1 1 1 I Introduction xxxv 



Whether a better result will be obtained by summing the first p terms of 

 ((1 x) 4- #) n or tne fi rst ( 1 terms of (a; 4- ( I x)) n by our present approximate 

 method largely depends on where the mode of the binomial is situated. We 

 must remember that if i be the greatest integer less than nu v, then the mode 

 of the binomial (u + v) n will be the (i + 1th) term, supposing u + v = 1. 



Now consider B. l2 (21, 81)/ (21, 81). 



The mode of the curve y = y Q a? s *(\ x) 90 is at *20 from the origin, and the 

 standard deviation of the curve is slightly less than '04. Hence if we place our 

 dichotomic ordinate at x = '12 it will be about twice the standard deviation from 

 the mode of the curve, i.e. in a position where Wishart's methods described later on 

 do not provide very good results. 



Here x = 12 and 1 - x = '88. Hence in the binomial ('88 + -12) 101 , the mode is 

 at the 13th term, and therefore within the sum of p = 2l terms. On the other 

 hand the mode of the binomial treated as (12 + -88) 101 is at the 89th term from 

 the start and not in the first 81 from the start. We therefore reduce the problem 

 to summing the first 81 terms of the binomial ('12 + '88) 101 by an approximative 

 method. 



The 81st term is u t , where t= 80, and is given by : 





or log us, = log T (102) + 21 log (12) + 80 log (-88) - log T (22) - log T (81). 



Using E. S. Pearson's Table of the F-function arid ten-figure logarithms we find 



MM = '00429,21344. 

 Then by formulae (v) and (vi) we have : 



n _ 80 /12\ _ _60_ _ /80 23 _ /920 

 y ~22 V88/ 121' ~ V 22 X 79~V 869' 



logio Q = T'695,3658,801, lo glo R = -01 2,3840,254, 

 logw QR = l- t T07,7499,055 6 , 2ca = - 2 



c = _ i = -67293,07108, = -05703,05448, 



- 2 



= Cl o- = 2-8178,4168, - = '23881,06882, 



<r 



= 7-9402,3173, o- = 41874,1727, 



= 63-0472,7993, ^ = -00325,24830. 



