MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
601 
235. The coordinates of the plane at infinity are clearly k Y , k 2 , Ic 5 , k 5 , and so the 
equation of the plane at infinity is 
d'ylr d\Ir Cyjr Cylr O'dr 
x &+ y sk+ z &.+ w dk l +v sr 
The Point.—% 236-238. 
236. The power of the point ( xyzwv ) with respect to the surface 
ax-\-by-\- C 2 + dw- f- ev= 0, 
ax + by + cz + dw + ev 
ak x -f- blc 2 -f- dkg "I" dk^ -f- dk^ 
will be equal to 
if the surface is a sphere, and will be equal to 
ax-\-by + cz + dw + ev 
b, c, d, e) ’ 
if the surface is a plane. 
237. The power of the two points [xyzwv), (x'y'zw'v) will be given by 
Hence, since 
,d\fe ,d\fr d-fr , d\fr , d^r 
—• 2tTr—x — \-y ~\~z ~ -J ~w ~— b v . 
dx J dy dz dw dv 
\jj(x, y, z, w, v) = ^(x, ij, z, id, v')=0, 
the distance 8 between the two points will be given by 
28 ^=\jj{x—x', y—y, z—z, w—io', v—v}. ..... (269) 
238. Let P, Q, It, S be any four points, then by equation (216) we have for the 
volume of the tetrahedron formed by them, 
288.{V(P, Q, R, B)}*=n(££2£!); 
and by § 204, 
16, V, Q, R, S\ [1, 2, 3. 4, 5\ j (6, P, Q, R, S\ 1 « 
V P. Q, K. 8/ x n Vl ; 2, 3, 4, 5r\ L \ 1, 2, 3, 4, 5 ) J * 
If then the coordinates of P, Q, Pt, S referred to (1, 2, 3, 4, 5) be (x\y\Z\W\V\) 
{x 2 y 2 z 2 w 2 v 2 ) {xyyyzywyv^x^y.yz we see that w 7 e shall have 
mdccclxxxvi. 4 H 
