These formulæ give the ratios of the six lines x, y, z; x', y', z', when the lines a, b, c; a', b', c', or their ratios, are known; when the angles a, B, 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.

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

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

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

17. Another expression analogous to that found may be had by substituting for y and their values in the formula

γ

It is remarkable, that the coefficients of sin a, sin ß, sin in these formulæ are the reciprocals of their coefficients in the preceding.

18. Other values of 2 may be obtained by putting the values of y and in terms of x in the formulæ

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

The three sets of fumulæ V, VI, VII, are remarkable for their symmetry and simplicity, qualities of great importance in analysis.; the last, viz. VII, seems however to be the most concise.

may be made by subsidiary angles; to determine these, let us assume that

Let be such, that sin 0 = √(2 sin & cos o cos a)

= a'2.

=

(sin 24 cos a),

CASE 1. When a 90°: find and 0, such, that

CASE 2. When a > 90°; find and 0, such, that

20. I shall now apply the formulæ to a case of Geodetic surveying, taking an example from Delambre's Methodes Analytiques pour la Determination d'un Arc du Meridien, (p. 141-2).

ABC (fig. 5.) is one of the triangles employed in measuring an arc of the Meridian in France; A is Villers-Bretonneux; B Vignacourt; C Sourdon; D is a station within the triangle. To determine its position, there are given,

Here a', b', c' are the sides, and A, B, C' the angles of the triangle (fig. 5. No. 2.) which is conjugate to ABC; as a', one of the sides of the conjugate triangle, may be any number, it may be considered as analogous to radius in Trigonometry, and its logarithm might be assumed = 10, or any other convenient number, by which negative or large indices may be avoided; it is here assumed to be 5. The lines to be determined are AD = x, BD = y, CD = %.