456 Prof. Sylvester on Hamilton's Quadratic Equation 



If the biquadratic has two equal roots, the given quadratic 

 will have two pairs of equal roots. 



If the biquadratic has two pairs of equal roots, the given 

 quadratic will have four equal roots. 



If the biquadratic has three equal roots, the quadratic will 

 have three pairs of equal roots. 



If the biquadratic has all its roots equal, the quadratic will 

 have all its roots equal. 



In the first case two of the three pairs of roots of the given 

 quadratic coincide, or merge into a single pair. 



In the second case, not only two pairs merge into one pair, 

 but the two roots of that pair coincide with one another. 



In the third case the three pairs merge into a single pair. 



In the fourth case the two members of that single pair coin- 

 cide with one another. 



So long as the equation in x remains regular, no kind of 

 equalities can exist between the roots other than those above 

 specified. 



For instance, let us consider the possibility of two values 

 of sj, and no more, becoming equal. First, let us inquire 

 what is the condition to be satisfied in order that the scalar 

 parts of two roots which belong to the same pair shall become 

 equal. It may be shown that the sufficient and necessary 

 condition that this may take place is that the irreducible sub- 

 invariant of degree 3 and weight (5 (i. e. the first coefficient of 

 the irreducible skew-covariant of the associated biquadratic 

 form [a, b, m, e, f ]) shall vanish. 



are imaginary. It may be shown that the roots of Hamilton's determining 



cubic are 



. (*+B)(y+b) , («+y)ft3+fi) , (*+a)08+y) 

 a . a- , a- _ , 



and these therefore are (as shown also by Hamilton) all of them real. 

 The biquadratic serves to determine the points in which the variable 

 conic associated to the equation pa?-\-qx-\-r (7. e. the determinant to 

 xp+yq-\-zr) is intersected by the absolute conic xz—y' 2 . Each root of 

 the given equation corresponds to a side of the complete quadrilateral 

 formed by the four points of intersection of these two conies ; and thus we 

 see that there are live cases to consider when the variable conic is a conic 

 proper, according as it intersects or touches the fixed conic (which can 

 h ippen in four different ways) ; and seven other cases where the conic 

 degenerates into two intersecting or two coincident lines (in which cases 

 the solution becomes irregular) ; viz. the intersecting lines may cut or 

 touch in one or two points the fixed one, and may cut or touch the conic 

 at their point of intersection, which gives five cases ; and the coincident 

 lines may cut or touch the fixed conic, which gives two more. Hence 

 there are in all twelve principal cases to consider in Hamilton's form of 

 the Quadratic Equation in Quaternions : or rather thirteen, for the case 

 of the variable and fixed conic coinciding must not be lost sight of. 



