118 On Continued Fractions in Quaternions. 



line CD, and if a semicircle (towards the proper hand) be con- 

 structed on that line as diameter, it will be possible to inscribe 

 a parallel chord ab, such that the given area shall be repre- 

 sented by the product of the diameter CD, and the distance of 

 this chord therefrom. We may also conceive that b is nearer 

 than A to c, so that abcd is an uncrossed trapezium inscribed 

 in a circle, and the angle abc is obtuse. This construction 

 being clearly understood, it becomes obvious, 1st, that because 

 the given area is equal to each of the two rectangles, ca . da 

 and CB .DB while the angles in the semicircle are right, then, 

 whether we begin by assuming the position of the point p to be 

 at the corner a, or at the corner b, of the trapezium, every one of 

 the derived points, q, r, s, t, &c., will coincide with the position 

 so assumed for p, however far the process of derivation may be 

 continued. But 1 also say, Ilnd, that if any other point in the 

 plane, except these two fixed points, a, b, be assumed for p, then 

 not only will its successive derivatives, Q, r, s, t, . . be all distinct 

 from it, and from each other, but they will ^^i? successively and 

 indefinitely to coincide with that one of the two fixed points which 

 has been above named b. I add, Ilird, that if, from any point 

 T, distinct from a and from b, we go back, by an inverse process 

 of derivation, to the next preceding point s of the recently con- 

 sidered series, and thence, by the same inverse law, to r, q, p, 

 &c., this process will produce an indefinite tendency to, and an 

 ultimate coincidence with, the other of the two fixed points, 

 namely, a. IVth. The common law of these two tendencies, 

 direct and inverse, is contained in the formula 



QB . PA CB ^ ^ 



= — = constant : 



QA . PB CA 



which may be variously transformed, and in which the constant 

 is independent of the position of p. Vth. The alternate points, 

 p, R, T, &c., are all contained on one common circular segment 

 APB ; and the other system of alternate points, Q, s, &c., has for 

 its locus another circular segment, aqb, on the same fixed base, 

 AB. Vlth. The relation between these two segments is expressed 

 by this other formula, connecting the angles in them, 



apb4-aqb = acb; 



the angles being here supposed to change signs, when their ver- 

 tices cross the fixed line ab. 

 Observatory, December 30, 1862. 



[To be continued.] 



