Arrangement of Successive Convergents in Order of Accuracy. 655 
Pi + \ ' - Pi = +1 
^ 1 
o-r ^1 
e. as x^^o — r > qjqi + i + r 
i.e. as r < (x^_^o - q/lqi + i) 
(Chrystars result, already referred to, follows easily.) 
We now observe that 
1 
^/ + 2 = «/ + 2 ,7- + • • • 
and - 1 4_ 1 + . . . 1. 
0/4-1 «/ «2 
In selecting the convergents nearest in absolute value to x, we seek to 
exclude those intermediate convergents which are further from x in absolute 
value than is lh + \lqi+i' i- e. the r's to be excluded satisfy the inequality 
Di > Do 
i.e. r< Ua^,, + ^ + • . . - + ^ . . . + M| 
The cases of a/ + o odd and Ov + o e^'^i^ must be distinguished. 
(1) ai + o odd = 2m + 1 (say). 
Since ^ + • • • and + . , . are both proper fractions, the 
inequality is satisfied by r = 1, 2, . . . ni only. 
(2) ai+o even = 2m (say). 
It is certain that values of r up to (ni — 1) are to be excluded. 
The value r = ni is to be excluded only if 
1,1 1^1 1 
i.e. if «./+;, < a/ + i 
or if + 8 — «7 + 1 ^11^ ^^^7 + i > «7 
or if a I + .5 = «7 + 1, cfv + 4 = C/, +5 < _ 1 
etc. 
The test is an easy one to apply in practice, for we are comparing 
partial quotients respectively right and left of a particular partial quotient. 
A difficulty arises when the comparison has to be carried so far that one 
of the partial fractions terminates ; this can be overcome by adding 00' s as 
partial quotients at the end, as many as necessary. Thus we may write 
