SINE-INTEGRAL, COSINE-INTEGEAL, AND EXPONENTIAL-INTEGEAL. 
371 
1 x 3, i 1 x 7 
3 ' 1 . 2.3 + 7 * 1. 2 ... 7 
1 x A 
y + log X + ~ • \23 
+ 
11 1 . 2 ... 
+ 1 
8 1 . 2 .. 
11 
. 8 
+ ... 
+ ... 
were formed, which, when suitably combined by additions and subtractions, gave the 
values of the functions. 
In Tables I. to IV. the highest power of x included was # 20 . 
In Table IX. # 63 was required for the value 15. In the intermediate Tables # 29 was 
the highest power included. 
The values of the functions when the argument was above 1G were calculated from 
formulae (2), the log sines and log cosines being taken from Taylor’s logarithms. 
As the formulae (2) are divergent, to remove every shade of doubt that might attach 
to their use, the functions for #=20 were computed from both formulae, and the agree- 
ment was perfect to the eighth place, which was as far as the second formulae could give 
correct results for this value of x. 
The calculation of the values from #=10 to #=20 was extremely difficult and labo- 
rious, owing to the great number of terms and high powers necessary to be included. 
The term involving # 76 was the first one rejected in the calculation for #=20; and to 
show how extremely unmanageable the formulae (1) had become, it may be stated that 
this value, calculated as before described, required the formation of about 22,000 figures 
exclusive of verifications. Great confidence may, however, be placed in the truth of 
these results (from #=10 to #=20), as they were also calculated entirely independently 
by deducing each term from its predecessor, and in addition the value of Ei (— #), which, 
on account of its extreme smallness, admitted of being obtained from formulae (2), 
served as a rigorous verification of the whole process, excepting the final additions and 
subtractions. 
The functions for #=20 were obtained correct to the 12th place, the values being 
Si20 = + 1-548 241 701 043 
Ci20 = + 0-044 419 820 845 
Ei20 = + 256 156 52-664 056 588 820 
Ei(— 20)=- 0-000 000 000 098. 
Having obtained the values for #=20, it was a matter of comparative ease to give the 
values of the functions for#=2 to a great many places; they are to 43 places as follows: 
Si 2 = + 1-605 412 976 802 694 848 576 720 148 198 588 940 848 5834 
Ci2 =+0-422 980 828 774 864 995 698 565 153 198 255 894 135 7378 
Ei2 = + 4-954 234 356 001 890 163 379 505 130 227 035 275 518 0536 
Ei(—2) = — 0-048 900 510 708 061 119 567 239 835 228 049 522 314 4922 
agreeing with Bretschaeider’s values to the first 20 places, which is as far as he has 
computed them. The value of y was taken from a paper by Mr. Shanks in No. 114 of 
the ‘Proceedings of the Royal Society.’ Bretschneider has calculated the functions for 
#=1 to 35 places. 
