26. This principle makes it possible to draw a line mathematically straight; for in the four linkages* shown below the point P inverts into with respect to the centre O, and if P move in a circle passing through the fixed point O, then Q will move in a straight line.

In each linkage the bars (links) denoted by the same letter are equal.
To prove the property of inversion:

In linkages (1) and (2)

OP=OM-PM, and OQ=0M+ PM; then OP.OQ=0M2— PM2. OM2-a - LM2 and PM2—b2 — LM2.

But

In linkage (3) the points O, P, Q divide the links in the same ratio.

* A linkage is a system of bars pivoted together.

The original account of linkages (1) and (2) was published in "Nouvelles Annales," 1873; of (3) in the "Report of the British Association," 1874; of (4) in the "Report of the British Association," 1884.

27. The inverse of a circle is a second circle if the centre of inversion is not on the first circle.

Hint.-Let C be the first circle, O the centre of inversion, Q the inverse of P. Draw QC' parallel to RC to meet OC in C'.

Since OP.OR is constant and OP.OQ is constant,

is constant.

OR

Therefore C' is a fixed point and C'Q a constant length.

28. Exercise.-The centre of inversion is a centre of similitude of a given circle and its inverse.

29. Exercise.-If two circles touch each other, their inverses also touch each other.

30. Exercise.-A circle can be inverted into itself.

Hint. The constant of inversion must be equal to the square of the tangent drawn to the circle from the centre of inversion.

31. The inverse of a sphere is a plane, if the centre of inversion is on the sphere.

Hint.-Every point on the sphere is on a great circle passing through the centre of inversion. and will invert into a point of the plane; compare with

32. The inverse of a sphere is a second sphere, if the centre of inversion is not on the first sphere.

о

FIG. 32

Hint.-Compare with § 27.

33. If two circles intersect, their angle of intersection is equal to the angle of intersection of their inverses.

FIG. 33

X'

Hint.

The circles X and Y invert into X' and Y'.

A circle can be described tangent to X at P and passing through Q. This circle inverts into itself and is therefore tangent to X' at Q. $$ 30, 29 Likewise a circle can be described tangent to Y at P and passing through Q. This circle is tangent to y' at Q.

The angle at which these tangent circles intersect is equal to the angle at which X and Y intersect and also to the angle at which X' and ' intersect.

34. Cor.—If a straight line and a circle, or two straight lines, intersect, their angle of intersection is equal to the angle of intersection of their inverses.

35. A single inversion may be found equivalent to any series of an odd number of inversions from the same centre.

Hint. If a invert into b, b into c, c into d, . . . m into n where the number of inversions is odd, find an inversion by which a inverts into ». Why does not this theorem apply to an even number of inversions?

RADICAL AXIS AND COAXAL CIRCLES

36. The locus of a point, from which tangents drawn to two circles are equal, is a straight line perpendicular to the line of centres.

M

FIG. 36 (1)

M

FIG. 36 (2)

N.

Hint.-(1.) If the circles intersect, the locus is the common chord. (2.) If the circles do not intersect, let A be the point in the line of centres from which tangents to the circles are equal. Erect the perpendicular AP.

37. Def. The straight line, which is the locus of the points from which tangents drawn to two circles are equal, is the radical axis of the circles.

38. The three radical axes of three circles meet in a point.

FIG. 38

Hint.-The tangents drawn to the three circles from the point of intersection of two of the radical axes are equal; hence the third radical axis must pass through the point..

39. The difference between the squares of the tangents drawn from any point to two circles is equal to twice the product of the distance of the point from the radical axis by the distance between the centres of the circles.

Hint.-Let C be the centre of 00', AB the radical axis, PR the perpendicular from P on 00'.

40. Cor.-The square of the tangent drawn from a point on one circle to another circle is equal to twice the product of the distance between the centres of the circles by the distance of the point from their radical axis.

41. Def.—A system of circles such that some line is a radical axis common to every pair of circles of the system is a coaxal system.

Thus if AB is the radical axis of the circles X and Y, X and Z, Y and Z, etc., the system of circles is coaxal.