of the Hyper geometrical Series. 237 



is especially adapted for fitting various types of frequency- 

 distribution. The relative magnitude of r and n is, indeed, 

 often a very good test of the " interdependence of contribu- 

 tory causes." 

 If we put 



a=—r,/3=—qn,y=pn — r+l ... (2) 



and denote by F(a, /3, y, x) the general hypergeometrical 

 series 



1+ ^ + ^±l»+i)^ +&c . . . . (3) 



1.7 1.2.7(7+1) 



we see that our series is a hypergeometrical series of the type 

 F(a, j3, 7, 1), or, as we shall denote it F 1 (a, ^7), multiplied 

 by a factor, which we may write A . 



If the successive terms of a hypergeometrical series be 

 plotted up as ordinates at intervals c, and the tops of these 

 ordinates be joined, we obtain a great variety of polygons, 

 which approximate to the interesting series of generalized 

 probability-curves with which I have already dealt. The 

 advantage of the hypergeometrical polygons over the curves 

 consists in the knowledge as to the nature of the chance 

 distribution indicated by the discovery of the actual values 

 of p, q, n, and r. The curves, however, possess continuity 

 and are easier of calculation. Clearly a knowledge of a, /3, 

 y, since 7 — a— /3 — l = n, gives n, and hence g, r, and p. 



We shall find it convenient to write 



m 1 = a + ^, m 2 = a/3. . . . . . (4) 



It is not, however, only in the question of distribution of 

 frequency that hypergeometrical series may be of service ; 

 it seems extremely probable that the three constants #, /3, 7 

 of Fj (a, (3, 7) may be of service in indicating close empirical 

 approximations to physical laws, owing to the great variety 

 of forms that the hypergeometrical polygon can take. 



Before we proceed to the fitting of hypergeometrical 

 polygons to given data, we require to demonstrate one or two 

 general propositions with regard to such figures. 



2. On the moments of Fi(a, /3, 7). — Let A= the area 

 of the polygon, thus if the ordinates are plotted at distance 

 c, we have A = cxF 1 . Let fx s K be the sth. moment of 

 the polygon about its centroid-vertical, the elements of area of 

 the polygon being concentrated along the ordinates. Let v 8 'k. 

 be the sth moment of the ordinates about a vertical parallel 

 and at a distance c from the first ordinate, i.e. 



