654 
Transactio7is of the Boyal Society of South Africa. 
I give below a proof of tlie rule and a method of arranging the con- 
vergents in one set so as to show the nearest in defect, the nearest in excess 
and the nearest in absolute value, satisfying the stated condition, 
§ 2. Consider the simple continued fraction 
11 1 ^ 
^ a.2 a.^ as 
and let the successive principal convergents be pjqi, p^/^o, • . pslg_>i • • • 
Arrange the sets of quantities : 
'^IhlhllvP-Jl:^ (A), 
llO,V.Jq.„V,ll, ■ ■ ■ ■ ■ ■ (B), 
and between any successive two 2^/1 (Ji and pi + 2I9.1 + 2 interpolate the set 
p! + P'+i ^ P' + 2f>! 4. 1 ^ _ Pi + (ofy + i - l )l^/ + i 
2/ + 5/ + 1 ' ^/ -t- + 1 ' ' ■ 2/ + (^/ + 1 - 1) + 1 
(intermediate convergents ) . 
When the sets are completed they give fractions increasing in com- 
plexity and respectively increasing and decreasing in magnitude, which 
continually approach x. These sets provide solutions to the problem of 
finding the fraction of given complexity (i. e. with denominator less than a 
given number) nearest in defect or nearest in excess to oc. A doubt is left as 
to which of the two is nearest in absolute value. 
Of the two fractions here mentioned it is known that one (at least) must 
be a principal convergent.* The final problem is thus restricted to a com- 
parison between a principal convergent of one set and an intermediate 
convergent of the other set. 
For precision let p>ilqi and jji + ^/qi + 2 be two successive principal odd 
convergents ; pi + ^/qi + i is a, principal even convergent, and the question is 
whether pi^ijqi^i or {p)^ ^P t ^) i {qi f ^'^z + i) is nearer x, where r can 
have any integral value from 1 to a/^2~ 1- 
Let Xi + 1 be defined by 
1 ,1,1 
X — a^ + -f . . . + ^- + 
so that Xi = a/^., + 
« 7 4- 3 
Then x = (pi + x^^o ' Pi + i)Kqi + ^1 + 2 ' ^i + i) 
Also, since I is supposed odd, 
i^/4i/^?/ + i > X > (p, nh+i)Kqi + nt-^i) 
The differences and Do of these fractions from x are 
i>i = ^ - (p/ + nh+dl{9.i + 
_ Pi + -Xz + o'Pl+l _ Pl+^'^Jli+l 
qi -t Xi^^' qi^^ q, -+- rqi^^ 
^ ^1 + 2 — 
{qi -i-^i + iqi + i) (2/+ r^z+i) 
* Chrystal, loc. cit., pp. 421-2. 
