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XLIII. On the Solution of a Class of Equations in Quaternions. 

 By J. J. Sylvester, Savilian Professor of Geometry in the 

 University of Oxford*. 



THE general equation of the degree co in Quaternions or 

 Binary Matrices is obviously o> 4 , but in certain cases 

 some of these roots evaporate and go off to infinity. The 

 only equation considered by Sir William Hamilton in his 

 Lectures is the Quadratic Equation of a form which I call 

 unilateral, because the quaternion coefficients in it are sup- 

 posed all to lie on the same side of the unknown quantity. I 

 propose here to show how Hamilton's equation, and indeed a 

 unilateral one of any order, may be solved by a general alge- 

 braical method and the number of its roots determined. 



It will be convenient to begin by setting out certain general 

 equations relating to any two binary matrices m, n. 



Writing the determinant of x+ym + zn under the form 



x 2 + 2bxy + 2cxz + dy 2 + 2eyz +fz 2 



(6, c, d, e,f thus constituting what I call the parameters of the 

 corpus m, n), we have universally 



m 2 — 2bm + d = 0, n 2 -2cn+f=0, d{m- 1 n) 2 — 2e(m'- 1 n)+f=0. 



Moreover if m, n receive the scalar increments /jl, v; d, e,f 

 become respectively 



d — 2/jib + fjL 2 , e—fic — vb + jjuv, f—2vc + v 2 . 



Let us begin with Hamilton's form, say 



«£ 2 — 2px + q=0, 

 and suppose 





^ 2 -2B^ + D = 



where B, D are scalars to be determined. 



Let 6 3 c, d, e, f be the five known parameters of the corpus 

 p, g. Then, since 



(p-B)-\ q -B) = 2.v, 

 we shall have 



4,(d + 2bB + B 2 )x 2 -4(e + bT) + cB + BV)x+f+c~D + I) 2 =0. 



Hence, writing B + b = u, D + c=v, 



d — b 2 = *, e — bc = fi, f—d 2 = y, 

 we have 



u 2 + a = X, uv + fi = 2\(u — b), v 2 + 7 = 4X(v — c). 



* Communicated by the Author 





