672 DR J. HALM 
we plot the successive = as ordinates, taking the corresponding as abscissz, we find 
that, owing to the continuous change of sign, the = are represented as the corner-points 
of a zigzag line; and it can be shown that for sufficiently great values of x this line 
oscillates along a straight line parallel to the axis of abscissee, and at a distance from it 
equal to the exact value of the series, 2.e. to a sin (,/e log (1+ /2)). Wet ud 
NE 
demonstrate this for a certain value of c,e.g.c=24. We find 
2(2) 
1 | +1:00000 | 12 ~ 2°43133 
» | —sileee7- [13 |) a ireeont 
5) | 253 108e3) aie) o 10566 
4 | —4:31251 | 15 | + 1-59235 
5 | 38897 | 16s 84708 
6 ~3-94285 | 17 | +1:36108 
7 | 397938. |) 180 |) = eeso62 
g. |. —3:s5e75 ‘|. 19) 1) -Pi-i7409 
Q. | o167 1/205 es iearoer 
10 ~ 284399 | 21 + 1:02088 
11 |) +2-24871 | 92 ~ 133133 
the horizontal line are obviously much smaller, and if we repeat this process of forming 
arithmetical means, the fourth operation of this kind will lead to the following set of 
values for 24(a) : 
© 2,(2) | x 3, (2) 
13 —0:18840 17 —~ 018841 
14 ~0°18842 18 — 018844 
15 —~ 0-18840 19 ~ 018842 
16 ~0'18844 | 20 ~ 018844 
Now we convince ourselves, by computing the higher terms of the series for x > 22, 
that however far we may extend the calculations, the values of 2,(x) will always be 
found between the last two figures of the preceding table, and will more and more 
converge towards their arithmetical mean, viz.,—0°18843. But the exact value of the 
series should be 
<a sin (24 log (1+ 2) 
or, if the angle be expressed in degrees, 
Ae sin (J24 x 50-499 ) = — 0-188438. 
We see, then, that in this case the first twenty terms of the series are quite 
sufficient to find the value of our series with an error of less than one unit of the fifth 
