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sin a cos y + cos a sin y = — sin (3, sin a cos 3 + cos a sin 3 = — sin y, sin a cos A –cos a sin A = sin (a-A), we obtain a” sin a + ary sin 3 + rz sin y = be sin (a – A). Hence we derive this elegant theorem,

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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 3; or into C and y, provided corresponding changes are made in the lines a, b, c and r, y, x: so that, on the whole, we may conclude that

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This is the property of the figure which I proposed to investigate; and it manifestly comprehends this other property, at 4: - by - C2. sin (a – A) T sin (3 – B) T sin (y – C).”

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These formulae, 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 A'B'C' (fig. 5. No. 2.) having remarkable relations with it, may be constructed as follows:

Let straight lines D'A', DB, D C meet in a point D, the angles A'D' B, BDC., A'D'C' being equal to ADB, BDC, ADC respectively. At A’ any point in D'A' nake the angles D'A'B' equal to DBA, and D'A'C' equal to DCA, thus forming two triangles D'A'B', D'A'C' (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, ea acq. 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

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13. The similarity of the partial triangles which constitute the two triangles ABC, A'B'C', besides the equal ratios a : y = y': a ', by which

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they were formed, give us also a c = y : c’, y : c = a c'; therefore a'a' = y/, # = , =%; and a like result for each pair of triangles:

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

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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 B'A'D', C.A'D' that make up A, hence we have this property:

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The affinity of the two triangles in respect of these, and other pro

perties 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, B = 3 B, B = 3 B, C = y C', C = y C. the formulae of Art. 11. gives us these,

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These formulae give the ratios of the six lines ar, y, x; a ', y, x', when the lines a, b, c, a, b, c, or their ratios, are known; when the angles a, 3, y contained by these lines are given, the lines themselves may be found by known propositions in Trigonometry. To these I shall in the sequel add others.

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These values of y and x being substituted in the above equation, it becomes

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Similar expressions may be had for y and x, and from these the following formulae have been obtained:

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By changing ar, y, x, a, b, c, A' into a ', y, z', a, b, c, A and the contrary, these formulae serve for the conjugate triangle A'B'C'.

17. Another expression analogous to that found may be had by substituting for y and z their values in the formula 3yx sin a + arz sin (3 + ay sin y = be sin A.

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It is remarkable, that the coefficients of sin a, sin 3, sin y in these formulae are the reciprocals of their coefficients in the preceding.

18. Other values of a may be obtained by putting the values of 9 and z in terms of a in the formulae a” – 24 y cos y + y' = co, a" – 2a : cos B + 2* = bo, 2 yx cos a + x = a.

Of these I shall only put down that deduced from the last, as the most symmetrical,

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