287 
KQ = 8Q— VQ, or, more concisely, K=s—v. (33) 
And if we further observe, that in general the product of 
two conjugate quaternions is equal to the square of their 
- common tensor, 
. @ X KQ= (sQ)’— (va)? = (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 + pa+ Bp — pB) = 1. (35) 
Again, by employing the principle, that TI] = [r, we 
y again decompose the first member of (35) into two fac- 
t rs, and may write the equation of an ellipsoid thus : 
T(a+P+oe).1tp=1, (36) 
if we introduce an auxiliary vector, «, connected with the vec- 
_ tor p by the relation 
3 o=p(a—f) p“', (37) 
which gives, by the same principle respecting the tensor of a 
product, 
ee Ts = T(a— 3); | (38) 
) that the auxiliary vector o has a constant length, although 
has by (37) a variable direction, depending on, and in its 
assisting to determine or construct the direction of the 
rp of theellipsoid ; for the same equation (37) gives for 
versor of that vector the expression 
up = + v(a—B +0). (39) 
, Henee, by the second general decomposition (20), and by 
the equation (36), the last mentioned vector p itself may be 
xpressed as follows : 
_v@-P+°), 
40 
~ t(a+ B+)’ (40) 
_ making then, in the notation of capital letters for points, 
VOL. Il. zZ 
