On Continued Fractions in Quaternions. 237 



gives the relations, 



{u^ — k)~^^k-{-i{uQ—k)~^kJ 



(^u^^k)-'-{uo-k)-' = k--i; 

 and generally, by an indefinite repetition of the last process, 



(W2n+, — ^)"^ — K— ^)~^=W(^ — 2*). 



There is no difficulty in hence inferring that 



whatever arbitrary quaternion (mq) may be assumed as the original 

 subject of the operation, which is thus indefinitely repeated. By 

 assuming for this original operand a vector pQ in the plane of ik, 

 some geometrical* theorems arise, less general indeed in their 

 import than the foregoing results respecting quaternions, yet 

 perhaps not uninteresting, as belonging to a somewhat novel 

 class, and coming fitly to be stated here, because they bear a 

 sort of limiting relation to the results recently published in the 

 Philosophical Magazine, as part of the present paper. 



6. Let c and d be the extremities, and e the summit of a 

 semicircle. Assume any point p in the same plane, and draw 

 CQ perpendicular to dp, so that the rectangle cq . dp may be 

 equal to the given square ce^. Then it is clear, 1st, that if the 

 hand (or direction of rotation) be duly attended to, in thus 

 drawing cq -i- dp, the point q will coincide with p, when the 

 latter point p is so assumed as to coincide with the given sum- 

 mit E. But I say also, TInd, that if the point p be taken any- 

 where else in the same plane, and if, after deriving q from it as 

 above, we derive r from q, &c., by repeating the same process, 

 these n£W or derivative points q, r, s, &c., will tend, successively 



* Note added during printing. — Since the foregoing communication was 

 forwarded, I have perceived that the theorem YIII. of art. 6, which pre- 

 sented itself to me as an interpretation of the expression for (mi — k)~^j 

 when A:=CE, i=DE, M(,=cp, u^-=cql, may be very simply proved by means 

 of the two similar triangles, qec, ecp'': and may then be employed to 

 deduce geometrically all the other theorems of that article. (Each of these 

 two triangles is similar to ep^c, if p, be on ep", and pp^ |1 do.) I see also 

 that the lately published results of art. 4 may all be deduced geometrically, 

 from the consideration of the two pairs of similar triangles, adp, qca, and 

 BDP, acB. These geometrical simplifications have only recently occurred 

 to me ; but it may have been perceived that, on the present occasion, geo- 

 metry has been employed merely to illustrate and exemplify the significa- 

 tion and validity of certain new symbolical expressions, and methods of 

 calculation ; some account of which expressions and methods I hope to be 

 permitted to continue. (March 16, 1853.) 



