163 
of Edinburgh, Session 1863-64. 
the penult and the antepenult, the terms of the progression would 
approach to some limit which, not being the root of a quadratic, 
may be the root of a cubic or of some higher equation ; such a pro- 
gression may be extended backwards, giving an approximation to 
another root ; and we may still farther complicate the recurrence 
by using four, five, or any number of fractions. 
While engaged in examining the nature of such progressions, 
and seeking for a demonstration of some general properties which 
they seemed to possess, I came upon a very simple theorem, which 
gives great facility in the search for the roots of equations. 
If we put an algebraic equation in the usual form— 
ax^-f . . . . , px-{-q = o, 
multiply each term by its exponent ; thus, 
nax'^-\-n~l , . . . _p + 0, 
and divide the expression so obtained by the original polynome, 
developing the quotient according to the descending powers of x, 
the resulting series takes the form 
or if we develop the quotient according to the ascending powers of 
X, and change the signs, we have, writing from right to left, 
+ &C. + Sx^ -f yx^ q- (3x^ -I" ax, 
and these two series conjoined make a duserr progression ; thus, 
+ &c. -i- + yx^ + fSx^^ q» aa? + n -f + Qxr-^ -j- J)x-^ •+• &c,, 
approaching on either side to a geometrical progression, the 
common ratio of which is a root of the equation ; that is to say, if 
we divide the coefficient of any term by that of the term to its left, 
we shall have an approximation to a root more and more close the 
farther we proceed along the series. The approximation on the 
right hand is to the root farthest from zero, that on the left hand 
to the root nearest to zero. 
If the equation 2x^ - llx'^-^lZx - 3 = 0 were proposed, we should 
form from it the expression 6ic3~22x2q-13a2-0, and, by division, 
thence form the progression 
^ 11 . 69 . 509 . 3937 . 30901 243657 
