453 
EXPRESSED IN TERMS OF THE ROOTS OF jx. 
but I believe that it has not been observed that this is only the extreme cases of a 
much more general equation, viz. 
rrii-, li—rtii 
where ^. 2 -, ••• (^i denote respectively the denominators to the convergents to the 
continued fractions formed with the quotients taken in a reverse order, i. e. the con- 
tinued fraction 
— ^ 1 1 
9'i+ §' 2 - 2 + $'1 
This is easily proved when f=l ; is of course (as usual) to be considered 1. So 
more simply for the improper continued fraction, 
k_ l_ I J_ 
^~qi- 
of which the convergents are supposed to be 
h. h. t 
and the reverse fraction 
1111 
5'i ^i-1 gi’ 
of which the convergents are supposed to be 
^ 2 ’ 
we have the more simple equation 
li . = 0. 
And it is well known, or at all events easily demonstrable, that 
4-1 i_ 1 1 2 
4 qi-i- ?i-2’'V/2 
2. 1 t 1 1 
mi ~qi— qi-i— qi-i"'q^ qi 
Art. (32.). If now. we use subscript indices to denote the degree in x of the quan- 
tities to which they are affixed, we have the general syzygetic equation 
Kr„_,._X— K4_,-_,.9„-f-K2=0, 
where K, a constant (which I have given the means of determining in the first 
section), being rightly assumed K.r„_,_j, become the numerator and deno- 
minator respectively of one of the convergents to^, expressed as an improper continued 
fraction, and K2- becomes the denominator to one of the convergents to or. 
* See Ijondon and Edinburgh Philosophical Magazine, “ On a Fundamental Theorem in the Theory of 
Continued Fractions,” October, 1853. 
