XLV XL VI] introduction cc\\x 



We may |.i / - <l_2,H)-2fe|l^* 



m, 



* 1 -" *> 



J?a, (1 TO, i) involves the tabulation of the incomplete B-function for a long aeries 

 of values of m between and 1, a difficult task, and one, as far as we are aware, 

 not yet attempted, but which we hope to undertake shortly. 



The use of Tables XLV XLVI is to obtain close approximations to 7,(n + 1), 

 where n is so considerable that it falls outside the Tables of tlie Incomplete B- 

 Function. The first method is to expand in terms of the even incomplete normal 

 moment functions. A table of these is given in Part I of this work, pp. 22 23, 

 up to Wio(tf), where the function m r (x) is defined by 



1 f * 

 m r (x) = -= 



V27T JO 



(viii), 



according as r is odd or even. In order to obtain I 9 (n + 1) to seven figure accuracy 

 it was found needful to table mw(x). This is provided in Table XIII, where TO U () 

 is also given. 



The formula for I (n + 1) is 

 I 9 (n + 1) = c WQ (#) C 4 w 4 (x) c 6 m 6 (x) + CgWs (#) + Ci Wio (x) c u m u (x) ... 



In this formula x = Vn sin 6, 



and the c's are provided by Table XLV for n= 100 to 400. They are functions of 

 l/n and its integer powers only. 



Illustration (i). What is the probability that an individual drawn at random 

 from the frequency distribution 



.Vo , v 



= 



y~ 



I 



1 + a' 



will lie between x= -faa ? Actually in a curve of the above type (i.e. (iii) above) 

 a? = (2m 3)a 2 , or for our particular case a= 12'974,596o-, and we are seeking the 

 chance that a single individual will lie between the limits 



1-297,460 x o-. 



Clearly tan = ^, and therefore sin = -r= = "0995,0372. Again the x with which 



to enter the Tables of the incomplete normal moment functions = Vn sin 0, which 

 is the value for formula (vi), i.e. in the case of I 6 (n + 1). Now n = 2m 3 = 168-34. 



Accordingly x = V2w - 3 sin = 1 2*974,596 x '0995,037 '2, 

 or this x for the normal moment functions 



= 1-291,02 



