202 
ON A CORRESPONDENCE OF POINTS 
[512 
and if we consider a second point P', the coordinates of which are x, y', z' and 
X', Y', we have the like relations between these quantities. Calling 8 the distance 
of the two points P, P', we may write 
S 2 = {«! (x - x') + a 2 (y - y') + a 3 (z - ft)} 2 
+ {ft (x - x) + ft (y - y) + ft 0 - z')Y 
- {(«I 2 + ft 2 ) 0-x) + (a 2 2 + ft 2 )(y ~ y ) + («3 2 + ft 2 ) (z - z')j 
X {(&• - x') + (y- y) + (z- z% 
the last term being in fact = 0; viz. this is 
& = ~ {(«2 - a 3 ) 2 + (ft - ft) 2 } (y - y') (z-z') 
~ {(«3 - «i) 2 + (ft - ft) 2 } (z - z') (,x - x') 
- {(«i - a 2 ) 2 + (ft - ft) 2 } (x - x) (y - y); 
or what is the same thing, the expression for the distance 8 2 of the two points P, P' is 
8 2 = — a 2 (y — y') (z — z) — b 2 (z — z) (x — x') — c 2 (x — x') (y — y'), 
which expression may be modified by means of the identical equations 
viz. writing 
we have 
and consequently 
l=x + y + z, 1 = oc + y' + / ; 
yZ - y'z, ZX - ZX, Xy - Xy = £ 7], £ 
X — x' = X (x' + y + z') — x' (x + y + z) = £ — 7), 
y-y = f- £ 
z-z' =V~Z' 
+ b 2 (- r) 2 + v t; - ££ + fy) 
+ c 2 (- £ 2 + vt + £? - Zv)- 
2. Treating x', y', 2! as constants and x, y, z as current coordinates, the formula 
for 8 2 is of course the equation of a circle, centre x, y, z! and radius 8. It thus 
also appears that the general equation of a circle is 
— a 2 yz — b 2 zx — c 2 xy + (Lx + My + Nz) (x + y + z) = 0; 
viz. writing — a 2 yz — b 2 zx — c 2 xy = U, and x + y + z = LL, this is 
U + (Lx + My + Nz) il = 0, 
where U= 0 is the circle circumscribed about the triangle ABC, and H = 0 is the 
line infinity. Of course the general equation of a circle passing through the points