_{1}

_{1}

_{1}

_{1}

_{1}

_{1}

_{1}of

_{1}

_{1}

_{1}

_{1}

_{1}

_{1}

_{2}

_{2}

_{2}

_{2}

_{1}

_{2}

_{2}

_{2}

_{1}

_{2}

_{2}

_{2}

_{1}

_{1}

_{2}

_{3}

_{3}

_{3}

_{3}

_{1}

_{2}

_{3}

_{3}

_{1}, n

_{2},

_{k}and real numbers $u$

_{1}, u

_{2},

_{k}that are members of the unit interval

_{1}, u

_{2},

_{k1}all positive

_{1}

_{2}

_{3}

_{k}

_{k}

_{1}, n

_{2}, n

_{3},

_{k}

_{k}]

_{k}

_{k}

_{k1}

_{k1}

_{1}, n

_{2},

_{1}, n

_{2}, ...] is called a

_{j}

_{j}

^{2}

_{1}, n

_{2},

_{1}

_{1}

_{1}

_{1}b

_{1}

_{1}

_{1}

_{1}

_{1}

_{1}gives

_{1}

_{2}

_{2}

_{2}

_{2}r

_{1}

_{1}

_{2}

_{2}

_{1}

_{2}

_{1}

_{1}, n

_{2},

_{1},

_{2},

_{r}]

_{1}, t

_{2},

_{r}with each $t$

_{j}

_{1}, t

_{2},

_{r}] is defined recursively by:

_{1}]

_{1}

_{1},t

_{2},

_{r}]

_{1}

_{2},

_{r}]

_{j}

_{2},

_{r}]

_{1}, t

_{2},

_{r}] is a rational number if each

_{j}

_{1}, t

_{2},

_{r}] is to be called a continued fraction, according to the convention of the first section, only when each

_{j}

_{1},

_{r}] suggests that the symbol should be computed in a particular case working from right to left

_{1}, t

_{2},

_{r}]

_{1}, t

_{2},

_{j}

_{j}p

_{j1}

_{j2}

_{0}

_{1}

_{j}

_{j}

_{j}q

_{j1}

_{j2}

_{0}

_{1}

_{j}

_{1}

_{1}

_{2}

_{1}t

_{2}

_{1}

_{2}

_{2}

_{3}

_{2}t

_{3}

_{j}

_{j}

_{j}

_{j1}

_{j1}

_{0}

_{1}

_{1}

_{j}

_{j}

_{j}

_{j1}

_{j}

_{j}

_{r}

_{r}

_{2}

_{1}

_{j}

_{j}

_{j}

_{r}

_{r}

_{r1}

_{r1}

_{r}

_{2}

_{1}

_{r}q

_{r1}

_{r}p

_{r1}

^{r}for each integer

_{r}q

_{r1}

_{r}p

_{r1}is the determinant of the matrix

_{r}

_{r}is the product of

_{r})

^{r}

_{r}

_{r}

_{1}

_{2}

_{r}

_{r}

_{r1}

_{r}

_{r1}

_{1}

_{2}

_{r}

_{j}

_{j}

_{j}

_{j}

_{1},

_{r}] is given by the formula

_{1},

_{2},

_{r}]

_{r}

_{r}

_{r}

_{r}

_{1},

_{r}] is defined by the

_{r}

_{r}

_{j}

_{1}

_{1}

_{1}

_{1}

_{1}and

_{1}

_{2},

_{r}]

_{2},

_{r}] must be equal to

_{2}

_{r}

_{1},

_{2},

_{r}]

_{1}

_{1}

_{1}

_{r}

_{r}

_{1}

_{1}

_{1}

_{1}

_{r}

_{r}

_{1}

_{1},

_{2},

_{r}]

_{1}, n

_{2},

_{j}

_{j}

_{1}, n

_{2},

_{r}

_{1}, n

_{2},

_{r}] formed with the first

^{th}

_{1}, n

_{2},

_{1}, p

_{2},

_{1}, q

_{2},

_{j}

_{j}

_{1}, n

_{2},

_{r}

_{r}

_{r}q

_{r1}

_{r}p

_{r1}

^{r}

_{r}

_{r}

^{r}

_{1}, n

_{2},

_{r}

_{r1}

^{r}

_{r}q

_{r1}

_{r}

_{r}

_{r}

_{r}

_{r}

_{r1}

_{r}

_{r}

_{r1}

_{r1}

_{r}q

_{r1}

_{r1}q

_{r}

_{r}q

_{r1}

^{r}

_{r}q

_{r1}

_{r}

_{1},

_{r}] where the

_{j}

_{j}

_{j}

_{1}, n

_{2},

_{r }

_{r}

_{r}

_{r}

_{1}

_{3}

_{5}

_{6}

_{4}

_{2}

_{j}q

_{j1}has infinite limit, and so the sequence of reciprocals $1q$

_{j}q

_{j1}must converge to zero

_{1}, n

_{2},

_{r }

_{r}

_{r}

_{1}, n

_{2},

_{1},

_{2},

_{r}

_{r}]

_{r}

_{r}

_{1}, n

_{2},

_{r}

_{r}] agree with those for the symbol $[n$

_{1}, n

_{2},

_{r}] except for the

^{th}

_{1},

_{2},

_{r}

_{r}]

_{r}

_{r}

_{r}

_{r}q

_{r1}

_{r2}

_{r}

_{r}

_{r}) q

_{r1}

_{r2}

_{r}

_{r}

_{r}q

_{r1}

_{r1}

^{r}

_{r}q

_{r1}

^{r}

_{r}q

_{r1}

_{r}q

_{r1}

^{2})

_{r1}

_{r}

_{1}

_{0}

^{th}

_{j2}

_{j1}

_{j}

_{j}

_{j}

_{j1}

_{j}

_{j2}

_{j}r

_{j1}

_{j}

_{j}

_{j1}

_{2}

_{1}

_{0}

_{j}

_{j}

_{j}

_{j}

_{j}p

_{j1}

_{j2}

_{0}

_{1}

_{j}

_{j}q

_{j1}

_{j2}

_{0}

_{1}

_{j}

_{1},

_{2},

_{j}]

_{j}

_{j}

^{th}

_{m}

_{j}

^{j1}

_{j}a

_{j}b

_{1}

_{1}a

_{1}b

_{1}

_{1}

_{1}

_{j}

_{j2}

_{j}r

_{j1}

_{j2}

_{j1}

_{j}

^{j3}(q

_{j2}a

_{j2}b)

_{j}

^{j2}(q

_{j1}a

_{j1}b)

^{j1}(q

_{j2}a

_{j2}b)

_{j}

^{j1}(q

_{j1}a

_{j1}b)

^{j1}

_{j2}a

_{j2}b)

_{j}(q

_{j1}a

_{j1}b)

^{j1}

_{j2}

_{j}q

_{j1})a

_{j2}

_{j}p

_{j1})b

^{j1}

_{j}a

_{j}b

^{m}(q

_{m1}a

_{m1}b)

_{m1}

_{m}

_{m}

_{m}

_{m}

_{m}

_{m}

_{m}

_{2}(

_{1}, n

_{2},

_{1}, n

_{2},

_{r}, n

_{r1},

_{r1},

_{1}

_{r}

_{r1},

_{1}, n

_{2},

_{r}, n

_{r1},

_{1}, n

_{2},

_{r}, z ]

_{1}, n

_{2},

_{r}, z ]

_{j}

_{j}

_{2}(

_{2}(

_{2}(

_{2}(

_{2}(

_{2}(

_{2}(

_{2}(

_{2}(

_{1}

_{2}

_{1}M

_{2})

^{1}

_{2}(

_{2}(

_{1},

_{r}, n

_{1},

_{r}, n

_{1},

_{r},

^{2}