Sir W. Rowan Hamilton on Qiiaterniotis. 285 



xp + pK = Oy (23.) 



which, when combined with (21.), gives 



(x2 - «2)2 ={i-K)p.p{i-x)- (» - x)p2(, _ x) = (. - x) V, 

 that is, 



/«2 _,2\2 



(24.) 



■=(^7 



but this is the equation of a sphere concentric with the ellip- 

 soid ; therefore the diametral plane (23.) cuts the ellipsoid in 

 a circle^ or the plane itself is a ajcHc plane. We see also that 

 the vector x, as being perpendicular to this plane (23.), is one 

 of the cj/clic normals, or normals to planes of circular section; 

 which agrees with the construction, since we saw, in article 36, 

 that the auxiliary or diacentric sphere, with centre c, touches 

 one cyclic plane at the centre a of the ellipsoid. The same 

 construction shows that the other cyclic plane ought to be 

 perpendicular to the vector i ; and accordingly the equation 



ip + pi = Q (25.) 



represents this second cyclic plane; for, when combined with 

 the equation (21.) of the ellipsoid, it gives 



and therefore conducts to the same equation (24.) of a con- 

 centric sphere as before; which sphere (24.) is thus seen to 

 contain the intersection of the ellipsoid (21.) with the plane 

 (25.), as well as that with the plane (23.). If we use the 

 form (9.), we have only to observe that whether we change 

 px to —xp, or ip to — f)», we are conducted in each case to the 

 following expression for the length of the radius vector of the 

 ellipsoid, which agrees with the equation (24.) : 



Tf='i^) (^«-) 



And because x^ — »^ denotes the square upon the tangent drawn 

 to the auxiliary sphere from the external point b, while 

 T(i — x) denotes the length of the side ba of the generating 

 triangle, we see by this easy calculation with quaternions, as 

 well as by the more purely geometrical reasoning which was 

 alluded to, and partly stated, in the 36th article, that the com- 

 mon radius of the two diametral and circular sections of the 

 ellipsoid is equal to the straight line which was there called 

 BG, and which had the direction of ba, while terminating, like 

 it, on the surface of the auxiliary sphere ; so that the two last 

 lines ba, and bg, were connected with that sphere and with 

 each other, in this or in the opposite order, as the whole se- 



