Sir W. Rowan Hamilton on Quaternions. 435 



(according to the rules of the present calculus), from the form 

 (204.) : 



T(Ai-6)=^+&~'S.6^ (207.) 



By a process exactly similar to the foregoing, we find also the 

 form 



T(Al + ^) = &-i-^S.s/>; . . . (208.) 



which differs from the equation last found, only by a change 

 of sign of the auxiliary and constant vector s : and hence, by 

 addition of the two last equations, we find still another form, 

 namely, 



T(Ai-6)+T(Ai + 6) = 26; . . . (209.) 



or substituting for Xj, e, and b their values, in terms of »), 9, 

 and p, and multiplying into T(jj4-9), 



= 2T.(»,-9)(,, + 6) (210.) 



79. The locus of the termination of the auxiliary and vari- 

 able vector Aj, which is derived from the vector p of the original 

 ellipsoid by the linear formula (193.), is expressed or repre- 

 sented by the equation (209.) ; it is therefore evidently a cer- 

 tain new ellipsoid, namely an ellipsoid of revolution, which has 

 the mean axis 2b of the old or given ellipsoid for its major 

 axis, or for its axis of revolution, while the vectors of its two 

 foci are denoted by ihe symbols +e and — e. If « denote the 

 greatest, and c the least semiaxis, of the original ellipsoid, 

 while b still denotes its mean semiaxis, then, by what has 

 been shown in former articles, we have the values, 



T,,=Ti=i(a + c); T9 = Tx=i(a-c); . (211.) 



and consequently (compare the note to art. 70), 



a=Tij + Tfl; c=Tti-TQ; . . . (212.) 



therefore 



«c=T)j2-Td2=62_;,2. .... (213.) 



also 



= 2T>j2 + 2Tfl2 = (T>j + Te)2 + (T)} - T&)% (214.) 

 and 



T(>) + d)2 = fl2_j2 + c2; .... (215.) 



whence, by (205.), 



T.W.-^-^ = (-^^^. . (...) 

 2F2 



