1 1 1 1 Introduction xxx\ ii 



But the area up to the mode is '4603,9845, hence the required area-ratio 

 ' -*)*><&/ (21, 81) --5005,3151. 



J-20 



Tin- true value is -5006,4116. 



The most difficult area-ratios for the incomplete B-function are those where 

 tin- dichotomic ordinate lies between 1*5 and 2'0 times the standard deviation from 

 the mode of the curve. In the present case the mode is at '20 and the standard 

 deviation slightly under '04. We are accordingly measuring in the difficult region; 

 the principal source of error is that the stump of the tail of our binomial starts 

 too near its maximum ordinate. 



In the above illustrations we have endeavoured to show the greatest accuracy 

 that can be obtained by this method, but for a large number of practical purposes 

 Table III will be adequate to provide the ratio (1 a Xj )/z Xl . 



(d) Sum of t + 1 terms from the start of the hypergeometrical series 



If the series be finite, or converging, then its sum = -,, \rWr - 7R. where F 



r(7-) r (7-p) 



denotes the complete F-function. Accordingly : 



aft q(a-H)0Q8 + l) \ 



'" 



F( 7 )F( 7 -a- 

 is a series the sum of which is unity, and if 



-- ... - ... ..... 



p F( 7 )F( 7 -a-/3) 



+ u tl ........................ (iv) 



will be the ratio of the sum of the first t + 1 terms of ^(a, j3, 7 , 1) to the total of all 

 the terms, or a "probability integral" of the hypergeometrical series ^ T (a,y3, 7 , 1). 

 To find the sum of the first ( + 1) terms of (ii) approximately, provided they do 

 not include the "mode" of the series, i.e. the maximum term*, we proceed as follows: 

 Compute 



I t y + t-la + t- 

 ~V t-l 



y + t- 



.(v). 



lu >-ilO <*?** * l "K10 -"' i J 



Ci= , - , -=-; -, whence - and cr 



Iog 10 e a 2 logioe o- 



and x\ = GI <r 



* The maximum term will be obtained by /,, when we give p the least integer value greater than 

 Y^^- If the t + 1 terms contain the mode, then we sum the remainder and subtract from unity 

 to obtain the desired approximation when the series is finite. 



