AM) IMAGINARY ROOTS OR EQUATIONS. 
59’ 
radius unity, and of points situated in pairs in the same radii at reciprocal distances from 
the centre. In a word, in each case we may say that the roots can be geometrically 
represented by points on a circle, and pairs of points electrical images of each other in 
respect to the circle, but the radius of the circle in the one case will be infinity, in the 
other unity. Conjugate like real equations will have all their invariants of an even 
degree real, and those of an odd degree will be pure imaginaries, or real quantities 
affected with the multiplier i. Their morphological derivatives (co variants, contra- 
variants, &c.) will be also conjugate forms. The whole doctrine of equations, as regards 
the separation of real from imaginary roots, and the determination of the limits within 
which the former lie, will reproduce itself with suitable modifications in the theory of 
conjugate equations, in which simple, on the one hand, and coupled or twin roots, on 
the other, will correspond respectively as analogues to the real and imaginary roots of 
the ordinary theory. Thus the following theorem may be demonstrated without diffi- 
culty, viz., in any conjugate equation the number of coupled roots is congruent to 0 in 
respect to the modulus 4 when the discriminant is positive, and to 2 in respect to the 
same modulus when the discriminant is negative ( 22 ). We see now how to interpret the 
( 2 ?) ( a ) A very simple linear transformation shows the immediate connexion between the solitary and asso- 
ciated roots of conjugate with the real and paired imaginary roots of ordinary equations. Ror if f(x, y)= 0 he 
a conjugate equation, writing 
y=v+iu, x=v — iw, 
f(x, y) becomes R(m, v), a real form in u, v. 
When u, v are real, we have 
y v+iu ( v\ . f v\ 
—— 1 — cosl tan -1 - l+tsml tan- 1 - 
X V — IW \ V W/ 
V 
when — =c+ iy, the two values correspond to 
y c+iy +i fy\' C—iy + i 
x~c+iy—i’ \xj —c—iy—i' 
Thus 
| : (|)' ::^+(y+if:^+(y-if; 
also 
y (y\ _$-l+y i +2ti 
x \xj c 2 — 1+y 2 — 2ci 
of which the modulus is obviously unity. 
(”) Now it is known that if t be the number of real, and r of imaginary roots in the real form, (u, v) n , its dis- 
*(*-D 
criminant, hears the sign (— ) 2 . Hence the sign of the discriminant of the conjugate form (x, y) n (since the 
determinant of v+iu, v — iu is 2i) will be ( — )?, where 
„_»(»-!) , t(t 1) (i + r)(J — 1 + r) +£(£ — 1) _ „ , . . 
^ 2 + 2 2 —t(t—l)+tr+ g • 
r(f— ij 
Hence since t and t(t— 1) are both even, (—)?=(—) 2 , and the sign of the discriminant of a conjugate 
form is + or — according as the number of imaginary roots does or does not contain 4 as a factor. 
It must he remembered that the sign of the discriminant is not in general the same as that of the zeta or 
squared product of differences of the roots. The sign of the zeta for real equations follows precisely the same 
law as the sign of the discriminant for conjugate ones. 
4 l 2 
