458 On Hamilton's Quadratic Equation. 



nated and x found in terms of known quantities : <$>x will have 

 six different values, which will give the six roots of fx—0. 

 It is far from improbable that a similar solution applies to a 

 unilateral equation fx=0 of any degree n in matrices of any 

 order co. 



Call ¥x the determinant of fx when x is regarded as an 

 ordinary quantity; then, if c\>x is an algebraical factor of the 

 degree co in x contained in Fx, it would seem to be in all 

 probability true that <f>x = is the identical equation to 

 one of the roots of fx = ; and, vice versa, that the func- 

 tion identically zero of any such root is a factor of Fx. 

 By combining the equations fx = 0, $x~0, all the powers 

 of x except the first may be eliminated, and thus every root 

 of x determined. The solution of the given equation will de- 

 pend upon the solution of an ordinary equation of the degree 

 nco, and the number of roots will be the number of ways 

 of combining nco things co and co together. Thus, for a cubic 



6*5 

 equation in quaternions the number of roots would be -^-, 



or 15. In the May Number of this Magazine it was supposed 

 to be shown to be 21 ; but it is quite conceivable that this 

 determination may be erroneous, especially as it was deduced 

 from general considerations of the degrees of a certain system 

 of equations without attention being paid to their particular 

 form, which might very well be such as occasion a fall in the 

 order of the system. I am strongly inclined, with the new 

 light I have gained on the subject, to believe that such must 

 be the case, and that the true number of roots for a unilateral 

 equation in quaternions of the degree n is 2n 2 — n * ; in which 

 case the theorem above stated, and which may be viewed as a 

 marvellous generalization of the already marvellous Hamilton- 

 Cayley Theorem of the identical equation, will be undoubtedly 

 true for all values of n and co. But I can only assert posi- 

 tively at present that it is true for the case of n—\ whatever 

 co may be, and for the case of w = 2, co = 2 f. 



* From the number 21 above referred to, now known to be erroneous, 

 the general value was inferred to be n? — n 2 +n, whereas it is demonstrably 

 2n 2 — n only for the general unilateral equation of degree n in quaternions, 

 as I proved it to be for the Jerrardian form of that equation. 



t I have since obtained an easy proof of the truth of the conjectural 

 theorem for all values of n and a ; see the Comptes Rendus of the Insti- 

 tute of France for October 20th last. 



