287 



KQ = SQ — VQ, or, more concisely, k = s — v. (33) 



And if we further observe, that in general the product of 

 two conjugate quaternioris is equal to the square of their 

 common tensor, 



Q X KQ= (sq)^- (vq)' = (TQ)% (34) 



we shall perceive that the equation (32) of an ellipsoid may 

 be put, by extraction of a square root, under this simpler, but 

 not less general form : 



T(ap + |oa+i3p-pi3)= 1. (35) 



Again, by employing the principle, that Til =■ IIt, we 

 may again decompose the first member of (35) into two fac- 

 tors, and may write the equation of an ellipsoid thus : 



T(a+/3 + <T).T/,= 1, (36) 



if we introduce an auxiliary vector, <t, connected with the vec- 

 tor p by the relation 



a = p{a-ft)p-\ (37) 



which gives, by the same principle respecting the tensor of a 

 product, 



T(7 = T (a - /3) ; (38) 



so that the auxiliary vector a has a constant length, although 

 it has by (37) a variable direction, depending on, and in its 

 turn assisting to determine or construct the direction of the 

 vector p of the ellipsoid ; for the same equation (37) gives for 

 the versor of that vector the expression 



u/j = ± u (a - /3 + (t). (39) 



Hence, by the second general decomposition (20), and by 

 the equation (36), the last mentioned vector p itself may be 

 expressed as follows : 



_ U (g - /3 + t) . ... 



^-T-(7+T+^)' ^ ^ 



making then, in the notation of capital letters for points, 



VOL. III. z 



