300 



converges to a limit, 



( h 

 » \a + 



the two quaternions a and b being supposed to be given and 

 real, then this limit u is equal to that one of the two real 

 roots of the quadratic equations in quaternions, 



u 9 + ua - h, 

 which has the lesser tensor ; and gave geometrical illustrations 

 of these results. 



The two real quaternion roots of the quadratic equation, 

 q^^qi+j, being, as in the abstract of December, 1851, 



q 1 = 1 (1 + i+j-k), q 2 = i(- 1 + i-j- k), 

 it is now shown that the four imaginary roots are 



?3=2 Z ( 1 +v / -3)-A, ^ = 1(1-^-3) -A, 



?6 =iO-+A)+i(l-i)V-3, g- 6 = i(i+A)-i(l-j)i/-3; 



but that in whatever manner we group them, two by two, even 

 by taking one real and one imaginary root, the formula 



u x = (l- v x ) x {v x q x - q 2 ), or - = v„ 



w j' + ^i 



where v x = q 2 x v Q\~ X ) v = — — — , and which is at once simpler 



and more general than the equations previously communicated, 

 conducts still to values of the continued fraction u x , or ( —- ) 0, 



which agree with those formerly found, and may be collected 

 into the following period of six terms, 



M = 0, «! = k, u 2 = j}(k-i), « 3 = k-i, u t = - i, 7i 5 = oo, 

 m 6 = 0, u-, = k, &c. 



In general it may be remembered that q x , q 2 , are roots of the 

 quadratic equation q 2 = qa + b. 



As an example of a continued fraction in quaternions which, 



