Sir W. Rowan Hamilton on Quaternions. 429 



But the versor of everi/ vector is, in this calculus, a square 

 root of negative unity ; we have therefore in particular, 



(U(n-9))2=-l; (162.) 



and under the sign TV, as under the sign T, it is allowed to 

 divide by —1, without affecting the value of the tensor: it is 

 therefore permitted to write the equation (160.) under the 

 form 



which form is thus demonstrated anew. 



75. A few connected transformations may conveniently be 

 noticed here. Since, for any quaternion Q, 



(TVa)2= -(VQ)2 = (TQ)2-(SQ)S . (163.) 



while the tensor of a product is the product of the tensors, 

 and the tensor of a versor is unity ; and since 



S.(p>j-6/j)(ij-fl) = S(p)j2-p»j6-ep>} + flp5) = ~2S.>jdp, (164.) 



because 



= S.pr=:S.QpQ,andS.pyiQ = SJpri=sS,ri$p; . (165.) 



we have therefore, generally, 



T.(/,,-5p)U(,,-fl) = T(p,,-fl|.); 1 . 



S.(p>}-fl^)U(»-9)=-2T(.j-d)-'S.)j9p;J * '' 



and there results the equation, 



TV.{pn-^p)\3{r,-^)=V{T{pn-^Y-^T{yi-^)-^{S.n^pf},{\61,) 



as a general formula of transformation, valid for any three 

 vectors, >j, $, p. We may also, by the general rules of the 

 present calculus, write the last result as follows, 



TV.(/pij-flp)U(>3-9)=i/{(p.j-9p)(>jp-p^) 



-^{n-^y^yfip-phf}', ..... (168.) 



the signs S and T thus disappearing from the expression of 

 the radical. For the ellipsoid, this radical, being thus equal 

 to the left-hand member of the formula (167.), or to that of 

 (168.), must, by (161.), receive the constant value &^ — r^; so 

 that, by squaring on both sides, we find as a new form of the 

 equation (161.) of the ellipsoid, the following: 



Or, by a partial reintroduction of the signs S and T, we find 

 this somewhat shorter form : 



T(f'J-flf)H4(>j-fl)-^(S.>)flp)2=(fl2-,,2)2. . (170.) 



