is 
389 
in November, 1843, while / is used as a temporary and abridged 
symbol for the old imaginary Y -1. In fact the rules of this 
calculus give 
(j+ hhy = j? + h (jk + hj) + heh’, 
=-1+0h+(-1)€¢ 1) =90, 
h being a free (or commutative) factor in any multiplication, 
as in algebra, but jk being = 7=- 4j, while 
WaPajp=kh=-1. 
Thus, at least for any numerical exponent x, we have the sim- 
plification, 
(L+j+hk)* =1+2(j+ hk), 
which Sir W. R. H. states that he has found useful in a part 
of a geometrical investigation, respecting the interpretation of 
certain continued fractions in quaternions, of the form 
Uy =| — } Uo 
a+ 
already mentioned by him to the Academy on a former occa- 
sion, and specially for the case when a‘ + 4(3? = 0, in the fraction 
(Barn 
where the vector 3 is supposed to be perpendicular to a and 
Po, and therefore also to pz. | 
By the investigation referred to, he has found, among others, 
the following results. Let C and D be two given points, and 
P an assumed point. Perpendicular to DP draw CQ, towards 
a given hand, and such that the rectangle CQ. DP may be 
equal to a given rectangle CC’D’D. From Q derive R, as 
Q has been derived from P, and conceive the process repeated 
without end. Then, I., the locus of the alternate points 
P, R, T, .. is one circle, and the locus of the other alternate 
points Q, S, U,.. is another circle. II. These two circular 
loci have the top C’D’ of the given rectangle for the common 
