216 Sir W. Rowan Hamilton on Quaternions. 



is evidently contained on the radius vector drawn from that 

 centre of the ellipsoid to the point .r 3/ g', or on that radius 

 vector prolonged. We see, also, that the length of this radius 

 vector of the ellipsoid, or the distance of the point x y z from 

 the origin of the co-ordinates, is inversely proportional to the 

 distance of the new point a^ y' z' of the spheric surface from 

 the point / m n, which latter is a certain fixed point upon the 

 surface of the ellipsoid. This result gives already an easy 

 and elementary mode of generating the latter surface, which 

 may however be reduced to a still greater degree of simplicity 

 by continuing the analysis as follows. 



35. Let the straight line which connects the two points 

 a' y z' and / m n be prolonged, if necessary, so as to cut the 

 same spheric surface again in another point a?" y ^": we shall 

 then have the equation 



from which the new co-ordinates a/', y", 2" may be eliminated 

 by substituting the expressions 



;r" = Z+^(^'-/), i/' = m + t{y'-m), z"=n + l{z'-n); 



and th^ root that is equal to unity is then to be rejected, in 

 the resulting quadratic for t. Taking therefore for t the pro- 

 duct of the roots of that quadratic, we find 



Z^ + m^ + n^ — 2(lp + mq-\- nr) 

 therefore also, by the last article, 



X^ -\- 7f -{■ Z^ 



consequently 



,2_ x^ + y^-\-z^ 



and finally, 



{x"-l)^-\-{y"-m)^ + {is"-n)^=x'' + y^ + z\ . (2.) 

 Denoting by a, b, c, the three fixed points of which the 

 co-ordinates are respectively (0, 0, 0), (/, m, w), (p, q, r) ; and 

 by D, d', e, the three variable points of which the co-ordinates 

 are (y, y, s'), (:i", y, z"), (xy y^z)', abed' may be regarded 

 as a plane quadrilateral, of which the diagonals ae and bd' 

 intersect each other in a point d on a fixed spheric sur- 

 face, which has its centre at c, and passes through a and d'; 

 so that one side d'a of the quadrilateral, adjacent to the fixed 

 side AB, is a chord of this fixed sphere. And the equation (2.) 

 expresses that the other side be of the same plane quadrilateral^ 

 adjacent to the same Jixed side ab, is a chord of ajixed ellipsoid. 



