76 
ME. A. catlet’s sixth MEMOIE ETOX QEAXTICS. 
191. If the point-equation of a conic be 
(d, h c,f, g, hXx, y, 2 )^= 0 , 
then its line-equation is 
or writing 
and to complete the system, 
I, ^ =0, 
I a, h, g 
fj h, b, f 
I ./*’ 
(^^ab —b? ■> 
f=gh~af, 
(B—hf—bg, 
-K=abc-af-bg^-ch^-{-2fgh, 
then the line-equation of the conic is 
192. The quantities 3, &c. satisfy the relations 
K''=9EBC— aiP— 38®”— Ci!'‘+2Jf©?S. 
a«+®A+®#=K> 
gA+®5+®/=0, 
ay +l/’+®c=0, 
®«+35i<+Jfy=0, 
®A+B5+Jf/=K, 
®y+S/'+jre = 0, 
0h+fb+€f=(», 
and moreover 
Ka=35C -4^^ K/=#il -91# , 
Kb =C91 -(§^ K^=1#-33(®, 
Kc=m -W, Kh=f0-€^. 
193 A system of points in a line is said to be a range, and a system of lines throng i 
a point is said to be a pencil. The theories of ranges and pencils, considered irrespect- 
ively of each other, are in fact a single theory, constituting the geometry ot one dimen- 
sion. It has been seen how in geometry of one dimension a range of points and a penci 
of lines, although considered (as they must be considered) as existing in distinct spaces, 
may nevertheless stand in certain relations to each other. In geometi) o tvo mien 
sions, the range and pencil may of course coexist in one and the same plane as t leii 
common locus in quo; and such coexistence occurs in fact very trequenty. tins i ue 
