Then eliminating b by uniting (1), (2), and (3), we have

cos c = m (cos a cos M+ sin a sin M),

(4)

ILLUSTRATION.

Equations (5) and (6) enable us to find c.

Given a= 75° 38′ 20′′, b = 54° 54' 38",

and

Second. To find ß.

The relation between a, b, y, and B is given by Art. 124, (equation 5), from which we have

cot B =

cos b sin a- sin b cos a cos SY.
sin b siny

Again using equations (2) and (3) we have

(7)

(8)

(9)

(10)

Equations (11) and (12) enable us to find ß.

Third. To find y.

By interchanging a and b and consequently a and ẞ in (11) and (12) we have, calling the auxiliary angle N,

159. Two parts required. It will be noticed that the same auxiliary quantity M is used to find both c and ß. We thus have a convenient method, much used in astronomy, for finding a side and an angle when two sides and the included angle are given.

For finding c and ẞ we have, collecting our formulas,

Dividing equation (10) of Art. 158 by equation (4), we have

An arc of 1' on the earth's surface is equal to one English geographical mile.

1. Find the distance between Boston, latitude 42° 21' N., longitude 71° 41' W., and San Francisco, latitude 37° 48' N. and longitude 122° 28' W.

SOLUTION. Let APD be the meridian of Greenwich from which longitude is measured, ABCD the equator, and P the north pole.

Let the positions of San Francisco and Boston be represented by E and F respectively. The desired distance is EF.

Then

A

B

angle DPE = 122° 28',

angle DPF = 71° 41',

P

F

D

2. Find the distance between New York (lat. 40° 43′ N., long. 74° 0′ W.) and San Francisco (lat. 37° 48′ N., long. 122° 28' W.).

3. Find the distance between Calcutta (lat. 22° 33′ N., long. 88° 19' E.) and Greenwich (lat. 51° 29' N.).

4. Find the distance between Baltimore (lat. 39° 17′ N., long. 76° 37' W.) and Calcutta.

161. Given two angles and the included side, to find any one of the remaining parts.

Let a, ẞ, c be the given parts.

First. To find y.

The relation between a, ß, c, and y is, Art. 121, eq. (3),

cos a cos ẞ+ sin a sin ẞ cos c.

cos y =

Let m sin M= cos α,

and m cos M = sin a cos c.

Uniting (1), (2), and (3)

(1)

(2)

(3)

Equations (5) and (6) enable us to find y.

Second. To find a.

(6)

The relation between a, ẞ, c, and a is given by Art. 124, equation (4), from which

or

cot a =

sin ẞ cos a + sin a cos c cos ß.
sin a sin c

Uniting equations (2), (3), and (8),

(7)

(8)

From (11) and (12) by interchanging a and ẞ, and consequently a and b, we have, calling the auxiliary quantity N,

162. Given two sides and an angle opposite one of them, to

find any one of the remaining parts.

Let a, b, a be the given parts.