Again, for the radii of the circles inscribed in the four triangles, taken in the aforesaid order, we write r, r,,,,,", whilst, for the radii of the circumscribed circles, we put R, R,, R,,, R,,,, respectively. The unaccented letters referring to the circles of the fundamental triangles.

These triangles possess many beautiful properties when considered in their mutual association, which render them worthy of greater attention than has yet been bestowed upon them. Indeed, till very recently, their existence has scarcely been alluded to by writers on spherical subjects, and even to the present day, not more than three of their propositions have, we believe, been published.

2. Let O be the centre of the circle inscribed in the fundamental triangle, and G H, K, its points of contact with the sides, join AO, BO, CO, and draw the radii to the points of contact. Then the tangents from A to the circle are equal; that is, AK=AH; ing( like manner BK = BG, and CG=CH.

b = a + y

c = a + y.

Again, in the right-angled triangle BOG, we have

that is,

tan. OG sin. BG tan. OBG,

tan.r sin. ẞ tan. B;

=

or by (1) just given and (3), upon page 77, applied to B, we have

Again, in the supplemental triangle BA'C, denoting the quantities BB', CH', and AK', by ẞ, y, a,, we shall have

Hence, in the right-angled triangle BOʻGʻ, we have

In exactly the same way we find the other associated inscribed radii,

and the whole tabulated gives

These formula were first given by Professor Lowry (1829), Leybourn's Repository, vol. v. p. 3.

Multiply these together, then we obtain

tan. r tan. r, tan. r,, tan. r,,, = sin. s sin. s—a sin. s —

b sin. sc...(4).

Divide (4) by the squares of each of the equations in art. (3), the first side by the first side, and the second by the second: then

sin.2 sc tan. r tan. r, tan. r,, cot. r,,,

which remarkable formulæ are due to Mr. Lowry (1819), vide Repository, ub. sup.

Again, by multiplication of the terms in (3), we have

tan. rtan. r,+tan. r,, tan. r,,,sin.s-b sin. s-c+sin. ssin. s-a

Taking also each of the other corresponding combinations, we obtain in all the three following equations,

Or, by addition, we have at once the following theorem.

tan. r tan. r, tan. r tan. r,, + tan. r tan. r,,, +tan. r, tan. r,,+

tan. r, tan. r,,, +tan. r,, tan. r,,

sin. a sin. b + sin. a sin. c + sin. b sin. c

(7).

"

That is, in words, the sum of the binary products of the tangents of the four inscribed radii are equal to the sum of the binary products of the sines of the sides.

We may notice one beautiful theorem more, which is due to Mr. Lowry, ubi. supra. It is

tan. r, tan. r,, +tan. r, tan. r,,,+tan. r,, tan. r, =

sin. s {sin. s asin. s- b+ sin. s c}

(8).

For the several consequences of these theorems, and a continuation of the inquiry, we must refer to the number xxiv. of Leybourn's Repository, now in the press, where expressions for the various trigonometrical functions of the sides and angles of the triangle, will be given in terms of the inscribed radii.

3. We now proceed to consider the circumscribed radii of the associated triangles. We shall immediately find these in terms of the angles, as we did those of the inscribed in terms of the sides.

Let Q be the centre of the circumscribing circle of the fundamental triangle, and draw the perpendiculars QM, QN, QP. Then M, N, P, bisect the sides a, b, c, respectively, and the several triangles BQC, CQA, AQB, are isosceles. Let the angles made by the radii QB, QC,