and the general Unilateral Equation in Matrices. 457 



If, now, the vectors as well as the scalars of the two roots are 

 to be equal, it may be shown that the second as well as the first 

 coefficient of the skew- covariant must vanish. But this cannot 

 happen without the discriminant vanishing*; for it may easily 

 be seen that the discriminant of a binary biquadratic with its 

 sign changed is equal to sixteen times the product of the first 

 and last coefficient, less the product of the second and penul- 

 timate coefficients of its irreducible skew-covariant. Hence 

 when two roots belonging to the same pair of the given quad- 

 ratic coincide, two values of X become equal, and therefore all 

 four roots belonging to two pairs merge into one. 



Again, it is not possible for two roots belonging to two 

 pairs corresponding to two different values of X to coincide; 

 for in such case the expression given for x shows that pq, p, 

 q, 1 would be connected by a linear equation. But when this 

 happens (as has been shown by me elsewhere), the invariant 

 of the associated ternary quartic vanishes and the equation 

 ceases to be regular. Thus, then, it appears that it is impos- 

 sible for a single relation of equality (and no more) to exist 

 between the roots of the given equation when its form is 

 regular. So, again, it may be shown that it is impossible 

 for four, and no more, relations of equality to exist between 

 the roots. 



It need hardly be added, that the equation px 2 4- qx + r — 

 ceases to be regular when q or r vanishes. 



The reader may satisfy himself as to the truth of what has 

 been alleged as to the relation of the discriminant of a binary 

 biquadratic to the coefficients of its skew-covariant by simple 

 verification of the identity 



16(a 2 d-Sabc + 2b d )(e 2 b-3edc + M) 



- (a 2 e + 2abd - dc 2 a + 6b 2 c) (e 2 a + 2edb- 9ec 2 + 6d 2 c) 

 = 27(ace + 2bcd-c 3 -b 2 e-ad 2 ) 2 -(ae-Md + Sc 2 ) B . 



The biquadratic equation in X, Y is what the determinant 

 of Xp + pq + vr becomes when X 2 , XY, Y 2 are substituted 

 therein for X, //,, v ; so that we may say that (a, b, m, e,f )(x, l) 4 

 is the determinant of px 2 + qr + r, when x is regarded as an 

 ordinary quantity. Let (f>x be any quadratic factor of this 

 biquadratic function in x : I have found that <px = will be 

 the identical equation to one of the roots of the given equation 

 fx = 0, where fx —px 2 + qx + r. 



Between the two equations fx = 0, cf>x=0, x 2 may be elimi- 



* The first two coefficients of the skew-covariant vanishing implies the 

 existence of two pairs of equal roots and vice versd. This is on the 

 supposition made that a, the first coefficient of the given quartic, is not 

 zero. 



Phil. Mag. S. 5. Vol. 18. No. 114. Nov. 1884. 2 H 



