368 
ME. J. W. L. GLAISHEE’S NUMEEICAL VALUES OF THE 
The sine-integral and cosine-integral occur in a Memoir by Bidone in the Turin 
Transactions for 1812, where they are expanded in series, the same as those marked (1) 
on the next page. 
From the moment of its introduction the logarithm-integral excited considerable 
interest, but it is only in the last twenty-five years that the other functions have become 
of importance. A complete list of all the memoirs in which these functions are consi- 
dered is given by Professor Bierens de Haan, on page 83 of his ‘ Supplement aux tables 
d’integrales definies,’ published in the tenth volume of the Transactions of the Royal 
Academy of Amsterdam ; and in the second volume reference is made to several other 
works in a memoir by the same author. 
Since 1845 the three integrals have been practically regarded as primary functions in 
the integral calculus ; and how well suited they are to this purpose is evident from the 
success which has attended the labours of those analysts who have sought to reduce more 
complicated integrals to dependence on them. 
Professor De Haan, in the fifth volume of the Amsterdam Transactions, has evaluated 
a very large number of integrals by means of them ; and in the great Tables* of the same 
author there are given nearly 450 functions dependent for their evaluation on that 
of these integrals. Considering therefore their extreme importance as a means of ex- 
tending the Integral Calculus, and the probable value of many of the integrals evaluated 
in physical inquiries, it seemed very desirable that they should be systematically tabu- 
lated, so as to be known, not only by convention, but in reality; and on this subject 
Professor De Haan has strongly expressed his opinion of the value of such Tables. 
Bretschnetder, in the memoir previously cited, has computed Si a, Ci a, Eia, Ei( — a) 
for the values 1, 2, 3 ... 10 of the argument to 20 places of decimals (except the values 
for a = l , which are extended to 35 places). This Table is reprinted by Schlomilcii at 
the end of his ‘ Analytisclie Studien,’ and a portion of it is quoted by the same author 
at the end of a paper in the thirty-third volume of Ceelle’s Journal. 
The Tables given in the present paper are the following : — 
Tables I., II., III., IV.— Si ay Cia, Ei x , Ei( — x) from a=0 to a=l at intervals of 
*01 to 18 places of decimals, with differences to the third order. 
Tables V., VI., VII., VIII. — Si ay Ciay Eiay Ei(— a) from a=l to a=5 at intervals 
of 0T to 11 places of decimals, with differences to the third order. 
Table IX. — Si a, Cia, Eia, and Ei( — a) from a=5 to a = 15 at intervals of unity, 
to 11 places of decimals. 
Table X. — Si a and Cia from a=20 to a=100 at intervals of 5, from a=100 to 
a=200 at intervals of 10, from a=200 to a=1000 at intervals of 100, and for several 
higher values of a to 7 places of decimals. 
Table XI. — Maxima and minima values of Si a to 7 places of decimals. 
Table XII. — Maxima and minima values of Ci a to 7 places of decimals. 
In the course of the work Bretschneider’s values have been verified as far as they 
* Nouvelles tables d’integrales definies, Leyden, 1867. 
