(230.) 



438 Sir W. Rowan Hamilton on Quaternions. 



them is bisected by Ur +Ud, or (see (221.)) by the axis major 

 of the ellipsoid j while the supplementary angle between on 

 and — ctt' is bisected by U>j — Ud, or (as is shown by (222.)) by 

 the axis minor. It is evident that —m and —aJ are also um- 

 bilicar vectors ; and it is clear, from what has been shown in 

 former articles, that the vectors )j and Q have the directions of 

 the axes of the two cylinders of revolution, which can be cir- 

 cumscribed about that given or original ellipsoid, to which all 

 the remarks of the present article relate. 



81. These remarks being premised, let us now resume the 

 consideration of the variable vector Aj, of art. 78, which has 

 been seen to terminate on the surface of a certain derived 

 ellipsoid of revolution. Writing, under a slightly altered form, 

 the expression (193.) for that vector Aj, and combining with 

 it three other analogous expressions, for three other vectors, 

 Ag, A3, A4, as follows, 



it is easy to prove that 



TAi = TA2 = TA3=:TA4; .... (231.) 

 and that 



S.)39Ai = S.>)5a2=S.>)9a3=S.>)9A4=S.»i9p; . (232.) 



i^hence it follows that the four vectors Aj, Ag, A3, A^, being sup- 

 posed to be all drawn from the centre a of the original ellip- 

 soid, terminate in four points, Lj, Lg, Lg, L4, which are the 

 corners of a quadrilateral inscribed in a circle of the derived 

 ellipsoid of revolution ; the plane of this circle being parallel 

 to the plane of the greatest and least axes of the original ellip- 

 soid, and passing through the point e of that ellipsoid, which 

 is the termination of the vector p. We shall have also the 

 equations, 



iC:£ = |:f=V-0; tsZl=^=V-'o; (233.) 



Aj— p b.dp ^4 — P 0.9 'p 



which show that the two opposite sides L^Lg, L3L4, of thisin- 

 scribed quadrilateral, being prolonged if necessary, intersect 

 in the lately- mentioned point e of the original ellipsoid. And 

 because the expressions (230.) give also 



V^=«. V;;^.=0, . . (234.) 

 these opposite sides LjLg, L3L4, of the plane quadrilateral thus 



