631] 
3 
631. 
SYNOPSIS OF THE THEORY OF EQUATIONS. 
[From the Messenger of Mathematics, vol. v. (1876), pp. 39—49.] 
The following was proposed as one of the subjects of a Dissertation for the 
Trinity Fellowships : 
Synopsis of the theory of equations; i.e. a statement in a logical order, of the 
divisions of the subject and the leading questions and theorems, but without demonstrations. 
In the subject “Theory of Equations,” the term equation is used to denote an 
equation of the form x n —p 1 x n ~ 1 + ... ±p n = 0, where p 1} p 2> .., p n are regarded as known, 
and x as a quantity to be determined ; for shortness, the equation is written f(x) = 0. 
The equation may be numerical; that is, the coefficients p x ,p 2 ,.., p n are then 
numbers ; understanding by number, a quantity of the form a + ¡3i, where a and /3 have 
any positive or negative real values whatever; or say, each of these is regarded as 
susceptible of continuous variation from an indefinitely large negative to an indefinitely 
large positive value : and i denotes V(~ !)• 
Or the equation may be algebraic; viz. the coefficients are then not restricted to 
denote, or are not explicitly considered as denoting, numbers. 
I. We consider first numerical equations. 
A number a (real or imaginary), such that substituted for x it makes the function 
x n — p 1 x n ~ l + ... ±p n to be =0, or say, such that it satisfies the equation, is said to 
be a root of the equation ; viz. a being a root, we have 
a n —p 1 a n ~ 1 + ... ±p n = 0, or say f(a) = 0 ; 
and it is then shown that x — a is a factor of the function f{x), viz. that we have 
/ (x) — (x — a) f {x), where f (x) is a function x n ~ l — q x x n ~- + ... ±q n -1, of the order n — 1, 
with numerical coefficients q x , q 2 ,.., q n -1- 
1—2