822 On Continued Fractions in Quaternions. . 



the vectors p', /o" being roots of the quadratic, to 



P^+po^-=^ (rji 



This last equation gave, by taking separately the scalar and 

 vector parts, 



p9 + S.pa=0; (8) 



y.pu=ff; (9) 



whereof the former (8) expressed that p terminated on a spheric 

 surface, passing through the origin, and having the vector —a 

 for its diameter ; while the latter (9) expressed that p terminated 

 on a riffht line, which was drawn through the extremity of the 

 vector ^a~S in a direction parallel to that diameter. Thus (9) 

 gave, by the rules of the present calculus, 



p = ^u-'+xa, p'=-l3'u-^-\-x^ot^, S.pa = a?a2; . (10) 



and therefore, by (8), I had the ordinary quadi-atic equation, 



.r2 + a7=/8^«-S or (2.2?+l)V=a4 + 4)8^>0, . (11) 



as in art. 4 (Phil. Mag. for February, 1 853) : the two values of 

 the vector p, which answer to the two values of the scalar coeffi- 

 cient X, being here supposed to be geometrically real and unequal; 

 or the right line (9) being supposed to meet the spheric surface 

 (8), in two distinct and real points, a, b. Hence by assuming 



p' = CA, p" = CB, Pq=C'P, pi = CQ, a = DC, /Q = CA. DA, (12) 



I was conducted with the greatest ease to the theorem^ of the 



last-cited article. 4 



9. But in the case of art. 5, namely when '' 



a4 + 4/32=0, (13) 



and when consequently 



a:=-i p"=p'=/3«--i«, . . (14) 



the equality of the two roots of the quadratic (11) in cc, or of the 

 two real and vector roots of the equation (7) in p, appeared to 

 reduce the formula (5) to an identity : and the simple process of 

 the article last cited did not immediately occur to me. I there- 

 fore had recourse to certain imaginary or purely symbolical solu- 

 tions, of that quadratic equation (7), or rather of the following, 

 by which we may here conveniently replace it, 



u^ + u[i-k)=j', (15) 



^he continued fraction to be studied being now, 



«.= (izfc)'«o. (16) 



where ijk are the usual symbols of this calculus, and Uq may 

 denote any arbitrarily assumed quaternion. By an application 



