angle 4 to D and E shall make equal angles BAD, CAE with the sides about the angle*."

The well known proposition, that the lines which bisect the three angles of a triangle meet in the same point is a particular case of this Theorem.

10. The Trigonometrical solution of the Geodetical Problem of Art. 1. which is deducible from the construction here given, is sufficiently obvious: I shall therefore, without at present entering into it, investigate another theorem which comprehends the former and various others.

Let the sides of any triangle be a, b, c, (fig. 5, 6).

Let straight lines be drawn from any point D to the angles of the triangle, and put

I owe this elegant proposition to T. Galloway, Esq. F. R. S. to whom it occurred when considering the Theorem.

where it must be observed that the angles a, ß, 7, must be so reckoned, that their sum is four right angles.

The condition that the triangle ABC is made up of the three triangles ADB, ADC, BDC (fig. 5), or else of the excess of two of them above the third (fig. 6), is expressed analytically by this other equation,

xy sin y+xz sin ẞ+yz sin a = bc sin A

(4).

These hold true, whatever be the position of the point D, observing always that the angles a, ß, y must satisfy the condition of making up four right angles.

By adding the first and second equations, and subtracting the third, we obtain,

2x2-2xy cosy - 2xz cos ẞ+2yz cos a = b2 + c2 — a2.

Let equation (5) be multiplied by sin a, and equation (4) by cos a: the

results are

x2 sin a-xy sin a cos y-ax sin a cos ẞ+yx sin a cos a = bc sin a cos A, xy cos a sin y + xx cos a sin ẞ+yz sin a cos a=be cos a sin A.

By subtracting the second of these equations from the first, and observing that

11. From the form of the function which is the first member of this equation, it will remain the same, although we change the angles A and a into B and ß; or into C and y, or into C and y, provided corresponding changes are made in the lines a, b, c we may conclude that

and x, y, ≈: so that, on the whole,

This is the property of the figure which I proposed to investigate; and it manifestly comprehends this other property,

These formulæ, which are remarkable for their symmetry and simplicity, suggest various solutions to the problem enunciated in Art. 1. Their evident analogy to the property of a triangle "that the sines of the angles are proportional to the opposite sides", has suggested another form under which they may be put.

12.

The hypothesis and notation of Art. 10. in regard to the triangle ABC (fig. 5. No. 1. and 6.) being retained, another triangle ABC (fig. 5. No. 2.) having remarkable relations with it, may be constructed as follows:

Let straight lines D'A', D'B', D'C' meet in a point D', the angles A'D'B', B'D'C', A'D'C' being equal to ADB, BDC, ADC respectively. At A' any point in D'A' make the angles D'A'B' equal to DBA, and D'A'C' equal to DCA, thus forming two triangles D'A'B', D'AC (fig. 5. No. 2.) similar to DBA, DCA (fig. 5. No. 1). Join B'C'; because DB: DA = D'A' : D'B' and DA : DC = D'C': D'A'; therefore, ex æq. DB: DC = D'C': D'B'; hence the triangles BDC, C ́D'B' are similar.

Let the lines and angles in the triangle A'B'C' be expressed by the same letters as are used for the triangle ABC, with the distinction of an accent over such as differ in magnitude, so that

BC' a', AC' = b', A'B' =

c', D'A' = x', D'B' = y', D'C' = 2.

Λ

The angles about D in the two triangles being equal; viz. y'' = a,

13. The similarity of the partial triangles which constitute the two triangles ABC, A'B'C', besides the equal ratios x y = y' a', by which they were formed, give us also xcy': c', y : c = x' : c'; therefore

and a like result for each pair of triangles;

hence the lines in the two triangles have the following properties :

14.

In the triangle ABC, the angle a is the sum of the three angles A, ABD, ACD, of which the last two are equal to the angles BA'D', C'A'D' that make up A', hence we have this property:

A+ A = α, B + B'

=

B, C + C = ~

II.

The affinity of the two triangles in respect of these, and other properties which are to follow, may not improperly be indicated by calling them Conjugate Triangles.

15. Because A = a A', and A' = a - A, also, similarly,

β

B = ß – B', B' = ẞ – B, C = y − C', C' = y − C,

the formulæ of Art. 11. gives us these,