286 Sir W. Rowan Hamilton on Quaternions. 



cant and the external part. In fact, as the point d, in the 

 construction approaches, in any direction, on tlie surface of the 

 auxiliary sphere, to a, the point d' approaches to g; and bd', 

 and therefore also ae, tends to become equal in length to Bci ; 

 while the direction of ae, being the same with that of ad, or 

 opposite thereto, tends to become tangential to the sphere, or 

 perpendicular to ac: the line bg is therefore equal to the 

 radius of that diametral and circular section of the ellipsoid 

 which is made by the plane that touches the auxiliary sphere 

 at A. And again, if we conceive the point d' to revolve on 

 the surface of the sphere from g to g again, in a plane per- 

 pendicular to Bc, then the lines ad and ae will revolve to- 

 gether in another plane parallel to that last mentioned, and 

 perpendicular likewise to bc ; while the length of ak will be 

 still equal to the same constant line bg as before : which line 

 is therefore found to be equal to the common radius of both 

 the diametral and circular sections of the ellipsoid, whether as 

 determined by the geometrical construction which the calculus 

 of quaternions suggested, or immediately by that calculus 

 itself. 



^5. We may write the equation (21.) of the ellipsoid as 

 follows: 



/(p) = l, ...... (27.) 



if we introduce a scalar function^/of the variable vector p, 

 defined as follows : 



[a^—i'^ffip) = {ip + px)(pi + xp) = ip^j -}- ipxp + pxpi + pK^p ; 



or thus, in virtue of article 20, 



(x2_,2^V(p) = (.2 4.x2)p2 + 2S..pxp. . . (28.) 



Let p+T denote another vector from the centre to the sur- 

 face of the same ellipsoid ; we shall have, in like manner, 



/(F + r) = l, (29.) 



where 



/(f-fr)=/(/») + 2S.VT+/(r), . . . (30.) 



if we introduce a new vector symbol v, defined by the equation 



{K^^i^fv={i^ + x^)p + ipK + xpi; . . . (31,) 



because generally, for any two vectors p and t, 



(p-f-T)' = p' + 2S.pr + T2, .... (32.) 



and, for any four vectors, i, x, p, t, 



S.iTXp = S.TXp»=S .xp»T=S .p<Tx; . . (33.) 



which last principle, respecting certain transpositions of vector 

 symbols, as factors of a product under the sign S., shows, 



