284 MESSRS. PROUDMAN, DOODSON, AND KENNEDY ON THE DIFFRACTION OF 
Checks were devised to secure accurate addition, and no errors were afterwards 
discovered in this part of the work. 
Tables* I. to VIII. were then constructed by taking anti-logarithms. This was a 
step the accuracy of which was not easily checked. The method used was to repeat 
the work with a different book of logarithms and under different conditions so as to 
avoid “ repetition ” errors. Only one or two errors were afterwards found in this part 
of the work. More errors were made in the summation of odd terms and of even 
terms of the series, the results of which are given at the feet of Tables I. to VIII. 
Only five or six, however, which escaped immediate detection by checking, were 
afterwards found in Table X. How these were detected will be described in the 
section on analysis of results. Except in the cases of kol = I, 2, one person computed 
A, B, 0, D and the other A', B', O', D'. The final work of combining these functions 
was done separately by each and the results compared. 
It has been the practice throughout the work to use more figures than were strictly 
necessary for the desired degree of accuracy. For example, many terms in the anti¬ 
logarithms are given to six figures, though only five-figure logarithms were used. 
This was found conducive to speed and cocuracy, but it is not intended that the 
tables should be regarded as accurate to the extent given, except when this is stated. 
The final results for Y,, Y 2 , Z ; , Z 2 are probably accurate to at least three decimal 
places, while Y^ + Y/ and Z^ + Z.f can safely be given to four significant figures, as is 
done in the tables. 
The derivatives cd', c, d were computed similarly. The gradients are probably 
accurate to the order given in the tables.! 
* The reason that 0 = 45° has been taken where a regular sequence would require 6 = 40°, 50°, appears 
from the account of the histoi'y of the work in § 1. 
f As stated at the end of § 1, these are not printed. The unprinted tables referring to this section are 
as follows :— 
Table IX. giving Ai + /xA' 2 - C 2 , A 2 + /xA'i - C'i, &c. 
terms of the series c. 
}J .») LO • 
3C'/3/x, 0D'/3 /x, &c. 
9Yi/3 p., 3y 2 /3/x, &c. 
3Yi/00, 0Y 2 /00, &c. 
0 AV + Y 2 *)/06>, 3 (Zj 2 + Z 2 2 )/00. 
XII. 
XIII. 
XIY. 
XY. 
XVI. 
XVII. 
XVIII. 
XIX. 
