xlii 
gy’ being a new disposable scalar, and 8 a new variable vector ; 
and, after having cleared the equation (15) of the radical 7 or 
v¥(— a’), by writing it as follows, 
2 re (p ae : ai ae (33) 
we get 
8 2 
0= (gc +8)? + (p + 9/e— =t#) (34) 
2 
— eo ae = ‘ ~ + 9"(8_ 2 £0) a | (p + g/e)? — Oe: 
if we make for abridgment 
g" = 9 — (p+ ge’). (35) 
If y’ or g’c is to be the constant vector of the centre of the 
locus, it is necessary that to every variable vector, 3, which 
satisfies the equation (34), should correspond another ve ctor 
— 6, equal in length but opposite in direction, and satisfying 
the same equation ; therefore the terms g’(d< + <d) must dis- 
appear, and we must have 
Fgh | AE: a vf = iy (36) 
the constant a being thus suggested by the search after a cen- 
tre, as well as by the search after a second focus. Making 
then g’=a in (34), we find the following equation of the 
surface, when referred to its centre, 
2 
0= 8 4 ms es + ap; (37) 
in which 
ap = a(P + ey = a1 4 2). (38) 
And because in general,-for any two vectors 6, «, the fol- 
lowing relation holds good, 
(2 + ay es = = + 82, (39) 
