514 
MR. R, LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
50. Again if L denote any straight line, since 7 r M = 0 , and 7 r LiL = — 2 ; we see that 
its coordinates (X, p, v, p ) must satisfy, (i) the homogeneous linear relation, 
or 
(L,l, 2,3,4\ 
>,'12,3,4] U ’ 
dyjr dy[r df fy_ n . 
X S 1 ‘ rf ‘0fc +! '07, i +P Si 1 “° ’ 
(72) 
and (ii) the non-homogeneous quadric relation, 
or 
XI ( L ’ 2 > 3 ' 4 )=o 
\L, 1, 2, 3, 4/ ’ 
ip(\, p, v, p)=— K. 
(73) 
The Circle .—§§ 51 - 55 . 
51. Let P be any point on a circle S, then 7 r S)I .=0 ; hence the equation 
leads to 
/p l 2 3 4\ 
n( ’ =o 
\s, 1, 2, 3, 4/ 
8-^Jr Bxfr Byfr 
0^+0^+0f + 0^=°- 
(71) 
Thus the equation of a circle is of the first degree. 
It follows that the general equation of the first degree 
ax-\-by-\-cz-\-dw— 0 
represents in general a circle, whose coordinates are given by, 
dx[r d-yjr 0\|r 
dg dt] 0f 0co Iv 
abed a\ + bk% + ck s + d/q 5 
by equation (71). 
52. Given any two circles whose coordinates are (£, rj, £, co ), (f, V, C, *'); 
power 7 t is given by 
'S, 1, 2, 3, 4' 
(75) 
their 
or 
ik g ; ^ 2 -’-)=o, 
rr 1 rfy d'lr ,df 'I 
"- K -£dj:+va -+^+ w a M 1 
-£d? +v d v ' +c dc +0) dco' 
(76) 
