Sir W. Rowan Hamilton on Qiiaternions. 459 



Joints, may thus be regarded as equivalent to the problem of 

 finding three constant vectors, «, /3, y, which shall, for nine 

 given values of the variable vector p, satisfy one equation of 

 the form 



{«0'-r) + (p-r)«}'±{^(p-r)-(p-7)/3]-=±i; • (9.) 



with suitable selections of the two ambiguous signs, depending 

 on, and in iheir turn determining, the particular species of 

 he surface 



30. The equation of the ellipsoid with three unequal axes, 

 referred to its centre as the origin of vectors, may thus be 

 presented under the following form (which was exhibited to 

 the Royal Irish Academy in December 1845): 



(«p+p«2)_(^^_p,3)2^1. .... (1.) 



and which decomposes itself into two factors, as follows : 



(«p+p« + /3p-p/3)(«p + p«-/3p + p/3) = l. . . (2.) 



These two factors are not only separately linear with respect 

 to the variable vector p, but are also (by art. 20, Phil. Mag. 

 for July 1846) conjugate quaternions', they have therefore a 

 common tensor, which must be equal to unity, so that we may 

 write the equation of the ellipsoid under this other form, 



T(«p + p« + ^p-p/3) = l; (3.) 



if we use, as in the 19th article, Phil. Mag., July 184-6, the 

 characteristic T to denote the operation of taking the tensor 

 of a quaternion. Let a- be an auxiliary vector, connected 

 with the vector p of the ellipsoid by the equation 



tr=p(«-/3)p-i; (4.) 



we shall then have, by (3.), and by the general law for the 

 tensor of a product, 



T(« + /3 + o-).Tp = l; (5.) 



but also 



(«-/3 + (7)p = («-/3)p + p(a-/3), . . . (6.) 



where the second member is scalar ; therefore, using the cha- 

 racteristic U to denote the operation of taking the vcrsor of a 

 quaternion, as in the same art. 19, we have the equation 



U(«-^ + (r).Up= + l; (7.) 



and the dependence of the variable vector p of the ellipsoid on 

 the auxiliary vector o- is expressed by the formula 

 U(«-/3 + .) 



''-±T(« + /3 + (r) V8.) 



Besides, the length of this auxiliary vector or is constant, and 

 ecjual to that of « — /3, because the equation (4.) gives 



T(r = T(«-/3); (9.) 



