The  converging  fractions  are  2?  i>  |j  t^>  2?>  yV^>  !zVt*

Art.  351.  To  show  that  the  difference  between  any  two  consecutive
converging  fractions  is  always  a  fraction  having  -|-1,  or — 1,/or
its  numerator,  according  as  the  fraction  subtracted  is  in  an  even  or
odd  place.
'     1_     b    _ah+l—ah  ^     +1

a     ab-{-l      a{ab-\-l)  a(ab-\-l)

I     ^      lcJ[-\       _lc{ab+l)-\-ab---{ab+l)(l)c-\-l)
ab+\     c(^ab+\)-\-a  (a&+l)[c(a6-(-l)+fl]

_,  --1

(a^;+l)[c(a6+l)+aj*


CONTINUED  FRACTIONS.  309

To  prove  the  property  in  ^a  general  manner,  let
P    Q     R

be  three  consecutive  converging  fractions,  corresponding  to  the

,  three  integral  fractions  _,  _,  _,    Then
a    b    c'

P__Cl_PQ^— P^Q
P'    Q'         P'Q'     '


and


a_R _r/q---rq;  ,

Q'    R'     "TrTQ?       '


but  R=Qc+P,  and  R'=Q'c+P',  (Art.  350.)

Substituting  these  values  in  the  last  equation,  we  find

Q  _R  ^(Q^c+PQQ— (Qc+P)Cl^^P^Q-~PCl'
Q'    R'  R'Q'  R'Cl'      •

But  the  numerator  of  this  result  P'Q — PQ'  is  the  same  with  a

P     Q

contrary  sign  as  the  numerator  of  — — ~,  which  we  have  before

shown  is  +1.  Hence,  the  difference  between  the  numerators  of
any  two  consecutive  approximating  fractions,  when  reduced  to  a
common  denominator,  is  the  same  with  a  contrary  sign,  as  that
which  exists  between  the  last  numerator  and  the  numerator  of
the  fraction  immediately  following.
But  it  has  been  already  shown  that  the  difference  of  the  nu-
merators of  the  1**  and  2"**  fractions  is  +1 ;  the  difference  of  the
numerators  of  the  2"'^  and  3"^  fractions  is  — 1 ;  therefore,  the  dif-
ference of  the  numerators  of  the  S'"'^  and  4^  is  -\-l,  and  so  on.
