238 On Continued Fractions in Quaternions. 



and indefinitely, to coincide with the point e. I add, Ilird, 

 that if, from an arbitrarily assumed point s, we go back, on the 

 same plan, to other points r, q, p, &c., these new points, thus 

 inversely derived, will also tend indefinitely to coincide with the 

 same fixed summit e. IVth. The alternate points p, r, t, . . . 

 are all contained on one common circular circumference ; and 

 the other alternate system of derived points q, s, u, . . are all 

 contained on another circular locus. Vth. These two new circles 

 touch each other and the given semicircle at the given summit e ; 

 and their centres are harmonic conjugates with respect to the 

 completed circle ced. (The same harmonic conjugation of the 

 centres of the two loci might easily have been derived for the 

 more general case considered in an earlier part of this paper, 

 from the last formula of art. 4 ; I have found that it holds good 

 also in another equally general case, hereafter to be considered, 

 when the given area of the rectangle under cq and dp is greater 

 than the square on the quadrantal chord ce, in which case there 

 can be no convergence to a limiting position, but there may be, 

 under certain conditions, circulation.) Vlth. If the chords pe, 

 RE, TE, . . of the one circular locus, and also the chords qe, se, 

 UE, . . of the second locus, be prolonged through the point of 

 contact E, so as to render the following rectangles equal to the 

 given square or area, 



pep'=rer'=tet'= . . =qeq'=ses'=ueu'= . . =CE^, 



then not only will the points p'rV . . be ranged on one straight 

 line, and the points q's'u' . . on another, but also the intervals 

 t'r', r't', . . q's', s'u', . . will all be equal to each other and to the 

 given diameter cd ; and will have the same direction as that dia- 

 meter. Thus the four points eprt, or the four points eqsu, 

 form what may be called an harmonic group, on the one or on 

 the other circular locus : and if, as in some modem methods, 

 the directions (and not merely the lengths) of lines be attended 

 to, the chords ep, er, et, . . or eq, es, eu, . . . may be said to 

 form, each set within the circle to which they belong, a species 

 of harmonical progression, Vllth. The orthogonal projection of 

 p'q', or q'r', &c., on cd, is equal in length and direction to the 

 half of that given diameter. Vlllth. If p'p" be so drawn as to 

 be perpendicularly bisected by the common tangent to the three 

 circles, the line p"q' will be equal in length and direction to the 

 given quadrantal chord ce. 



Observatory, February 19, 1863. 



[To be continued.] 



