Manchester Memoirs, Vol. Ixvi. (1922), No. 1 5 



as ^iq -^|x the integrand tends to zero when ''^^5, and this is 

 true also when w = 5, though for a more subtle reason, i.e., on 

 account of the way in which fe tends to i. Thus, when n^ 5 

 the integrand is zero at both ends of the range. 



(5) As regards the actual work of computation, in applying 

 Simpson's or Weddle's rule the integration in the transformed 

 integrals, for Ii and I2 as well as for 6, was divided into ten 

 equal parts. Thus the integrand in (4.1) and (4.2) was 

 calculated for the nine values 9°, 18°, 27°, . . ., 81° of ^^^ the 

 end values (for 0° and 90°) being zero, as just explained. This 

 rendered it necessary to calculate 6 for each of these values of 

 )jo ; in so doing, cp was computed for the ten values o, o'l, 0*2, 

 . . . , o"9 of z. AH this was done for five different values of n, 

 viz., 5, 7, 9, II, and 15, it being convenient to choose odd 

 values. 



For each of these values of n and ^oj the limiting value of 9 

 for ^ = was calculated by means of (3.9) ; the logarithm of the 

 last factor in {3.9), used in this calculation, was also tabulated 

 separately, as it occurs again in the integrands of (4.2) and 



(4-3). 



In computing cp for other values of z, the expression used 

 was (c/. ^.%, 3.7), 



(5.1) cp = 2{ \-{sm\x^lsinx^Y-^{cosy^^jcosVi^Y]-h 



^z (i -cos^[L)~^=2/sin\L, 

 where 



(5.2) cos^ = { {sinXxo/sinyoY-'^ (cosxo/cos\xoY }* 



These formulae are suited to logarithmic calculation ; thus 

 (5-3) log- cos^= ^^ (log. sinXxo - log. sinxo) 



+ (log. COSXo - log. COSXy^,), 



which is the more readily calculable since ^ (n-i) is a small 



whole number ; a single reference to a table of logarithms of 



trigonometric functions then suffices to give [l and log. sin\K, 



for use in the formula 



(5.4) log. 9 = log. z - log. sin^i.. 



Seven figure tables were used in these computations, but in 



order to obtain log. cp correct to five significant figures (as 



desired), only five figures were read out, except in dealing with 



the smaller values of z, i.e., o'l, o'2, and o'3- 



The graph of the function cp with respect to z was a smooth 

 curve to which it seemed legitimate to apply Simpson's or 

 Weddle's rule for numericalquadrature ; as an example, the 

 computed values of cp are given in the following table (to three 



