Resolution of Algebraic Equations. 277 
with opposite signs (=*= a, =±= 6 &c. ) ; that those whose terms 
on both sides the middle term are alike (which are generally 
called recurring equations) arise from pairs of roots, of which 
each pair contains a quantity and its reciprocal fa A, b, — &c. ] ; 
together with Maclaurin’s demonstration of the particularities 
of the coefficients when an equation has equal roots.* And the 
extent to which these notices might easily be carried, from 
observation of the effects of the different sorts of proportion, 
and all other relations, is prodigious. But my present con- 
cern is merely with the result, supposing from any means a 
relation to be previously discovered affecting any number of 
the roots. For example, — suppose, in the above given equation, 
(x” — px n ~ l -f qx n ~ 2 — rx n ~ 3 -j- sx n ~±&c\ = o), whose roots 
we called (a, b , c, d &c. we happened to know that two 
of the number ( a and b ) were equal ; then, since they might 
both be expressed by the same character, the («) roots of the 
equation might now be represented by only ( n -—1) distinct 
characters ; and therefore, of the subordinate equations derived 
from the construction of the coefficients, two might be employed 
to determine one root. ( a and b ) being equal, the equation fur- 
nished by the value of the coefficient (y>), and also that fur- 
nished by the coefficient (g), may be both together used to 
determine the same quantity. But, if any quantity (a) be a root 
of an equation, the simple equation (u: — a — o) must be a di- 
visor of that equation;^ therefore here (x — a) must be a 
common divisor of the two equations furnished by [p and q ). 
* Vide Ma cl a t/rin’s Algebra, chap. iv. p. 162, et infra. 
+ Vide Sanderson’s Algebra, Vol. ii. p. 679, 680, Art. 432, and all algebra; 
on the method of divisors. 
O o 
MDCCXCIX. 
