(^.(^Q^f^^^i^^f'VMtiqm m Quaternions. 3?g 



therefore ,r^ >.,> >v^ 



^'" ; ={{uo-k)-' + n{k--i)}-'; . . (36) 



or more simply, 



(u,n-k)-'-{uo-k)-' = n{k-i). . . . (37) 



It was in this way that I was originally led to the formula of 

 art. 5, namely, 



{u,n^^-k)-'^{u,,-k)-' = n{k-^i); . . (38) 



but having once come to see that this result held good, it was 

 easy then to pass to a much more simple proof, such as that 

 given in the last-cited article, which was entirely independent of 

 the imaginary symbol here called h, and employed only real 

 quaternions. " .iiiij^ji Oo 



12. It may be regarded as still more remarkable, that Itie 

 same real results are obtained, when we combine a real root with 

 an imaginary one, instead of combining two real roots or ttvo 

 imaginary ones. Thus the quadratic* equation (15) has on£ 

 root, namely k, which must be considered as real in this theory, 

 whether by contrast to the symbol k (or to the old imaginary of 

 algebra), or because in the geometrical interpretation it is con- 

 structed by a real line, namely by the chord ce drawn to the 

 point of contact e of the spheric surface (8) with the right line 

 (9), under the condition (13); a and ^ being then for conve- 

 nience replaced, as in art. 5, by the more special symbols i—k 

 andj. Now if we adopt this real root k as the value of u', but 

 retain the second of the two imaginary or biquaternion roots 

 (17), as being still the expression for w", the numerators of the 

 formula (21) will remain unchanged, but the denominators will 

 be altered ; and instead of (25) we shall have this other formula, 



* An equation of the wth dimension in quaternions has generally n* roots, 

 real or imaginary ; hecause it may be generally resolved into a system of 

 four ordinary and algebraical equations, which are each of the wth degree. 

 However, it is shown in my Lectures that for the particular form (3), 

 u^-\-ua=.b (or q^=qa-\-b), which occurs in the present investigation, only 

 six (out of the sixteen) roots are finite; and that of these six, two are gene- 

 rally real, and four imaginary. In the particular case of the equation (15), 

 Jc is by this theory a quadruple root, representing at once two real and two 

 imaginary solutions, which have all become equal to each other, by the 

 vanishing of certain radicals. Thus there remain in this case only three 

 distinct roots of the quadratic (15), namely the one real root k, and the two 

 imaginary roots (17) : and what appears to me remarkable in the analysis 

 of the present article 12, although otherwise exemplified in my Lectures, 

 is the mixture of these two classes of solution of an equation in quaternions, 

 a root of one kind being combined with a root of the other kind, so as to 

 conduct to a correct determination of the value of a certain continued frac- 

 tion, regarded as a real quaternion, which admits (as in art. 6) of being 

 geometrically interpreted. 



