490 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
and any straight line 
u.— uiynz—0 ) 
then the power of the line and conic is and the power of this straight 
line and the line 
a!=Vx + my + nz =0, 
is W mm -\-nn. 
Again, we must define the power of S and u to be a, and the power of S and a to 
be zero ; the power of S with respect to S to be unity. 
15. Exactly as in § 8 we can show that for any two systems of four conics inscribed 
in S, 
(1, 2, 3, 4\ 
\5, 6, 7, 8 1 
= n 
b 2, 3, 4 
b 2, 3, 4 
X 
(5, 6, 7, 8\1 
\5, 6, 7, 8/J 
Chapter III.— Special Systems of Circles. 
Circles Touching three Straight Lines .—§§ 16, 17. 
16. Denoting the four circles which touch the sides of a triangle by (1, 2, 3, 4), and 
the sides of the triangle by a, h, c, we have, if x denote any other circle, 
i.e.. 
and since 
we have at once 
/6, a, b, c, x\ 
U b 2, 3, 4 ) 
= 0, 
0 , 
1 , 
b 
b 
T 
± 
0 , 
Tiff, ]5 
7 Ta, 2> 
77V, 35 
TTa, 4 
0 , 
TfylJ 
Wb 5 
17 6 , s, 
TTb, 4 
0 , 
77" Cl 1, 
7 Tg t g> 
7 Tc,3) 
TT c \ 4 
b 
7D;, 1 j 
7 r.c, o> 
7 D, 3’ 
^,4 
l 
TT a , i = 777, )3 =— &C., 
= 0; 
bV 
r. 
Vo r 
Th is theorem will be subsequently extended. 
17. Again let x denote the nine-points circle of the triangle, z the inscribed circle, 
and let (1, 2, 3) denote the mid-points of the sides; then since 
77 V, i — 77".r t 2 = — 7 T. V) 3 = 0 
= — c) 3 ; 77 % 0 = (c — a) 3 ; Tr :/i =(ct — b ) 2 ; 
