[158 
158] 
A SIXTH MEMOIR UPON QUANTICS. 
587 
adrantal 
distance 
sider a 
ith the 
ne con- 
and in 
ents of 
to such 
f lines, 
listance 
rder to 
sary to 
y stems 
in the 
7r, the 
or the 
points, 
e case 
be an 
le, the 
on not 
points, 
Dole of 
n like 
)ly the 
[ual to 
istance 
ad we 
îe, viz. 
ne, or 
pole 
zero, 
216. These properties of the circle lead immediately to the analytical expressions 
for the distances of points or lines in terms of the coordinates. In fact, take 
{a, b, c, /, g, hjx, y, zf = 0 
for the point-equation of the Absolute; its line-equation will be 
(a §, mm y, &=q. 
The point-equation of the circle having the point (x', y', z') for its centre, is 
(a, ...$#, y, zf (a, y', z'f cos 2 0 - {(ft, y, z\x, y', z')} 2 = 0, 
or 
(a, ...$#, y, zf (a, ...$V, y', zf sin 2 6 — (a ...\yz' — y’z, zx - zx, xy' — x'yf — O, 
from which (by the same reasoning as for the case of geometry of one dimension) it 
follows that the distance of the points (x, y, z), {x, y, z') is 
cos - 
(a, y, z\x, y, z) 
V(ft, ...$#, y, zf \/(a, y', zf ’ 
or what is the same thing, 
sin-1 V(a .yz , ZX - z’x, xy' - x'yf . 
' V(a, ...$#, y, zf V (ft, y, /) 2 
and it appears from the cosine formula (see ante, No. 208), that if P, P', P" be 
points on the same line, then we have, as we ought to have, 
Dist. (P, P') + Dist. (P', P") = Dist. (P, P"). 
217. In like manner, the line-equation of the same circle, the line-coordinates of 
the axis being (£', y, £"), is 
(a. v, rmv, v, r$r, v. o)*=». 
(a, ,, ...m (yco*e-K(a, sr-rf. fu'-W-o, 
from which it follows that the distance of the lines (£, ??, £) and (£', 7/, £') is 
cos - 
(a, y, nr, y, r> 
or what is the same thing 
■■■$<- V?, ¡Tf- rf, ft' 
?>»v<a,v, n* 
sin 
74—2