\ X V] Introduct Ion ex\\\\ 



Tht- riMjuiivd t ransformation is 



,., . *'3- 



, or - 



Now in! lies between and 1, and accordingly to obtain the probability integral 

 of the curve (i) we have only to add unity to the B-function ratio I x (\, (I -7/<i 

 and divide by two. 



Since mi can lie only between and 1, this involves the tabulation of /*($,(! mi)) 

 for small ranges of mi ; but this range of the B-function has not yet been adequately 

 computed, and we cannot at present provide a table of the probability integral of 

 the symmetrical curve (i). 



Meanwhile, until the required table be provided, a good method to determine 

 ^*(i (1 m i)) is to use the formula provided by Soper* for the integral 



when p and q are small. 



We shall not consider further the probability integral of the curve (i). 

 For (ii) we have to make the same transformation, 



- _ 



' ~ a* 



and have 2 P*=| (1 +/*($, (1 +,))}. 



Table XXV gives the value of 



and accordingly we must take n= 2m 2 + 3 ; it runs from m t = $ to m^= 14. 



When m 2 = l4>, fi t = 2'818,182, and we are not yet close enough to the normal 

 curve (/3 2 = 3) to use its probability integral as anything but a rough approximation. 



For (iii) the requisite transformation is 

 x'* x 



or x = 



a -, > * * -- / o 9 i 



a 3 2 1 - x x* + a 8 2 



and we have $P X ' = | {1 + I x ( \, (w 8 J))} ; 



our table will accordingly give S P X ' from w 3 = 15'5 to w 3 = 2'5, or from ^=3*230,769 

 to 2 = The former value of /3 2 is still too far from fit = 3 to allow anything 

 but a rough approximation to be obtained from the normal curve. 



7/A 



If we choose our curve to be y = . /, a \/i 



\i ~T X ) ' 



as is frequently done, then n = 2m 3 , and 



or a = , __ if we take a = 1. 



Vn-3 



* Tracts for Computers, No. vn. pp. 2122, Cambridge University Press. See also the present 

 work under Table XLVIII (Introduction). 



9* 



