Sir W. Rowan Hamilton on Qtiaternions. 371 



other, while the points L and ji are situated on the same inde- 

 finite straight line as a, b ; that is, on the axis of that circum- 

 scribed cylinder of revolution which has been considered in 

 former articles. The vectoi's of the points d, e, in the same 

 construction of the ellipsoid, (if drawn from its centre as their 

 origin,) having been seen to be respectively <r — x and p, (com- 

 pare article 40,) the equation 



(rp+px = (16,), art. il, 



combined with (Si.) and (86.), gives for their product the ex- 

 pressions : 



(<r-x)^ = ^(.-x) = A(»-x)2; . . . (102.) 



and in general if two pairs of co-initial vectors, ashereo- — x, p, 

 and A, » — X, give, when respectively multiplied, one common 

 scalar product, they terminate on four concircular points: the 

 four points d, e, l, b are therefore contained on the circum- 

 ference of one common circle : and consequently the point L 

 may be found by an elementary construction, derived from this 

 simple calculation with quaternions, namely as the second 

 point of intersection of the circle bde with the straight line 

 AB (which is situated in the plane of that circle). Again, the 

 equations (85.) and (90.) give 



T(p-X) = T(p-^); .... (103.) 

 therefore the point e of the ellipsoid is the vertex of an isosceles 

 triangle, constructed on lm as base; and the point m may thus 

 be found as the intersection of the same straight line ab or al, 

 with a circle described round the point E as centre, and having 

 its radius =el = 6= the mean semiaxis of the ellipsoid. When 

 the two points l and m have thus been found, the third point 

 X can then be deduced from them, in an equally simple geo- 

 metrical manner, by drawing parallels ln, mn to the sides 

 AC, BC of the generating triangle abc, from which the ellipsoid 

 itself has been constructed; these sides ln, mn, of the new 

 and variable triangle lmn, will thus be parallel to the two 

 cyclic normals of the ellipsoid; and the foregoing analysis 

 shows that they will be portions of the axes of the two circles, 

 which are contained upon the surface of that ellipsoid, and 

 pass through the point e on that surface: while the point N, 

 of intersection of those two axes, is the centre of that common 

 sphere (97.), which contains both those two circular sections. 

 It is evident that this common sphere must touch the ellipsoid 

 at v., since it is itself touched at that point by the two distinct 

 tangents to the two circular sections of the surface ; and hence 

 we might inter that tlie semidiamcter ne ov ^ — p of the sphere, 

 of which the length ?• has been assigned in the formula (99.), 

 2 B 2 



