Series to Frequency Distributions in Space. 391 



§8. This solution fails for symmetrical distributions. 

 If r=r' , it follows that f =tj and from (64) that y = l, so 

 that the value of n given by (66) becomes 



Now for nearly symmetrical distributions ft would be 

 nearly = ft' and ft = ft', so that it would be the ratio of two 

 very small numbers and liable to an exceedingly high error; 

 in fact n is quite indeterminate in the exactly symmetrical 

 case. 



In this case, however, the solution can be completed by 

 the use of marginal moments alone, and the formula given 

 by Pearson for n (eqn. 32 I. c.) can be adopted. In fact let 

 ft= IhojPzoPzo, ft' = PQ0/P02P03, w e must have approximately 



4ft' - 10ft/ + 6ft' + 2 _ 4ft- 10ft + 6ft + 2 _ 

 ft'-4ft' + 3ft' + 2 ■ ft -4ft + 3ft + 2 ~ n - 



We may therefore use 



B =V 



(4/8,'- 10 /3 / + 6/8/ + 2)(4/8„- 10 A + 6/9, + 2) ,,..,. 

 08,'-4A' + 3/3 1 ' + 2)08,-4/9, + 3/8 1 + 2) ' ^ S; 



The other constants can be determined as before, or if we 

 are at an early stage of the calculations convinced of the 

 symmetry of the distribution, the heavy work of calculating 

 the product moments may be omitted and the whole solution 

 carried out by Pearson's method. 



It may be observed that the solution by marginal mo- 

 ments only can be used even in unsymmetrical cases if (6S) 

 is approximately true. On the other hand, the higher 

 moments being subject to very high variations in individual 

 samples, a more accurate fit is to be expected from the use 

 of p2i*Pi2,Pu than, from ft, ft' which depend on p sg , /> ur > 

 and which replace these moments in the " marginal " 

 solution. 



By whichever of the above methods n is determined, its 



probable error, since it depends on the higher moments (the 



fourth or fifth), is much greater than the probable errors 



r v 

 of - , — in the determination of which no moment higher 

 n n 



than the third is employed. The quantities g, c, c' are 

 derived from n and are subject to similar variations. 



§9. Another solution involving no moments higher than 

 the third can be obtained for distributions of discrete 



