converges to a limit, 
b \o 
2 — ———— 
a (- + ) 0, 
the two quaternions a and b being supposed to be given and 
real, then this limit w is equal to that one of the two real 
roots of the quadratic equations in quaternions, 
w+ ua=b, 
which has the lesser tensor ; and gave geometrical illustrations 
of these results. 
The two real quaternion roots of the quadratic equation, 
g’=qi+j, being, as in the abstract of December, 1851, 
n=s(1+i+j-h), g2=3(-1+7-j-h), 
it is now shown that the four imaginary roots are 
z z 
ee aie eit ead al 
a that in whatever manner we group og two by two, even 
by taking one real and one imaginary root, the formula 
a Uyt Gz 
tiy=(1—P2) (Ye gu 9), oF EL = 04, 
z 1 
_ Ut Ge cape. ; 
where vy= 927 Up 91-75 Vo= a ; , and which is at once simpler 
0 1 
and more general than the equations previously communicated, 
> \z 
conducts still to values of the continued fraction w,, or (4) 0, 
Z 
which agree with those formerly found, and may be collected 
into the following period of six terms, 
U=0, m=h, U=$(k-2), Us=h-1t, u=-1t, us=c0, 
Us = 0, Uy — kh, &e. 
In general it may be remembered that 9, g., are roots of the 
quadratic equation g*=qa+b. 
As an example ofa continued fraction in quaternions which, 
