of the Hypergeometrical Series. 243 



Thus, all the constants p, g, r, and n of the series (1) are 

 determined. 



The base unit c is given by (12), or 



e=w Je&=R (40) 



To obtain the successive ordinates of the hypergeometrical 

 frequency-polygon we must, if A be the total number of 

 observations, take the successive terms of 



1 + r qn + T(r " 1] - 9<gn-l) &c> 



pn — r+1 1.2 (pn — r+1) [pn— r + 2) 



multiplied by A, or 



Apn(pn — l)(pn — 2) . . . (pn— r + 1) 

 c n(n — l)(n — 2) . . . (n — r + 1) 



The position of the first ordinate is at a distance d^v x —c 

 from the mean (or centroid- vertical) of the series, i. e. 



d=cm 2 /n (41) 



Thus the solution is fully determined. Its possibility 

 depends on positive and real values for n and r, and for p 

 and q. 



As an illustration I take the following data provided for 

 me several years ago by members of my class on the theory 

 of chance at Gresham College. 



In a certain 18,600 trials the distribution of frequency was 



759 cases of occurrence, 



3277 



V 



1 



» 



5607 



>> 



2 



Dccurrences, 



5157 



>J 



3 



>> 



2701 



J) 



4 



» 



907 



» 



5 



it 



165 



JJ 



6 



>> 



24 



;■) 



7 



•» 



1 



r> 



8 



j> 







)? 



9 



55 







»» 



10 



?» 



Taking moments round the point corresponding to three 

 occurrences I find 



/V=- -501,4516, /V= 8-433,6021, 



/*,'= 1-815,5376, /*,'= -10-504,6774. 



^/= -1-948,5484 



