Series to Frequency Distributions in Space, 399 



The various terms o£ the double series are given by 



/ ^_rt.oQ 7 /ll-263\/ll-04\(13'399)(12-399)... 04-399-5-/) 

 <iO_5 897^ § ^ gt J (20 .2i)(21-21)...(i9-21 + , + O — ' 



[When r + r' is not an integer the value of Z(s, s') given in 

 § 7 is written in the form 



NQm)^ ( !)(/') (g"W 

 (n) r+ r , (jm — r — r' + l], +s ," 



The factor / \, +r can be evaluated by Gamma Functions 



•or by treating it as a constant which is to be determined by 

 •equating the total of the calculated values to N. Gamma 

 functions were employed to obtain 5*897.] 



In comparing the actual frequencies with the theoretical 

 frequencies deduced from this formula, we must not forget 

 that the starting points are different. 



In the annexed table the actual frequencies are given in 

 small type, the calculated terms of the series in large type. 

 They cannot be compared directly as the start of the 

 theoretical values is at ('187, "48 6) and the c, c' differ 

 slightly from unity. 



Some idea of the goodness of fit can be obtained by an 

 examination of the model of which photographs are oiven 

 in PI. VII. 



The white rectangles represent the actual frequencies, and 

 the black the calculated ones. The theoretical frequencies 

 interpolate with the actual ones in a very continuous manner. 

 This is particularly noticeable in diagonal sections where 

 theoretical and actual frequency ordinates are very nearly 

 coplanar. 



§ 12. When the starting point of the double series is 

 known a priori, the solution admits of much simplification. 

 Consider the case of discrete data, like the whist table, 

 where (with perfect shuffling) we would expect the starting- 

 point to be at (0, 0) and c = c' = l. 



Let m, m' be the means of the actual data. 



From (7) and (8) we obtain 



c(l + rq)=c + m } 



6 .'(l + r'?):=c' + m', 

 or rq = m, r'q = m' (74) 



