On Continued Fractions in Quaternions. 328 



of a general process (described in art. 649 of my unpublished 

 Lectures on Quaternions), I found that the quadratic (15) might 

 be symbolically satisfied by the two following imaginary quater- 

 nions, or biquaternion expressions : 



u'=-i-h{l-j);u"=-i + h{l-j); . . (17) 



where h is used as a temporary and abridged symbol for the old 

 and ordinary imaginary of common algebra, denoted usually by 

 \^ — Ij and regarded as being always a free or commutative factor 

 in any multiplication : so that 



A^=— 1, hi=ihj hj=jh, hk—kh, . . (18) 

 although ji = — ^;, &c. In fact the first of these expressions (1 7) 

 gives, 



u\u^ + i-k) = {i + h[\--j)}{k^-h{\-j)} 



=^ik + h{i[l-j) + {l--j)k}^h\\^jf 



=z-j + h{i-k+k~i) + h%l'-2j-l) 



= '-j + 0h + 2j=j; ...... (19) 



and the second expression (17) gives, in like manner, 



u'i{u"-\-i-k)=j: (20) 



so that, without entering at present into any account of the pro- 

 cess which enabled me to find the biquaternions (17), it has been 

 now proved, a posteriori, by actual substitution, that those expres- 

 sions do in fact symbolically satisfy the quadratic equation (15). 

 And because they are unequal roots of that equation, as differing 

 by the sign of h, I saw that they might be employed in the 

 general formula (2), without being liable to the practical objec- 

 tion that lay against the employment of the two real but equal 

 roots, p', p", of the equation (7). 



10. Introducing therefore into the formula (2), or into the 

 following, which is a transformation thereof, 



u-u" _ u'f^{u,^u") (21) 



%-u' u'%Uo-y')' 

 the values (17), or these which are equivalent, 



u'z=--h{l-j-hi), u" = h{l-j + hi)', . . (22) 

 and observing thnt 



U±k^r=f±hU^ + ^f|-^^=0, .... (33) 

 and that therefore* 



(l—j^:hiY=l—aij + a!hi; (24) 



* More generally, with these rules of combination of the symbols hijk, if 

 /be any algebraic function, and/' the derived function, 



A^+tj±thi)=f{\)+tf{\)U±hi) ; 



because (^jiiAi)2=0, if f be any scalar coefficient. 



