454 Prof. Sylvester on Hamilton's Quadratic Equation 



(1) The avoiding of the principal errors of weights simply 

 hung on*, or the uncertainty resulting from the possible hete- 

 rogeneous nature of solid bodies used as loads, as, for example, 

 Lamont's ring. 



(2) The avoiding of variations of a magnetic directive force 

 by the temperature and terrestrial magnetic variations. 



(3) The greater simplicity of repeating a determination by 

 means of a single oscillation-period when the moment of 

 inertia of suspension has been determined once for all. 



(4) The avoiding of the influence of layers of air vibrating 

 with the body, Or of any possible magnetic induction upon the 

 weights used as load. 



L. On Hamilton's Quadratic Equation and the general Uni- 

 lateral Equation in Matrices. EyJ. J. Sylvester, F.R.S., 

 Savilian Professor of Geometry in the University of Oxford^. 



IN the Philosophical Magazine of May last I gave a purely 

 algebraical method of solving Hamilton's equation in 

 Quaternions, but did not carry out the calculations to the full 

 extent that I have since found is desirable. The completed 

 solution presents some such very beautiful features, that I 

 think no apology will be required for occupying a short space 

 of the Magazine with a succinct account of it. 



Hamilton was led to this equation as a means of calculating 

 a continued fraction in quaternions, and there is every reason 

 for believing that the Gaussian theory of Quadratic Forms 

 in the theory of numbers may be extended to quaternions or 

 binary matrices, in which case the properties of the equation 

 w T ith which I am about to deal will form an essential part of 

 such extended theory J. Let us take a form slightly more 

 general than that before considered, viz. the form 



px 2 -\-qx + r = 0, 



with the understanding that the determinant of p (if we are 

 dealing with matrices), or its tensor if with quaternions, differs 



* Compare Dora, Wied. Ann. xvii. p. 788 (1882), and 0. Beling, 'On 

 the Theory of Bifilar Suspension ' (Breslau, 1881). 



t Communicated by the Author. 



% I have found, and stated, I believe, in the form of a question in the 

 1 Educational Times ' some years ago, that any fraction whose terms are 

 real integer quaternions may be expressed as a finite continued fraction, 

 the greatest-common-measure process being applicable to its two terms, 

 provided both their Moduli are not odd multiples of an odd power of 2, 

 which can always be guarded against by a previous preparation of the 

 fraction. 



