656 Transactions of the Eoyal Society of South Africa. 
«2 = «o + CO • ■ * ' ^^^^ ^^^^^ purposes of the 
comparison we replace by 'X> . 
An interesting special case occurs when the two partial fractions compared 
are actually equal ; this requires that all the corresponding partial quotients 
be equal in pairs : 
i.e. au + i — au-i, ai,+^ = ak-c^ ... «2/;-2=%- 
In this case the intermediate convergent and the principal convergent 
under comparison are equidistant in value from x. The value of x is thus 
half the sum of these tw^o convergents, and we may write 
' ^ rto «/ '2m ai'-- a, 2 \q/^^ qi -\- mqi + ^ j 
a result which can be got by applying the properties of symmetrical con- 
tinuants.* 
The reasoning, used above for I odd, gives identical results for I even. 
§ 3. To obtain a convenient scheme of arrangement take an example 
2 + i + -i + i+ i + i. 
Write down the principal odd and the principal even convergents, and 
interpolate the intermediate convergents. We have the two sets, 
0/1; 2/1,(7/3), 12/5; 17/7,90/37; 163/67. 
1/0; 5/2, (22/9), (39/16), 66/23; 73/30, (236/97), (399/164), 562/231, 
725/298; 888/365. 
Certain of the intermediate convergents may now be removed as being 
further from the true value than is a principal convergent with a smaller 
denominator. 
In the set between 2/1 and 17/7 = 3, and we remove 7/3 only. 
In the set between 5/2 and 73/30 a^^-^ = 4, and we remove 22/9 cer- 
tainly and possibly 39/16 ; here o^+o is 2 and «^ is 3, i.e. ai+o < ai and so 
39/16 must be removed ; and so on. 
Those to be removed are enclosed by brackets in the above list, and the 
fractions left have the property that between two of them consecutive 
in complexity no simpler fraction can Ije inserted as near in absolute value 
to the fraction as the less complex of the two. 
The follow^ing arrangement in groups which end instead of begin with a 
principal convergent has advantages : 
Fractions in defect. Fractions in excess. 
0/1 1/0 
(1/1), 2/1 (3/1), 5/2, 
(7/3), 12/5, 17/7 (22/9), (39/16), 56/23, 73/30 
90/37, 163/67 (236/97), (399/164), 562/231, 
725/298, 888/365 
The method of construction is as follows : 
* Muir, Proc. Eoy. Soc. Edin., 1873-4. 
