494 Sir W. Rowan Hamilton on Qiiaternions, 



The equation (192.) will then give, 



T(p>j-6p)2=V()3 + 5)2. / • • • ^ 

 We have also the identity, 



(92_^2)2^(^__5)2(^4.e)S+(^9_5^)2. . . (195.) 



which may be shown to be such, by observing that 



(,,_e)2(^ + fi)2=(,2 4.e2_2S.>j9)(,,2 + 9H2S.>)5) 



= (^2 + fi2)2_4(S.^9)2^(^2_52)2^4.(X.,,fl)2_4(S.>)fl)2 



= (,,2_fl2)2_4.(V.>j9)2=(62_»,2)2_(^9_e^)2. , . (igg.) 



or by remarking that (see equations (152.) (163.)), 



,^-feS.(,,-fl)(>, + e), >]9-fl>,=V.(),-S)(,, + 9),-| 



and(>)-S)2(n + e)2=(T.(^-fl)(l + 5))^; -J ^ 



or in several other ways. Introducing then a new vector e, 

 such tliat 



,,9-fi>j = sT(»i4-S), or, £=2V.>i3.T(*) + 5)-'; . (198.) 



and that therefore 



(,,9-e,,)2=_s2(^ + 5)2, .... (199.) 

 and 



2S.o36p = S.£p.T(*)+9), 4(S.r)9^)2=-(S.e^)2(») + 6)2; (200.) 



while, by (135.), 



T(*j-fi)=5, ()j-6)^=-&^; . . . (201.) 

 we find that the equation (170.) of the ellipsoid, after being 

 divided by (>) + 5)^ assumes the following form : 



Ai2^&-2(S.ep)2 + 6^ + 82=o. . . . (202.) 

 But also, by (193.), (198.), 



S.eXi = S.g^; (203.) 



the equation (202.) may therefore be also written thus : 



0=(Xi-e)2+ (6 + 6-^8. sp)2; . . . (204.) 



and the scalar b + b~^^.sp is positive, even at an extremity of 

 the mean axis of the ellipsoid, because, by (195.) (199.) (201.), 

 we have 



(^2-»,2)2=-.(^,2 + s2)(^^g)2=(^,2_X62)T()3 + e)2, . (205.) 



and therefore 



Ts<6 (206.) 



We have then this new form of the equation of the ellip- 

 soid, deduced by transposition and extraction of square roots 



