XLV \LV1J I ulruJm-linii rrxxiii 



Similarly Iroin (xi) a* adequate for tin- remaining <'* we find 



It is mm. ressary to find fa(x) and fa(x), UH by examining Table XLVI we see 

 that quantities of their order divided by //* and p* are not significant to eight 

 decimal places when p is l(j!)-34. Accordingly we ha\- 



'40:ilS,<i,S:i<j - -UOO-J.t.rj.H + 'OOOO/H U ' 



4020,0311. 



But Ie (169-34) = - " - GO x -4026,0311 



= -9970,4299 x c x -4026,0311, 



and Co for n=p I = 168'34 has already been found in Illustration (i) from 

 Table XLV to have the value 1'004,4477. Hence 



/ (169-34) = -4031,9797, 

 practically identical with the value found in Illustration (i). 



If second differences only are used for <fa(x\ and first only for </>i(#) and <fa(z), 

 when p is large (over 100), then we obtain six-figure accuracy. 



Illustration (iii). Let us apply Table XLVI to find the integral of a much 



lower power of cos 0. Required 



r30 



cos 15 6W. 

 Jo 



Here we are taking a very low value of p, and n = p 1 = 14 lies far outside the 

 range of Table XLV. Hence that Table cannot be used either to determine or to 

 check the value of the integral. We must use the second method, i.e. that of 

 Table XLVI. 



Now 



......... (xvi), 



where x = %Jp tan \ 6. 



In our case x = 2 v/15 tan 15 



= 2 x 3-8729,8335 x -267,9492 

 = 2-075,5256. 

 Therefore, for interpolation in Table XLVI, 



= -755,256, = -244,744, 00 = '0308,0740, 



^0$ (1 + 6) (! + </>) = -0033,6547. 

 To determine ^>Q(X), we apply (xv) and find 



</>o (x) = -4809,3983 + -0000,9145 - -0000,0020 

 = -4810,3108. 



