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

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8. When (~-+) i.s cxpanded according to the form in Art. 11,
a( t) -11+X+~ +~ &C. - I + &C. +2 I +&= 5
— 1+ 2 12 +- XI + &-.c. + 2 x
| 2 +- 1.2 3&. }
X.-2 PxIX X." ~
XVI.
1. Transpose y to the left side of the equation and square each side. If cx=2,
then y =; therelure y must be less than 4 in order that ex may le less than 2.
2. First find the value of x, next the value of 1-x.
3. Let x be the common ratio; then c —, and -, also =l 1- -- -,' 1-^ cca 1-x c c
ands -a ) S1 -.). - ".S a s S- - ".
1-a a 1-x C6 Cc c c o
IHence 1-1_a(1-= and log (i —) - log (i-).
4. (1) Let x, y be respectively the first terms and common difference of the
arithmetic series, and the first term and common ratio of the geometric series.
Then x+(p —l).y=a, x+(q-l)y=b, x+Q('-l)y=c:
also xyl -1 = a, xyq-1 = b, xy-1 = c.
From these equations respectively, both expressions may be found.
5. First (logey)x=x, then logey=x', and loge(logey)=-~g, also y=e"'
x-1
next (logey)' -x, logry-=ac, (log,)^e -l=T1 =.
6. See Art. 9, and eliminate logax, loggy, ancd logcz
7. First, taking the first and second expressions for the first equation. Clearing
from fractions, it becomes x log y(y + z- x) = y I og x(z + x - y),
whence may be found -/ lo(yy x) (1)
log 1 z log (I.'y')Y
Next, taking the second and third expressions for the second equation, and
reducing as before, there results
log xy log( (zyy)y (2)
lo gy: log (zyyz)z
Thirdly, taking the first and third expressions for the third equation, and from
it results
log x- log(zx'z)z (3)
log z,' -log (zx2x)
clultiplying these three resulting equations
lo(yx, logzxy logx a2 log(ayT )X' log(__y')y log (X.')2
log x -' leog y' log z~Y log (yyy ') log', )z log (zx:lo og(.1 xx) log(z'sy)Y- log(xzx) 1
log (T,-); log (Yxay) lo log (;y —)=
log(y.ax:) — log(z'ly:!)y-, log(a-zg' z
log y(z-xZ)X iog, — an d 1 og — (-. ~
Therefore log(yx Y)x=log(^,,z)X, (y '2xy)x- (zzt), and yxY= r —%xz;
Similarly log (xsy)Y =log (yx.'1)Y, (x)Y-yz) =(? s.ay)y, and xyyz =.')JY;
and log (xzYx)z = log(y') ( xy), =)zz (z"y-), and xzx4 = zy'.
Wherefore xy=z =,:x:.= x.
8. See Art. 9, and assume logaN=x, and lhogbN=y.