and a hyperboloid of two sheets, with the. same asymptotic- 

 cone (26), and with the tangent planes (24), is represented 

 by the formula 



(ap+pa)^ + Op - p(5r = 1. (27) 



By changing p to p — y, in which y is a 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 7ime 

 given points, may thus be regarded as equivalent to the pro- 

 blem of finding three constant vectors, a, /3, 7, which shall, 

 for nine given values of the variable vector p, satisfy one 

 equation of the form 



{«(p-7) + (p-y)«}'- ± ^|3(p-7)-(p-7)/3}^= ± 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 -H pa = 2s . ap, /3p — p/3 = 2v . j3p ; (29) 



so that, by making for abridgment 2a = a', 2j3 = [5', the 

 equation (21) of the ellipsoid (for example) will take the 

 shorter form, 



(s.a'p)^-(v./3'p)^=l. (30) 



Another modification of the notation, which, from its geo- 

 metrical character, will often be found useful, or at least illus- 



