[Let p be the angular radius of the variable circle, d1 and d2 the distances of its pole from the poles of given circles, t1 and to the tangents from its pole to them; then

Therefore

and

Similarly

cos 81 = cos p cos r1 + sin p sin r1 cos a.

cos ti = cos p + sin p tan ri cos a;

cos T1 = 1 + tan p tan ri cos a.

cos T2 = 1 + tan p tan r2 cos B. Therefore, &c.]

15. If a circle touch two fixed circles, the ratio of the sines of the halves of the tangents to it from their limiting points is constant.

SECTION IV.

Centres and Axes of Similitude on the Sphere.

161. Problem.-To draw a great circle touching two small circles on a sphere.

Let TT (fig. 65) be a great circle touching two small circles at T and T2, respectively. Let P, and P2 be the poles of the small circles; then PT, and PT, meet in

2

2

P, the pole of TT2. Now, the sides PP, and PP, of the triangle PPP, are 90 - r, and 90 - 2, where r and r2 are the radii of the circles; and therefore if the small circles be given, the point P is at once determined, and the points T1 and T2, where PP, and PP, produced meet the circles, give the points of contact of the required common tangent.

Cor.-It is always possible to draw a pair of direct common tangents, except when one circle is entirely within the other; and a pair of transverse common tangents when one circle is entirely outside the other.

162. Centres of Similitude.—The point of intersection of the direct common tangents to two small circles is called their External Centre of Similitude, and the point of intersection of the transverse common tangents their Internal Centre of Similitude.

It is clear that the centres of similitude of two circles lie on the great circle joining their poles. For OP bisects the angle between the direct common tangents, and so also does OP2. Therefore O lies on P1P2; similarly lies on P1P2.

Cor. The point of contact of two circles which touch externally is their internal centre of similitude, and the point of contact of two which touch internally is their external centre of similitude.

163. Theorem.-The centres of similitude divide the arc joining the poles of the circles internally and externally into segments, whose sines are in the ratio of the sines of the spherical radii of the circles.

For (fig. 65) if r1 and r2 denote the spherical radii,

and similarly for the internal centre; therefore, &c.

Cor. 1.-Any great circle through a centre of similitude makes equal angles with the radii to the points where it cuts the small circles.

[For (see fig. 65), sin OP, sin P,OB, = sin r, sin OB,P1,

and sin OP, sin P,OB1 = sin ra sin P2A,B2.

2

1

P1A,B1 = P1BA1 = P2A,B2 = P2B2A2.]

Cor. 2.-Conversely-If an arc of a great circle cuts two small circles, and makes equal angles with the radii to the points of section, it passes through a centre of similitude.

Cor. 3.-Hence, if a small circle touch two other small circles, the arc joining the points of contact passes through a centre of similitude.

Cor. 4.-The centres of similitude and the poles of the given circles form a harmonic row.

[Art. 136].

Definition.—Of the four points A1, B1, A2, B2 (fig. 65), in which any great circle through a centre of similitude of two circles meets them, the points A, and A2, or B1 and B2, are said to correspond directly, and the points A, and B2, or B1 and A2, are said to correspond inversely.

164. Theorem.—If two great circles be drawn through a centre of similitude of two small circles, any pair of the points in which they intersect the small circles are concyclic with their inversely corresponding points.

2

Consider the quadrilateral (fig. 65) B‚ÂÂ1⁄2 ́B ́ formed by the points B1, Bí, and their inversely corresponding points A2, A2. Denote each of the equal angles of the triangle P,B,B by x, and those of PA,A by y, and

let denote the equal angles P,B,0 and P2A,B2, and φ the angles P.BO and P2AB. Then, for the angles of the quadrilateral, we have

and therefore the quadrilateral B1A,A,B is such that the sum of one pair of opposite angles is equal to the sum of the other pair; that is, it is inscribable in a small circle.

Cor. 1.-Hence,

tan OB, tan OA, = tan OB, tan

[Art. 19 (4).]

OA?

=tan OT, tan OT,.

Therefore-The product of the tangents of the halves of the arcs intercepted between either centre of similitude and a pair of points inversely corresponding is constant. Cor. 2.-Combining Cor. 1 with Art. 43, we find

tan 04, tan OB1 tanOT

=

=

tan OA2 tan OB2 tan OT

Therefore-The tangents of the halves of the arcs between either centre of similitude and a pair of corresponding points are in a constant ratio.

Cor. 3—If B,B1, and Ad be produced to meet at R, we have

tan RB, tan RB, tan RA, tan RA,

=

and therefore R is on the radical axis of the two circles.

Therefore-The arc joining a pair of points meets the arc