SINE-INTEGRAL, COSINE-INTEGRAL, AND EXPONENTIAL-INTEGRAL. 
373 
Si 7 r was calculated both from Si 3 and Si 3.1, thus verifying the formula as well as the 
value of Si w. 
In all cases the values calculated from the difference formulae on the preceding page 
were verified to as many places as could be obtained from formulae (2). 
In Table XI. Si (xn) — is tabulated in preference to Si xr, as the changes in the 
function are thus rendered more apparent. 
The curves y=Si x, y=Cix, y=Ei x were easily drawn from the values in the Tables. 
It is thus seen how rapidly the two former curves become flattened, and it is worth notice 
that the radius of curvature at any maximum or minimum point is equal to the abscissa 
of that point. 
The point where the exponential-integral curve cuts 
the axis of x has the abscissa 0-37249680 . . ., in other 
words, this is the only real root of the equation Ei x=0. 
It is my intention to determine the points where the 
sine-integral and cosine-integral curves cut the lines 
with which they ultimately coincide, and with this 
view I have already determined the ordinates corre- 
sponding to points midway between the maxima and 
minima values as data for a first approximation. This 
will amount to finding the roots of the equations 
Sia’=-|x; Cia’=0. 
In Tables I. to IV., the work has been performed 
correct to the 20th place, and the last two figures have 
been finally rejected ; in Tables V. to IX., the twelfth 
place has been corrected throughout the work, and the 
last figure only rejected. In the other Tables two 
figures have been generally rejected. 
In Tables I. to VIII., all the results were verified as 
far as 7-figure logarithms were available, and in some 
cases 10-figure logarithms have been used. A very 
large portion of the work was done in duplicate, and every figure has been carefully 
examined either by my father or myself; in all cases great pains were taken to ensure 
accuracy, by independent methods as by logarithms, or in the way by which the values 
from 10 to 20 were verified. 
It may be mentioned also that differences as far as the ninth order have been taken 
of the numbers in Tables I. to IV., thus affording a rigid test of the accuracy of more 
than the first ten, and in the sine-integral of the whole eighteen figures. 
The Exponential-integral Curve, 
t/=Ei x. 
