1891-92.] Dr Muir on Convergents to ilio Boots of a Number. 19 
and 
(^7i+ir^+(-iro-ir+ 
r 
In conclusion, and in a line or two, it may be pointed out how 
the expressions for the coefficients corresponding to A and B in the 
higher cases may be found. Taking the case of + + 
and denoting the rational integral functions which these coefficients 
are of n by (^, we have 
(n^ + ni + 1)’'= cf)(n).7i^ + i 
and if in this we write anl for and multiply both sides by a, a 
being one of the prime cube roots of unity, there results 
a{ahis + an^ -f If = cf>(n).n^ + + ij/(u).a ; 
also, by writing ahi^ for n^, and multiplying both sides by a^, we 
obtain 
a\an^ + -h 1)'' = (fi(7i).n^ + ')fn).an^ + ip{n).a^ . 
Now from these three by addition x(^) and if/{n) are eliminated, and 
we determine 
2 n-1 
-1 2 71-1 
. -1 2 71-1 
-1 2 
. {m + Tis + 1 )^ + a(a^7is + a?il + 1 )’* + a^iam + + 1 )’' 

Similarly 
X(n) 
{ni -\-nl -{-ly + a^{a^ni + ard + !)’*+ a{a7ii + adnl -f 1 
3?^3 
'/'(«) = 
{rd + Til +!)’■+ (a%i + anl +!)’'+ [ani + ahil + 1 )’’ 
. 
and 
