and the general Unilateral Equation in Matrices. 455 



from zero. Let us construct the ternary quadratic 



aa 2 + 2buv + 2cuw + dv 2 + 2evw +fw 2 , 



defined as the determinant of up + vq + wr, on the one suppo- 

 sition, or by means of the equations 



a=Tp 2 , d = Tq 2 , /=Tr' 2 ; b=$p&q-$VpYq, 



c =SpSr-SYpYr, e=SqSr-SYqYr, 



on the other supposition. 



On referring to the article of May, it will be seen that the 

 solution of the equation may be made to depend on the roots 

 of a cubic equation in the quantity therein called X. When 

 fully worked out, this equation will be found to take the 

 remarkable form e Kn . 1 = 0, where I is the invariant of the 

 ternary quadratic above written, and Q, = 2a8 c —a8 d . It may 

 also be shown that 



(p + b — u)(q — c—u) 

 X ~ 2X ' 



where u is a two-valued function of X, and v a linear function 

 of u. 



I shall suppose that I, the final term in the equation in X, 

 differs from zero: the solution of the given equation in x will 

 then be what may be termed regular, and will consist of three 

 pairs of actual and determinate roots. When 1 = 0, the solu- 

 tion ceases to be regular ; some of the roots may disappear 

 from the sphere of actuality, or may remain actual but become 

 indeterminate, or these two states of things may coexist. The 

 first coefficient of the equation in X is a, the determinant of p 

 (or its squared tensor), which also must not be zero, as in 

 that case one root at least of X would be infinite. Let us sup- 

 pose, then, that neither a nor I vanishes. The very interesting 

 question presents itself as to what kind of equalities can arise 

 among the three pairs of roots, and what are the conditions 

 of such arising. 



This equation admits of an extremely interesting and succinct 



answer as follows: — Let m represent — ^ — ; the equalities 



between the roots of the given equation in x will be com- 

 pletely governed, and are definable by the equalities existing 

 between those of the biquadratic binary form 



(a, b, m, e,f)(X, Y) 1 *. 



* Iftbe equation is regarded as one in quaternions, the determining 

 biquadratic is the modulus of x 2 -\-xp-\~q\ from which it follows imme- 

 diately that, if p, q are real quaternions, all the four roots, say ec, /3, y, d, 



