XLV XLVI] Introduction ccxxi 



Clearly with n as high as 168-34, c u m w (a;) and CuWiuOr) contribute nothing to 

 the value required. 



Finally, substituting in formula (ix), there results 



l e ( n + 1 ) = I 6 ( 1 00-34) = -4034,3808 - -0002,3028 

 + -0000,0024 - -0000,0106 

 + -0000,0000 - -0000,0000 

 = -4034,3808 - -0002,4010 

 = -4031,0708. 



We must double this value to obtain the probability that a single individual on 

 random sampling will lie between + -faa. Hence 



P^ a = '8063,0506. 



Let us ask what the result would have been had we assumed that since 

 2 = 3-036,071, we might have used a normal curve. We have seen that the limits 

 are l-207,460<r. Hence from Table II of Part I we have \ (1 + a x ) = z x and 



2o = -001,4747, z l = -003,1 995, 



# = 746, = -254, 00 = -0315,8067, 

 and by the central difference formula 



^=002,7635. 



Therefore \a x = -402,7635, 



and Pi-2w,460a = a* = '805,5270. 



We see therefore that the leptokurtosis modifies the probability in the third 

 decimal place, the value of it being reduced. 



Dr Wishart in his paper has taken several cases of I 9 (n + l) when n lies 

 between 100 and 400 and found that the present Tables give results for /,(n+ 1), 

 checking to seven figures with those obtained by quadrature or by the expansion 

 of the incomplete B-function B a (\n + 1, ). The labour is considerable, as twelve 

 interpolations have to be made, six of them involving the use of second central 

 differences. Further the limits of n in the Table are somewhat narrow. Accordingly 

 Dr Wishart has provided a second table of functions 0o(#), 0i(#), </*(#), #s(#) and 

 04 (x) (see Table XLVI), which give the cos ^-integral by means of the equation 



cos* 0<20 



= ^ {0o (*) - i 0i (x) + ^ 2 (x) - 03 (*) + ^ 04 (*) - 



where p = n + 1 and x = 2 p tan %0. 



The great gain here is that the expansion may be used for values of p less than 

 100 even down as low as p = 9, provided that we are dealing with points withiu the 

 range of three times the standard deviation on either side of the mean. The actual 



