SYNOPSIS OF THE THEORY OF EQUATIONS. 
5 
[631 
it may be so, viz. 
contain the factor 
not a root of the 
equation. Similarly, 
en x = a is said to 
»sitive integer which 
ifferent from a, we 
a multiple one) of 
here /3 is a positive 
re original equation 
b y and so on. 
las in every case n 
i distinct, but they 
ted; the case is at 
ration. It is there- 
i oration has all its 
or more of them 
laginary parts, may 
i function with real 
maginary coefficients 
>ot a, this must be 
inary as real roots; 
naginary roots enter 
631] 
This is solved theoretically by means of a theorem of Cauchy’s, viz. writing in 
the original equation x + iy in place of x, the function f(x + iy) becomes —P + iQ, 
where P and Q are each of them a rational and integral function (with real coefficients) 
of (x, y). Imagining the point (x, y) to travel along the contour, and considering the 
P 
number of changes of sign from — to + and from + to — of the fraction j? corre- 
sponding to passages of the fraction through zero (that is, to values for which P 
becomes = 0, disregarding those for which Q becomes = 0), the difference of these 
numbers determines the number of roots within the contour. The investigation leads 
to a proof of the before-mentioned theorem, that a numerical equation of the order 
n has precisely n roots. 
But, for the actual determination, it is necessary to consider a rectangular contour, 
and to apply to each of its sides separately a method such as that of Sturm’s 
theorem; and thus the actual determination ultimately depends on a method such as 
that of Sturm’s theorem. 
Recurring to the case of an equation with real coefficients, it is important to 
separate the real roots, viz. to determine limits, such that each real root lies alone 
by itself between two limits l and m. This can be done (with more or less difficulty 
according to the nearness of the real roots) by repeated applications of Sturm’s 
theorem, or otherwise. 
The same thing would be useful, and can theoretically be effected, in regard to 
the roots of an equation generally, viz. we may, by lines parallel to the axes of 
x and y respectively, divide the plane into rectangles such that each (real or imaginary) 
root lies alone by itself in a given rectangle; but the ulterior theory, even as regards 
the imaginary roots of an equation with real coefficients, has not been developed, and 
the remarks which immediately follow have reference only to equations with real 
coefficients, and to the real roots of such equations. 
bs as to the number 
means of Sturms 
, (x) (a constant) of 
rein for x any two 
these functions how 
ty by other means, 
be real) coefficients, 
jinary) roots, viz. if 
root a + /3» as the 
•oot by such point ; 
estion is how many 
Supposing the roots separated as above, so that a certain root is known to lie 
alone by itself between two given limits, then it is possible by various processes 
(Horner’s, or Lagrange’s method of continued fractions) to obtain to any degree of 
approximation the numerical value of the real root in question, and thus to obtain 
(approximately as above) the values of the several real roots. 
The real roots can also frequently be obtained, without the necessity of a previous 
separation of the roots, by other processes of approximation—Newton’s, as completed 
by Fourier, or by a method given by Encke—and the problem of their determination 
to any degree of approximation may be regarded as completely solved. But this is 
far from being practically the case even as regards the imaginary roots of such 
equations, or as regards the roots of an equation with imaginary coefficients. 
A class of numerical equations which need to be considered, are the binomial 
equations x n -a = 0, where a, =a+/3i, is a complex number. The foregoing conclusions 
apply, viz. there are always n roots, which it may be shown are all unequal. Supposing