236 On Continued Fractions in Quaternions. 



quite transparent; and all of them are moveable, descending 

 pretty rapidly through the fluid when the specimen is inverted. 

 One of the moveable crystals shown at a is a congeries of small 

 crystals. The other transparent crystals are shown at b, c and d. 

 Three little black fragments are shown at 1, 2 and 3. These 

 three descend with much less velocity in the fluid than the transpa- 

 rent crystals, and have therefore a lower specific gravity. The fluid 

 is not quite transparent, but has a slight degree of milkiness in 

 its aspect ; and it contains a good deal of a darkish flocculent 

 matter, which descends through the fluid with a motion just 

 perceptible. The sides of the cavity are lined with a sort of 

 powdery deposit or thin crust. While some of the black frag- 

 ments descend slowly in the fluid, as already stated, others ascend, 

 a fact which I have never observed in the many hundreds of 

 cavities of topaz which I have had occasion to examine. The 

 milky colour of the fluid, the flocculent matter, and the powdery 

 crust coexisting with the fluid*, are facts which I had never 

 before observed. 



St. Leonard's College, St. Andrews, 

 February 26, 1863. 



XXXIX. OnContinued Fractions in Quaternions. By Sir Willi am 

 Rowan Hamilton, LL.D., M.R.I. A., F.R.A.S. ^c, Andrews' 

 Professor of Astronomy in the University of Dublinj and Royal 

 Astronomer of Ireland. 



[Continued from p. 118.] 



5. T ET us now consider the continued fraction, 



where Uq and w, are quaternions, and a, ^ are two rectangular 

 vectors, connected by the relation, 



and, as a sufficient exemplification of the question, let it be sup- 

 posed that a, /3 have the values 



az=i-k, fi=j. - 



It may easily be shown, by the rules of the present Calculus, 

 that the expression, 



u,= ■ > 

 * i—k-^UQ* 



* In many cavities in topaz in which the fluid had disappeared I had 

 found them lined with a crust. See Edinb. Trans, vol. x. pp. 22, 23. 



