ccxxiv Tables for Statisticians and Biometricians [XLV XLVI 



For the remaining </>'s we need use only (xi) and have 



X (a-) = -0961,8806 + '0000,5064 = -0962,3870, 

 < 2 O) = -0262,5465 + -0000,3535 = -0262,9000, 

 03 O) = -0149,7249 + -0000,2276 = -0149,9525, 

 < 4 (x) = -0064,0052 + -0000,2205 = -0064,2257. 



We have #i(a?)/15 =-0064,1591, 2 (a)/225 =-0001,1684, 



</> 3 (a;)/3375 = -0000,0444, 4 (#)/50625 = -0000,0013 

 Hence 



f cos 

 Jo 



r^ (-4810,3108 - -0064,1591 + -0001,1684 - -0000,0444 + -0000,0013) 

 lo 



?f x -4747,2770, 

 15 



/2-7T 



and since A / ~ = '6472,0864, 



V 15 



we have I " cos 15 0d0 = -3072,4787. 



re 



I COS 15 



Jo 



The correct value of the integral may be found by putting x = sin 2 0, when it 

 becomes 



o 

 and hence by expanding the binomial and integrating out we have 



f cos 15 0d0= -307 2,4784, 

 Jo 



or, only a difference of three in the eighth decimal place. But the method is better 

 than this, for we have not for a small p, $1 (x) exact enough by using only S 2 . If 

 we proceed to 5 4 , we have, for ^ (x), 



z Q = -092,3170, *! = -097,4425, 

 8% = -5381, S 2 ^ = -5549, 

 8% = + 214, 8 4 ^ = + 189, 



which by (xv) gives fa (x) = -0962,3904 instead of -0962,3870. We have thus 

 fa(x)jp = -0064,1594 instead of -0064,1591, 



and thus f * cos 15 6d6 = -6472,0864 x -4747,2767 



Jo 



= -3072,4784, 



agreeing exactly with the correct value. The reader must therefore bear in mind 

 that, if extreme accuracy be required, it is advisable for low values of p to proceed 

 with the interpolation formula to 8* for </>i(#) as well as for </>o(#). It is not 



