108 Samuel Karp, Jack Kotik, and Jerome Lurye 
The second integral on the left of (B13) is given by (B6) with i=n+1. The first in- 
tegral in (B13) can be written (since sin 8 ,, = 1) 
Bnet Bn F 
|, H(B')Y, 5 |1 - sin e'\) dBi | H(B') (Fl - sin 6'|) dp' 
1 
Bn+1 F 
¥ |, me'y¥,($ - sin 6'|) dp’. (B14) 
n 
The first integral on the right of (B14) contains no singularities and extends over an 
odd number of subintervals; we therefore evaluate it by the combination of the trapezoidal 
and Simpson’s rules. The second integral must be specially dealt with because the first 
derivative of 1- sin B' with respect to 8B’ vanishes at B' = 6,,, = 7/2. If we are to main- 
tain consistency in order of accuracy to which the integral is evaluated, we cannot merely 
replace H(B') by H, 4, in the interval B, <B'< B41- Rather, a linear approximation to 
H(B') must be employed, viz., 
: : Hae a H, 
H(B') %H_ + Bisa aS GS ere) 
n 
= H+ yy H(A + -F) (BS BY < Byyy)- BD) 
Upon substituting from (B15) into the integral and expanding Y, about the point 
B' = 7/2, we obtain the following approximation: 
Bnei 
: ’ F ‘ , ; alte y' Fr? a2 
j, H(B DY. (Fla - sin B ) dB © On toe 22) 1 = a: 
n 
H " Fr2 2 
AS | jigs (—)- 3-— (B16) 
2n 32n2 288n2 
Addition of this result to the first integral on the right of (B14), followed by substitu- 
tion from (B14) into (B13), gives the final algebraic equation for the quantities H;. This 
equation corresponds toi =n+1. All our calculations were made with n = 6. 
Having obtained g,(x) the computer finds 
I cf) = | 6,(x;f) dx 
