lix 
and a hyperboloid of two sheets, with the. same asymptotic 
cone (26), and with the tangent planes (24), is represented 
by the formula 
(ap + oa)’ + (Bo — pf)’ = 1. (27) 
By changing p tog — y, in which y isa third arbitrary 
but constant vector, we introduce an arbitrary origin of vec- 
tors, or an arbitrary position of the centre of the surface as 
referred to such an origin; and the general problem of deter- 
mining that individual surface of the second order ( supposed 
to have a centre, until the calculation shall show in any parti- 
cular question that it has none), which shall pass through nine 
given points, may thus be regarded as equivalent to the pro- 
blem of finding three constant vectors, a, B; y, which shall, 
for nine given values of the variable vector g, satisfy one 
equation of the form 
talo—y)+(o—y)a}> + {B(o—y)—(o—7) B}?= + 1; (28) 
with suitable selections of the two ambiguous signs, depending 
on, and in their turn determining, the particular nature of the 
surface. It is not difficult to transform the equation (28), or 
those which it includes, so as to put in evidence some of the 
chief properties of surfaces of the second order, with respect 
to their circular sections. 
The recent expressions may be abridged, if we agree to 
employ the letters s and v as characteristics of the operations 
of taking separately the scalar and vector parts of any quater- 
nion to which they are prefixed ; for then we shall have 
ap + pa = 28. ao, Bo — eB = 2v. Bo; (29) 
so that, by making for abridgment 2a = a’, 23 =)’, the 
equation (21) of the ellipsoid (for example) will take the 
shorter form, 
(s. a’)? — (v. B’p)? = 1. (30) 
Another modification of the notation, which, from its geo- 
metrical character, will often be found useful, or at least illus- 
