389 



in November, 1843, while h is used as a temporary and abridged 

 symbol for the old imaginary yf - 1. In fact the rules of this 

 calculus give 



(J + hky =f + h (jk + kj) + h*k\ 



= _ i + oh + (- 1) (- 1) = 0, 



h being a free (or commutative) factor in any multiplication, 

 as in algebra, butjk being = i = - hj, while 



h* = & =f = £ 2 = - 1. 



Thus, at least for any numerical exponent x, we have the sim- 

 plification, 



(l+j + hky = 1 + x (j+ hk), 

 which Sir W. K. 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 



b V 



a +, 



already mentioned by him to the Academy on a former occa- 

 sion, and specially for the case when a 4 + 4j3 2 = 0, in the fraction 



P,= (— )po, 



where the vector /3 is supposed to be perpendicular to a and 

 p , and therefore also to p x . 



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, L, 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 CD' of the given rectangle for the common 



