460 Sir W. Rowan Hamilton on Quaterniojis, 



we may therefore regard a— /3 as the vector of the centre C 

 of a certain auxiliary sphere, of which the surface passes 

 through the centre A of the ellipsoid ; and may regard the vec- 

 tor « — ^ + (ras a variable and auxiliary guidc-c/iord AD oi the 

 same guide-sp/iefe, which chord determines the (exactly similar 

 or exactly opposite) direction of the variable radius vector 

 AE (or p) of the ellipsoid. At the same time, the constant 

 vector —2/3, drawn from the same constant origin as before, 

 namely the centre A of the ellipsoid, will determine the posi- 

 tion of a certain fixed point B, having this remarkable pro- 

 perty, that its distance from the extremity D of the variable 

 guide-chord drawn from A, will represent the reciprocal of' 

 the length of the radius vector p, or the proximity (AE)~' of 

 the point E on the surface of the ellipsoid to the centre (the 

 use of this word " proximity " being borrowed from Sir John 

 Herschel). Supposing then, for simplicity, that the fixed 

 point B is external to the fixed sphere, which does not essen- 

 tially diminish the generality of the question; and taking, for 

 the unit of length, the length of a tangent to that sphere from 

 that point ; we may regard AE and BD' as two equally long 

 lines, or may write the equation 



AE = BD', (10.) 



if D' be the other point of intersection of the straight line BD 

 with the sphere. 



31. Hence follows this very simple construction^ for an 

 ellipsoid (with three unequal axes), by means of a sphere and 

 an external point, to which the author was led by the fore- 

 going process, but which may also be deduced from principles 

 more generally known. From a fixed point A on the surface 

 of a sphere, draw a variable chord AD ; let D' be the second 

 point of intersection of the spheric surface with the secant BD, 

 drawn to the variable extremity D of this chord AD from a 

 fixed external point B; take the radius vector AE equal in 

 length to BD', and in direction either coincident with, or op- 

 posite to, the chord A D ; the locus of the point E, thus con- 

 structed^mll be a7i ellipsoid,wh[ch will pass through the point B. 



* This construction has already been printed in the Proceedings of the 

 Royal Irish Academy for 1846 ; but it is conceived that its being reprinted 

 here may be acceptable to some of the readers of the London, Edinburgh, 

 and Dublin Philosophical Magazine; in which periodical (namely in the 

 Number for July 1844) the first jyrinUd publication of the fundamental 

 equations of the theory of quaternions {i'=j-:=k-z= — 1, ij^i,jlc=.i, iii=J, 

 ji=: — lc,kj-^ — i, »'/:=— _yj took place, although those equations had been 

 communicated to the Royal Irish Academy in November 1843, and had 

 been exhibited at a meeting of the Council during the preceding month.^ 



