and since, by table page 24 (art, 27 and 9),

But unity being successively added to and subtracted from each

sin b

of the members of the equation

sin a

sin B
sin A

then, one of the

two results being divided by the other, we arrive at the equation

which may be transformed into the following, by the formulas

and dividing the equation (a) by this last, we have

A)

which will make known two sides of a spherical triangle, in which we have the third side and the two angles adjacent to it; since, if we designate by b' and a' the values of the arcs b + a there will result

and b

b = } (b' + a'),

a = 1 (b' — a'). 4. Again, taking in article 54 the equations cos A+ cos B cos C = sin B sin C cos a, cos B+cos A cos C = sin A sin C cos b, latter, we shall find

and, dividing the former by the

Unity being successively added to and subtracted from each member of this equation, and one of the results being then divided by the other, we shall derive from it, as above (a),

1
cos B
Х

cos A+ cos B
tang (B

=

cos C

1+ cos C

A) tang ¦ (B + A)

sin (b

sin (ba)'

sin

(b

-

sin

--

a) cos (b
(ba) cos † (b + a)
sin b sin a

a)

(b);

and, as the equation

sin b + sin a

sin B sin A

the preceding transformation, may be written thus,

tang (BA) cot (B+ A)

employed in

by multiplying and dividing the equation (b) by this last, we

sin B + sin A

formulas, which will supply the place of the preceding, when we

know two sides and the angle contained by them.

60. By taking all the variations, of which the equations found above are susceptible, we have

* In order to deduce these formulas from those analogous to them in the preceding article, it is necessary to observe that

and that the sin (p—q)=—sin (q—p), and cos (p—q) = cos (q—p). Trig.

From the last twelve formulas we deduce the following, which will serve to find the third angle or the third side of a triangle, when two sides and the two opposite angles are known.

If to these equations, we add the equations (A) which are applicable to the case in which two sides and one of the opposite angles, or two angles and one of the opposite sides, are known, we shall have every thing necessary for the solution of spherical triangles. The preceding part of this chapter may therefore be regarded as a complete treatise on spherical trigonometry. By combining together the different formulas successively obtained, we may deduce from them a great many others of very frequent use in astronomical calculations. We are indebted in

* These formulas and the preceding are known under the name of the analogies of Napier, because they are deduced from the rules given by that geometer for the solution of spherical triangles. (Logarithmorum canonis descriptio.)

this respect to M. Delambre, for very elegant and very numerous results, and for important applications of approximate methods and series, to the cases which are susceptible of them.

Recapitulation of the Formulas necessary for the solution of any Spherical Triangle.

61. NEGLECTING the variations which the same case may present, we find only the six following formulas,

1. Given the three sides, (a, b, c), to find one of the angles (A).

tang A =

b)

sin (a + b c) sin (a + c
sin (ba) sin (a+b+c)

2. Given the three angles (A, B, C), to find one of the sides (a).

3. Given two sides (b, c), and the contained angle (A), to find the

other angles (B, C).

To find afterwards the third side (a), see the formula for case 6. 4. Given two angles (B, C), and the included side (a), to find the

In order to find the third angle (A), see the formula for case 5. 5. Given two sides (a, c), and one of the opposite angles (C), to find the other opposite angle (A).

* Instead of this formula and the preceding, we often make use of the following, namely,

obtained in the note to page 53, and which are analogous to that employed for the corresponding case of plane trigonometry (58).