MR, R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
489 
Let us define, then, the power of the two conics crS — L, and a' 2 S—M 2 , as the 
expression ad and let us denote this by tt. Thus 
7T =ad + hh '; 
the conic S being supposed reduced to its standard form, and f, g, h being the co¬ 
ordinates of the pole of L with respect to S. 
, General Theorem .—§§ 13-15. 
13. If we have any two systems of conics, say (1, 2, 3, 4, 5), (6, 7, 8, 9, 10), 
inscribed in the same conic S, the powers of the conics of one system with respect to 
the conics of the other system are connected by the relation 
n 
=o. 
T, 2, 3, 4, 5 
7, 8, 9, 10, 
i.e., the same equation as in § G. 
Thus taking the same equations as in § 10, by multiplying the matrices 
( 12 ) 
cq, 
fl> 
9i> 
K 
«6> 
J 6, 
9b, 
h 6 
fi> 
fh, 
h 
« 7 , 
A 
97, 
h 7 1 
a 3, 
/s» 
9z, 
h 3 
«S> 
a 
9s, 
h 8 
A 
9r 
K 
«0, 
A 
9o, 
h, 
«5> 
A 
9b, 
h 
«10> 
flQ’ 
9w> 
K i 
we have at once the relation 
^l.O’ 7r J,7’ 
77 -2,6’ 77 2,7, 
n 3, 6 > U 3,7’ 
6> 7r 4,7» 
n o,Q’ n b,7’ 
77 1, 8> 7T 1,0> 7r l, 10 
ir-2,6, n ~, i0 
7*3,8> 17 3, 0> 77 3, 10 
77 i, 8? 0> 77 J, 10 
77 o, 8’ 77 b, S’ 77 o, 10 
(13) 
14. This equation has been proved for two systems of conics inscribed m the same 
conic. The result is also true if any of the conics be replaced by straight lines, 
provided that we define the power of a straight line and a conic of the system to be 
the power of the straight line and the chord of contact of the conic and the conic S ; 
the power of two straight lines being defined as the perpendicular distance from the 
pole of one line with respect to S to the other line. Thus let any conic of the system 
be 
it = a 2 S — (fx-\-gy+hz)~=0, 
3 R 
MDCCCLXXXVI. 
