = 1) then a reasoning like 

 that of the preceding articles will lead to a binomial of 



the form p{r ^ s)" (3) 



aiid we will hâve, instead of (2): 



déviation — rzA in pr" cases 



— (n — 2)A „ pnr"-^s 



/ ^. . n{n — 1) , o 1 



- (n - 4)A „ P"Y~2 r s~ » '^ . . . (4) 



+ (n — 2)A „ pnrs""'^ 

 + nA „ ps" 



The corresponding point-binomial is again obtained by 

 taking the déviations as abscissae and the number of cases 

 as ordinates. 



Now when r and s are very différent, if we construct 

 thèse point-binomials for very moderate values of n, we 

 will find that they give a dissymmetrical arrangement for 

 the déviations and this must be the cause why Que tel et 

 and Pearson hâve started from this form, to get an 

 analytical représentation of skew curves. 



In order however to make the point-binominal approach 

 a continuous curve and also in other respects to come 

 nearer to the case of nature, we hâve to take n very 

 considérable. 



Now, as soon as we do this, we find that the point- 

 binomial converges very rapidly, not to a skeiv curve, 

 but to the normal one. The démonstration is not much 

 more difflcult than in the case of the symmetrical point- 

 binomials and is virtually contained in Laplace: Theor. 

 Ànalyt. des Prob. p. 301 etc. It is owing to this that 

 Pearson does not obtain his curves by determining the 

 limit of the point-binomial (4) for n = oo , but by some 

 indirect device. 



If now, as before, we extend our considération to causes 



