Sir W. Rowan Hamilton on Quaternions, 279 



and making 



'^+^=jz:;^9? «-/3=;2Z^> • • • (2-) 



and 



!t±£^,=Q, f^=Q', .... (8.) 



the two linear factors of" the first member of the equation (1.) 

 become the two conjugate quaternions Q and Q', so that the 

 equation itself becomes 



QQ'=1 (4.) 



But by articles 19 and 20 (Phil. Mag. for July 1846), the 

 product of any two conjugate quaternions is equal to the square 

 of their common tensor; this common tensor of the two qua- 

 ternions Q and Q' is therefore equal to unity. Using, there- 

 fore, as in those articles, the letter T as the characteristic of 

 the operation o^ taking the teiisor of a quaternion, the fequaiion 

 of the ellipsoid reduces itself to the form 



TQ = 1; (5.) 



or, substituting for Q its expression (3.), 





(6.) 



which latter form might also have been obtained, by the sub- 

 stitutions (2.), from the equation (3.) of the 30th article (Phil. 

 Mag., June 1847), namely from the following*: 



T(«p + pa + /3p-p/3)=l (7.) 



38. In the geometrical construction or generation of the 

 ellipsoid, which was assigned in the preceding articles of this 

 paper (see the Numbers of the Philosophical Magazine for 

 June and September 1847), the significations of some of the 

 recent symbols are the following. The two constant vectors 

 « and X may be regarded as denoting, respectively, (in lengths 

 and in directions,) the two sides of the generating triangle 

 ABC, which are drawn from the centre c of the auxiliary and 

 diacentric sphere, to the fixed superficial point b of the ellip- 

 soid, and to the centre a of the same ellipsoid ; the third side 

 of the triangle, or the vector from a to b, being therefore de- 

 noted (in length and in direction) by » — x: while p is the 

 radius vector of the ellipsoid, drawn from the centre a to a 



• See equation (35.) of the Abstract in the Proceedings of the Royal 

 Irish Academy for July 1846. The equation of the ellipsoid marked (1.) 

 in article 37 of the present paper, was communicated to the Academy in 

 December 1845, and is numbered (31.) in the Proceedings of that date. 



