a Class of Equations in Quaternions. 395 



will be found to be the determinant 



• B -B 3 -B 5 -B 7 

 . B 3 • B 5 -B 7 -B 9 



• B 5 • B 7 • B 9 • B 11 



• B 7 B 9 -B 11 • B 13 * 



and a similar determinant will fix the degree of B in the 

 resolving equation for any value of co. Hence the number of 

 solutions of the unilateral equation in quaternions of the Jer- 

 rardian form of the degree co is q)(2cd — 1) or 2o> 2 — co, and the 

 evaporation will accordingly be o> 4 — 2co 2 + co, or 



(ft) 2 -~G))(a) 2 + a)~l). 



Moreover the same method with a slight addition will serve 

 to determine the roots of the general unilateral equation in 

 quaternions, the number of which will be a cubic function of 

 co, as I propose to show and to give its precise value in some 

 future communication, either in this Journal, or at all events 

 in the memoir on Universal Algebra now in the course of 

 publication, under the form of lectures, in the i American 

 Journal of Mathematics' f. 



I very much question whether the old method of Hamilton, 

 as taught by its most consummate masters, Tait in this 

 country, or the late Prof. Benjamin Pierce in America, would 

 be found sufficiently plastic to deal effectually with an analy- 

 tical investigation in quaternions of this degree of complexity, 

 so as to lead to the formula for the number of solutions of the 

 unilateral equation of the Jerrardian form above given. 



I invite my much esteemed and most capable former col- 

 league and former pupil, Dr. Story, of the Johns Hopkins, 

 and Prof. Stringham, of the University of California, who 

 carry on the traditions of the Harvard School, to put the 

 power of the old method as compared with the new to this 

 practical test. 



Postscript. — If 



a? — ?>px 2 + 3q%—r=0, 



[where p, q, r are perfectly general matrices of the second 

 order which satisfy the general equations 



* It may readily be seen that the highest term in the equation for 

 finding B is identical with the resultant of 



D 4 - 24B 2 D 3 + 80B 4 D 2 and 4BD 3 - 40B 3 D 2 + 64B 5 D - 64B 7 , 

 i. e. will be 2 18 . 3 . 7 . 19B 28 ; and that the last term (at all events to the 

 sign pres) will be 6 4 8 2 , which is of 4 . 3+2 .2.4 (»". e. of 28) dimensions 

 in x, and is therefore codimensional (as it ought to be) with B 28 . 

 t It is given in the Postscript below. 



2 D2 



