SCALES OF NOTATION : 37

If |gn| = |#m41]|, gn = = 7mi1 and therfor gny = + 1.
Then |gmi1| = 1 < |?my2| and therfor |gmi1| < |7my2]
Hence in this case also our theorem is tru and # = m 4 1.

96. Th. Ifa,ry, re, 73, ***, Qo, q1, G2, G3, *** are as in § 95,
then for some value of n, where n = 0, either g1 must be 0 or
the value 0 may be chosen for it.

By § 95 there is a value of # for which |¢.| < |#n41]

If this g, is zero, it is exactly divisible by 7,41 and the quo-
tient ¢,41 will be zero. § 60.

If this ¢, is not zero, since |g.| < |?n41], it is not exactly
divisible by 7,41 (§ 73) and either the lower or the upper
approximate quotient got by dividing g, by 7,41 will be 0.

We may choose the 0 value for g,41.

(In case @ = 0, the first quotient g is 0, but the first number
in the series qo, ¢1, g2, g3, *** to be 0 is not a quotient, but g,
that is, a itself.)

97. Th. Ifa,r, 7,73 -+, Qo q1, @2, G3, * * * are as in §§ 95,
96, Qny1, where n = 0, being the first zero quotient, and a,, a,
as, as, *-- are as in § 92; then all remainders are numerically
less than the corresponding divisors (that is, |ar—| < |ri|); all
quotients after g4y will be zero;

if a = 0, n = 0, that 1is, the first zero quotient is q1, and all
the remainders are zero;

if a —= 0, no quotient need be zero and an infinit number of
sets of remainders can be found, the remainder a, in each set
corresponding to the first zero quotient in that set being equal to
g» and therfor not zero;

and, if for convenience we set ro = 1,

lé‘:) {almliIO <rm) }

Q
Il

= Qg?o + Q17?1 + Qarori?s + Qstorirels + ¢+ o - QuPotifels cc c Ty

That all remainders are numerically less than the corre-
sponding divisors follows from § 80.