Ill] Introduction xxxi 



The area of the tail of the normal curvi * 



A a = A \(\-a^ = A^z,f t 



A 



but y x = -z* = y a . 



<T 



Hence A a =>y a <rffl* (i). 



It remains to determine the <r and x of the auxiliary curve. I)ittTentiating 

 the normal curve twice we have 



x ayd | yd' ^ , x* 



or - , and ! *=r- - = 1 -- ^ . 



a- y a y a <r a* 



Hence we deduce <r-. ...(ii), 



Jj. '2 _ u " 



v y Maya 



and *' = * = ...(iii). 



- 



The latter equation enables us to find x' and so $4 from the table, the former 

 gives us cr, and then substituting in (i) we have A a as an approximation to the 

 tail area. 



The process must be modified when the frequency curve is replaced by a 

 discrete series such as the binomial or hypergeometrical. We then proceed in 

 a manner to be described later. 



Illustration (i). Suppose we require to find the integral of the incomplete 

 F-function, i.e. 



I(a >P )=t V P- 1 e 



J a 



where a is large. Here we have y = v p ~ 1 e~". 

 Hence by differentiating twice (ii) above gives 





x a 



i ,--\ 



and (111) a; = - = . , 



" v p 1 



and thus by (i) A a - af~ 1 e~ a -~ 



v 



1 j>"~~ t 



and accordingly I(a,p) = ^ i - r -\ ^ 



1 (P) vp1 



* x' is written for the x\a of the auxiliary curve. 



