596 
PROFESSOR SYLVESTER ON THE REAL 
In general, let 
[a-\-ia, b-\-ifi, c-\-iy, , c — iy, b—ifi, a — iafx, 0 
be an equation in which all the coefficients, reckoning simultaneously from the two ends, 
are conjugate to one another, and the central coefficient, if there is one, which can only 
be when n is even, real. 
Let fy satisfy this equation. Then evidently |=p— ig will also satisfy it; or, 
which is the same thing, will satisfy it. 
Now either this root will be identical with the former one, or a distinct root ; in the 
former case we must have y> 2 -|-y 2 :=l, and the root will be of the form cos a-\-i sin a ; in 
the second casejp 2 +# 2 will differ from unity, and there will be a pair of imaginary roots 
of the form £>(cos sin cc), -(cos a-j-i sin a), in which the real parts g, ^ are reciprocal 
to one another, and the directive parts e~ ia identical. Moreover, if we write the given 
equation under the form U +«V = 0, and suppose, as can always be done, that U and V 
have been divested of any algebraical common factor, it may easily be shown that the 
equation so prepared, and which may be called a Conjugate Equation proper, can have 
no real roots and no pairs of imaginary roots in the sense in which that term is employed 
in the theory of equations with real coefficients ; but the distinction between simple or 
solitary and twin or associated roots reappears in the theory of conjugate equations, 
under a different form. It will of course be understood that the class of simple roots 
for which the modulus is unity is quite as general as that of twin roots, for each of 
which the modulus may be anything different from unity, just as in the ordinary theory 
the case of real is quite as general as that of imaginary roots, although the former may 
be represented by points on a fixed straight line, whilst the points representing the 
latter may be anywhere in the plane, this liberty of displacement being balanced, so to 
say, by the constraint of coupling. The general geometrical representation of the roots 
of a real equation is a system of points in a line, and a system of pairs of points at equal 
distances on opposite sides of the line. So the general geometrical representation of the 
roots of a conjugate equation will be system of points in the circumference of a circle to 
through A at a point very close indeed to the horizontal extremity of A. It may he noticed that when 
j>= — f , the smaller ordinates of R and A are each — gj, the ordinate of K and the larger ordinate of A being 
each — 
I have found the points of contact of K with A by actually substituting <p — <f, i. e. y>(p+l) 2 for a in A=0. 
This gives the equation 
2064p 4 + 7352p 3 + 9823p 2 +583% + 1296=0, 
one factor of which is 4p+3, dividing out which we have 
516p 3 + l-451p 2 + 1368p + 432= 0. 
The Newtonian criterion applied to the three first coefficients of the above gives — 1362^, showing that two of 
the roots are impossible ; the remaining real root I find to be ’8946, &c. It does not appear to be a rational 
number. 
