[ 367 ] 
XX. Tables of the Numerical Values of the Sine-integral , Cosine-integral , and Expo- 
nential-integral. By J. W. L. Glaisiier, Trinity College , Cambridge. Communi- 
cated by Professor Cayley, F.B.S. 
Received February 10, — Read March 10, 1870. 
It has for a long time been evident that the extension of the Integral Calculus would 
require the introduction of new functions ; or, rather, that certain functions should be 
regarded as primary, so that forms reduced to dependence on them might be considered 
known. 
Thus, in the evaluation of Definite Integrals, the three transcendents 
called the sine-integral, the cosine-integral, and the exponential-integral, have become 
recognized elementary functions, and great use has been made of them to express the 
values of more complicated forms. They were introduced by Schlomilch to evaluate 
Arndt also employed them in a similar manner about the same time. 
The first two functions had, however, previously received some attention from Bret- 
SCHNEIDER, who appears to have been led to their consideration by their analogy with the 
logarithm-integral, as in a paper in the 17th volume of Crelle’s Journal he announces 
his intention of not only tabulating this integral, but also of forming “ tabulas aliarum 
quarundam functionum, cum logarithmo integrali arete junctarum.” The Tables are 
published in the third volume of Grunert’s ‘ Archiv der Mathematik und Physik ’ for 
1843, and contain ten positive and negative values of the logarithm-integral, and ten 
values of the sine-integral and cosine-integral, besides tables of several other functions. 
The exponential-integral was introduced in its present form by Schlomilch, though 
for all real values it is the same 
between the two forms being 
li e x =VAx. 
The logarithm-integral appears to have been first discussed by Mascheroni f , and in 
1809 a work was published by Soldner at Munich concerning its theory, which also 
contained a Table of its values. This Table is reprinted in De Morgan’s ‘ Differential 
and Integral Calculus,’ p. 662. 
* Ceelle's Journal, vol. xxxiii. p. 316. 
f Referred to by Beetschneidek, Ceelle’s Journal, vol. xvii. p. 257. 
3 D 
MBCCCLYX. 
