SCALES OF NOTATION 43

From these equalities a;_; is the remainder got by dividing
gr—1 by 7y taking gi for quotient. Thus the hypothesis of § 92
is satisfied by the numbers ao, a1, @2, - -+, a» with go in place
of a. Therfor the conclusion of §92 follows and, putting

k=h+1, h l
Qo = 2. {alH (?‘m)}
m=0

1=0

Hence by the hypothesis ¢, = a.

Since ax—; is positiv or zero and less than 7, and

Qrk—1 = Qp—1 + G’x,

gr is the exact or lower approximate quotient got by
dividing its predecessor ¢— in the series qo, ¢1, @2, q3, **+ by
7w and a;—; is the corresponding positiv or zero remainder. § 81.

Hence, since g, = a, this set of integers, ao, a1, as, **+, a,
is the same as the set shown to exist in § 99.

Therfor there is only one such set.

101. The special case of the preceding two theorems when
1 =1, =r3= +++ = r gives us the following two theorems.

Th. If a is any positiv or zero integer, v an integer greater
than 1, if qr, where k = 1, is the exact or lower approximate
quotient obtaind by dividing its predecessor in the series a, qi, @z,
gs, =+ by r, if also ay_ 1s the remainder corresponding to the
quotient qi, then a quotient g,i1, where n = 0, will be arrived at
whose value is zero, this being the first quotient in case a = 0;
if we stop the process of division as soon as we obtain a zero
quotient, there is only one set of remainders, all positiv or zero
and each less than r;

if @ = 0, there is only one remainder, which is 0;

if a —= 0, the last remainder a, is equal to q, and therfor not
zero;
finally, whether a = 0 or a —= 0,
n+1
a=Y gartt=atar+arit- - +apar"t e
k=1
n+1

=2 GGt ¢ D =aur"+ apar" + oo fasr? +air +ao
k=1