Elementary arithmetic, with brief notices of its history... by Robert Potts.

33
S,=S,+C=-S (1+ ),&c.
Take logarithms of all the equations and add the results.
XXI.
See Arts. 9, 11.
2. Let logc =x, logc =y, logab =z, by first eliminating a, b, c, and next, y and z
from the equations involving x, y, z, there results the equation 2x 3 - 3x2 - 6x +1 = 0,
or 2x(x22-1) +3x(x-I)- (x-1)=0. Of the three values of x, take x = (/33-5),
then y = (/33 + 1), and z-=(V/33 + 7), whence it may be shewn that the common
difference is 2.
3. Take three terms of the series, and shew that the second is a geometric mean
between the first and third.
4, 5, and 6 may be shewn to be true in the same manner as directed in the preceding example.
XXII.
1. See Art. 11, p. 9.
2. Since -x -=-1;.o-1-s x-+l
*lo -1 I_ xx1 log ( — l)= + loge /-1
and by effecting the expansions by Art. 14, p. 11, the required expression will be
found.
3. Expand loge(x +yN/ —1) and make the sum of the possible parts equalto a,
and the impossible parts equal to b.
4. The expansion of log (a + b/ —l) —, or of its equivalent
logea+ loge(l +- — 1)-', will consist of two series, one of terms involving the
symbol V/ —. It may readily be shewn that this part of the series is equivalent
to the expansion of 1/-1 loges/a2 +b2.
6. Since 1 +x+x2 --—. O
6. Since 1-a+-2= L —; log,(l-+-+-2a)=loge(l —;x)-loge(1 —x).
l-x3 1+x3 n1+x+x2 1-X3 1+x
7. Since 1 +x+2= —, and 1-+x2 — +: then +X+X2 1 + —X.,
and loge I -+X l-oe llog 3 + loge 1 L
XXIII.
1. Art. 17, note. 2. Art. 18. The rest of the examples require the application
of the Articles 17 and 18.
XXIV.
1. Art. 16. 2. Art. 18.
The following six examples require no remark.
10. See Section I., pp. 3 and 12.