634 MESSRS A. T. DOODSON, R. M. CAREY, AND R. BALDWIN, THEORETICAL 
the multiplications was checked by differencing. Then the Simpson integration 
was performed, giving 3l , 1 (0, x), the work being checked at three points by 
the interpolation formula. A second integration gave 3 2 I 1 (0, x), and was checked 
as before. 
The next procedure was to multiply 3 2 I;i_(0, x) by w(x), and this was checked 
by differencing ; errors might have arisen either in the multiplication or in the 
copying of 3 2 I 1 (0, x) from the calculating machine. This differencing was essential, 
as the interpolation formula only checks the summation process arid does not 
check clerical errors. 
Successive application of the integration yielded ultimately 3 2w I„(0, x) and 
3 2n_i r„(0, x), each multiplied by powers of 10. 
Similar processes were carried out in the e-system. 
The values of these functions for x = 1 in both systems were then extracted, 
and since in all cases interpolation showed slight differences between the odd and 
even series, the final values were corrected so as to distribute this error equally 
between the two series. The values so obtained can be written as k -, T„( 0, l) and 
K~ n I' n (0, l). The factor 10 15 tT 3 in w(x) has also to be accounted for, and also the 
factor 3 2 in the integration process. The values obtained for I' n (0, l) should be 
multiplied by 3 if they are to be written in the form 0, l). Allowing for 
powers of 10 introduced in the integration formula, and also for any powers of 
10 introduced to avoid an undue number of decimal places in the integration 
processes, we have ultimately 
/c=9~ x x 10 -16 x w 3 = 55 - 096 .... (26) 
The corresponding values of K _n I m (0, x) and *" n I' n (0, x) at intervals of 0T in x 
are given in Tables IV to VII for the western and eastern systems. 
7. Extrapolation Methods for the Larger Values of n. 
It was found, when the period equation (20) was formed, that too few values 
of I n (0, l) had been calculated, and as the extra terms required were of no great 
importance, extrapolation was used so as to avoid further integrations. The most 
useful procedure was to construct the ratio of «:"”I n (0, l) to ic 1} I }1 _ x (0, l) for suc- 
cessive values of n. When the ratios were differenced, extrapolation was easily 
carried out to the required degree of accuracy, and the resulting values of «~ n I„(0, l) 
are indicated by square brackets in the table of the next paragraph. In addition, 
this method of differencing provided a very useful check on the accuracy of the 
values of k“ 5 T„(0, l). 
8. Computation of k _? T„(0, 2). 
The evaluation of the components «- _n I„(0, l) and k _ 7 T„(0, l) in both systems 
having been satisfactorily performed, they were arranged as in the table below. 
