cos. A cos. B

=

sin. A sin. B

=cot. A cot. B, which is formula (5.)

We have thus proved the truth of the results derived from the application of Napier's rules, and may therefore apply these rules without scruple to the solution of various cases of rightangled triangles.

Let us then take each combination of the two data, and determine in each case the other three quantities, adapting our formulæ to computation by tables.

2. Given a, b, required A, B, c,

R sina cot. B tan. b .. cot, B=R sin. a cot. b

R sin. bcot. A tan. a
R cos. ccos, a cos. b

.. cot. A=R sin. b cot. a

3. Given a, c, required A, B, b

R sin. a sin. A sin. c .. sin. A=R

R cos. ccos. a cos. b .. cos. b=R

R cos. B-tan. a cot. c

(9)

5. Given A, c, required B, a, b.

R cos. A tan. b cot. c .. tan. b=R 'cos, a tan. c
R cos. ccot. A cot. B.. cot. B=R tan. A cos. c

R sin. asin. A sin. c

6. Given B, c, required A, a, b.

R cos. B cot. c tan. a .. tan. a=)
=R cos. B tan. c - (16)

8. Given B, a,

R cos. ccot. A cot. B .. cot. A=R tan. B cos. c -
R sin. b sin. B sin. c

7. Given A, b, required B, c, a.

R cos. A cot. c tan. b .. cot. c-R cos. A cot. b
R sin. bcot. A tan. a .. tan. a=R tan. A sin. b
R cos. B sin. A cos. b

required A, c, b.

(17)

(18)

R cos. B

R sin. a

(22)

tan. b=R tan. B sin. a

(23)

(24)

ON THE SOLUTION OF OBLIQUE-ANGLED SPHERICAL TRIANGLES.

The different cases which present themselves are contained in the following enumerations.

1. When two sides and the included angle are given. 2. When two angles and the side between them are given. 3. When two sides and the angle opposite to one of them are given.

4. When two angles and the side opposite to one of them are given.

5. When three sides are given.

6. When three angles are given.

I.

When two sides and the included angle are given. The remaining angles may be determined from the formula (.)

Thus, let a, b, C, be given, A, B, c, required.

A and B being known, c may be obtained from (ε.)

sin. c sin. C

For

And, in like manner, if any two other sides and the included angle be given, the remaining parts may be determined.

II. When two angles and the side between them are given. The remaining sides may be determined from the formula('.) Thus, let A, B, c, be given; a, b, C, required.

A-B

a and b being known, C may be obtained by (s.)

For

sin. C sin. c

=

sin. A sin. a

And, in like manner, if any two other angles and the included side are given, the remaining parts may be determined. III. When two sides and the angle opposite to one of them are given.

The angle opposite to the other side may be found from formula (ε.)

Thus, let a, b, A be given, B, C, c, required.

sin. B

sin. b

The angle B being determined, the remaining angle C will be found from (σ.)

The angle C being determined, the remaining side c will be found from (8.)

sin. c

sin. C

For

or c may be found from (.)

And, in like manner, if any other two sides and the angle opposite to one of them be given, the remaining parts may be determined.

IV. When two angles and the side opposite to one of them are given.

The side opposite to the other angle may be found from formula (ɛ.)

Thus, let A, B, a, be given; b, c, C, required.

sin. b

sin. B

sin. A

The side b being determined, the remaining side c will be found from (')

A-B

The side c being determined, the remaining angle C will be found from (.)

sin. C

For

sin. c

or c may be found from (r.)

And, in like manner, any other two sides being given and the angle opposite to one of them, the remaining parts may be determined.

V. When three sides are given.

The three angles may be immediately determined from any one of the formulæ (7 1,) (y 2,) (y 3,) (y 4.)

The choice of the formula, which it will be advantageous to employ in practice, will depend upon the consideration already noticed in the solution of the analogous case in plane trigonometry.

VI. When three angles are given.

The three sides may be immediately determined from any of the groups of formulæ (8 1,) (§ 2,) (§ 3,) (§ 4.)