SINE-INTEGRAL, COSINE-INTEGRAL, AND EXPONENTIAL-INTEGRAL. 369 
coincided with those in the present paper, though from the great care he used, and the 
mode of verification he adopted, there was little doubt of their accuracy. One error 
was detected in Ei( — 5), which should be ‘00114 .... instead of -00144 .... This has 
doubtless arisen in the final copying or printing. 
Expressed in series the functions are: 
Si x=x—-. 
l 
3 1.2.3 5 1.2. 3. 4. 5 7 1 -2. ..7 
1 
Ci *= 7+i l0g,(O— . —A 
Ei *= T + Log, (x‘) + .r+ 1 . jA + 1 . _ 
3.4 6 1 . 2 ... 6 
yA 
+ • 
2.3 ' 4 1 .2.3 .4 
( 1 ) 
y being Euler’s constant 0-5772156 . . . 
From these expressions it is evident that Si#, Cix, and Ei# are connected by the 
relation 
Ei(a\/ — l)=Ci#+>/ — 1 Si x. 
The logarithms are written as above to indicate that they are real when x is negative, 
or of the form ci\/ — 1. The logarithm-integral differs in this respect only from the 
exponential-integral, for 
li^= y + hog e (# 2 3 )+# + l.^+l.^ 
2 . 3 
or 
1 . 2 . 3 . 4 
The series in (1) are clearly always convergent, however large x may be. 
The following series are easily obtained by integration by parts : 
c - nr [11.2,1 .2.3.4 1.2. ..6 
Si x= — cos ad-- 
2 \x x a 
y,3 
or 
x 
,7 
IT 1 .2.3 , 1 .2. 3. 4. 5 1 .2.. .7 , 1 
— sm x\-v — — 2 — H s — 1 • • • r 
[a.’ 2 x 4 x 6 x 8 J 
■ j 1 1 . 2 , 1 . 2 . 3 . 4 1 . 2 ... 6 
Lix= sina:-- - — =-4- s — = — - 
X X u 
■cos X 
{ 1 1.2. 3 ,1.2. 
[a ; 2 x 4 
3.4.5 1.2. ..7 
+ • • 
Ei,u= 
1 , 1 , 1.2 , 1.2.3 , 1.2. 3. 4 
( 2 ) 
These series are ultimately divergent, though for values of x greater than unity they 
begin by converging, and when x =17, seven places of decimals are obtainable from them 
for Si x and Ci x. 
Formulae (1) were used for values where the argument was less than 16, formulae (2) 
where it was greater. 
Tables I. to IX. were calculated in the following manner. The denominators of the 
terms in the series (1) were first computed, the first 20 figures of which, as far as the 
71st power, and the logarithms of their reciprocals, are given in the following Table. 
3 d 2 
