428 Sir W. Rowan Hamilton on Quaternions. 



tion, which is made by the cyclic plane having for equation 



S.{6~ri)p = 0; (153.) 



this plane being found by the consideration that vj — Q has the 

 direction of the cyclic normal i, or by making the coefficient 

 ^=0, in the formula (148.). 



74. The equation (149.) of the ellipsoid may be successively 

 transformed as follows : 



= T{&^p-ri{6p-{-p^)+r,pri} 



= Ty{{Q-r,)&P^np{d-ri)} 



= TV.{pd-np){Q-ri) 



= TV.(>,p-p9)(,,-9); (154.) 



and by a similar series of transformations, performed on the 

 equation (150.), we find also (remembering that fl^— >3^, being 

 equal to x^—t,% is positive), 



b{l^^n^)^Ty.{pyi-Qp){yi-6). . . . (155.) 



The same result (155.) may also be obtained by interchanging 

 vj and 9 in either of the two last transformed expressions (154.), 

 for the positive product b{Q'^—ry^) ; and we may otherwise 

 establish the agreement of these recent results, by observing 

 that, in general, if Q and Q' be any two conjugate quaternions 

 (see Phil. Mag. for July 1846), such as are here rip—pQ and 

 pYi — &p, and if a be any vector, then 



TV.Qa=TV.Q'a; (156.) 



for 



and because 



V.Qa=:«SQ-V.«VQ, n 



V.Q'«=aSQ + V.«VQ;/ • • • ^ '•-' 



= S.aV.aVQ, (158.) 



the common value of the two members of the formula (156.) is 



TV.Q«=V'{(TV.«VQ)2 + (T«.SQ)2}. . (159.) 



If then we substitute for b its value, 



b=T{r}-$), eq. (135.), art. 70, 



and divide on both sides by this value of b, we see, from (154.), 

 (155.), that the equation of the ellipsoid may be put under 

 either of these two other forms : 



TV.()jp-pd)U()j-5)=a2-,,2, . . . (160.) 



TV.(p)j-9p)U(*3-fl) = fl2-)j2. . . . (161.) 



