UNDER THE FORM OF AN IMPROPER CONTINUED FRACTION. 
473 
■P^ QQ 
If we call the ith simplified denominator to theSturmian convergents to Di(^), 
and if we call the ith simplified quotient Xi(a7), we have 
If we construct the numerators and denominators of the convergents to 
_J. l_ j_ 
Q.J GI2 Gig Glj 
according to the general rule for continued fractions as functions of Qi, Q2, Q3, &c., so 
that calling the denominators Aj, A2, A3, &c. A;, 
we have 
Aj — Q, A2 — Q1Q2 — 1 
Z 2 rj 
— rji rj2 ry2 
Z -_ i . Zi _ 3 ... Z ( i ) 
A, = QA-1 
being in fact the multiplier of f'x in the equation which connects fx and f'x 
with the z— 1th complete residue, and consequently retaining Q(a?) to designate the 
complete fth quotient, we have 
Z 2 r74 fyA ry 
( 0+1 
Z 6 ryS ryS 
i-l 
Z 2 ry^ ry% 
A,)}, 
which equation gives the connexion between the form of any quotient and that of the 
immediately preceding convergent denominator of the continued fraction which ex- 
fx 
presses 
Art. (g.) I have found that the coefficients of the n factors of fx in the expression 
above given for the quotients possess the property that the sum of their square roots 
taken with the proper signs is zero for each quotient except the first (the coefficients 
for thefirst being all units), i. e. Di/i,+Di^2+**-IlA=0 for all valuesof i except ^=L 
Moreover I find that the determinant formed by the n sets of the n coefficients of the 
factors oifx in the complete set of n quotients is identically zero, i. e. the Deter- 
minant represented by the square matrix 
r 1 1 1 1 ' 
{B.hf ...{B.KY 
YD,hY {Tt.hY ...{Tt.KY I _o 
V(^n-AY{^n-X2Y{^n-AY-A^n-Xf J 
Art. {h.) It should be observed that U* is the form of the simplified quotients for all 
the quotients except the wth {i. e. the last), for which the simplified form is not U„, 
but which arises from the circumstance of the last divisor, which is 
the final Sturmian residue, not containing x; it being evidently the case that the division 
3 Q 2 
