118 The Circle.

12. A coaxal system having real limiting points is the inverse of a con-
centric system, and a system having imaginary limiting points the inverse
of a pencil of lines.

13. If a variable circle cut two given circles of a coaxal system at given
angles, it cuts every circle of the system at a constant angle. This may
be seen at once by inversion : or without inversion, as follows :—If 8 = #2
+ y2 + Zgx + 2/y 4- c = 0 cuts S' ss #2 + y2 + Zg'x + 2fy + cr = 0 and
S" zz x2 + y~ 4- Zg"% + 2/"y 4- c" = 0 at angles <£', <£", it cuts the circle
/S' — faS" = 0 at the angle

"^{''""jtv-™*"}' (282)

where E denotes the radius of 8' — kS" = 0.

14. The radical axes of the circles of a coaxal system and a circle which
is not one of the system are concurrent.

15. The circles & + y» - Ihx -f 52 = 0, z2 4- y- - 2% - b2 = 0 cut
orthogonally.

DEF. — I%e two points wJiicJi divide tlie distances between the centres of two
circles internally and externally in the ratio of their radii are called the
centres of similitude of the circles.

Thus if x2 4- y2 + 2p? -f 2/y + tf = 0, ^ + ?/2 + ty'x + 2/'y + c' = 0 be
two circles, their centres of similitude are —

internal, the point

'

r + r '

(283)
and eternal, --

16. If S, S' be two circles whose radii are r} r', prove that their internal
centre of similitude is the centre of — -1- -r = 0, and the external one, the
r r
* S S/ n
centre of - — 7=0.
r r
O CV
17. If S, S' be two circles, - ± ~ = 0 will invert one into the other:
in what respect do these inversions differ ?