where S denotes the area of the plane triangle whose sides are

S

therefore

is approximately equal to the spherical excess of the

spherical triangle, and thus the theorem is established.

It will be seen that in the above approximation the area of the spherical triangle is considered equal to the area of the plane triangle which can be formed with sides of the same length.

107. Legendre's Theorem may be used for the approximate solution of spherical triangles in the following manner.

(1) Suppose the three sides of a spherical triangle known; then the values of a, ß, y are known, and by the formulæ of Plane Trigonometry we can calculate S and A', B', C'; then A, B, C are known from the formulæ

(2) Suppose two sides and the included angle of a spherical triangle known, for example A, b, c. Then

Then A' is known from the formula A'= A

plane triangle two sides and the included angle are known; therefore its remaining parts can be calculated, and then those of the spherical triangle become known.

(3) Suppose two sides and the angle opposite to one of them in a spherical triangle known, for example A, a, b. Then

=T

and C'π- A' - B'=π- A - B' approximately; then S=aßsin C'. Hence A' is known and the plane triangle can be solved, since two sides and the angle opposite to one of them are known.

(4) Suppose two angles and the included side of a spherical triangle known, for example, A, B, c.

Then S=

y2 sin A' sin B'
2 sin (4' + B')

y2 sin A sin B
2 sin (A + B)

nearly.

Hence in the plane triangle two angles and the included side are known.

(5) Suppose two angles and the side opposite to one of them in a spherical triangle known, for example A, B, a. Then

C'π-A'B'π-A-B, approximately, and

S=

=

a2 sin B' sin C'
2 sin (B' + C')'

which can be calculated, since B' and Care approximately known.

108. The importance of Legendre's Theorem in the application of Spherical Trigonometry to the measurement of the Earth's surface has given rise to various developments of it which enable us to test the degree of exactness of the approximation. We shall finish the present Chapter with some of these developments, which will serve as exercises for the student. We have seen that ap

S proximately the spherical excess is equal to and we shall 22,

begin with investigating a closer approximate formula for the spherical excess.

109. To find an approximate value of the spherical excess.

Let E denote the spherical excess; then

αβ

sin } E = sin C (1-4) (1 – A)(1-7)TM

=

4r2

= sin caß
4r2

sin C = sin (C'+3E)=sin C+ } E cos C"

= sin C" + sin "cos Cag = sin C (1 +

3

From (1) and (2)

Caß
272

C'

a2 ß3

..(1),

12r2

Hence to this order of approximation the area of the spherical triangle exceeds that of the plane triangle by the fraction

a2 + B2 + y2

of the latter.

2472

111. To express cot B - cot A approximately.

cos A

B2 + y3 — a3
2βγ

+

Now we have shewn in Art. 106, that approximately

a2 + ß* + y* − 2a3ß3 — 2ß3y2 — 2y3a2

24Byr

112.

a2 - B2

ay sin Bay sin B

The approximations in Arts. 109 and 110 are true so far as terms involving r1; that in Art. 111 is true so far as terms involving r2, and it will be seen that we are thus able to carry the approximations in the following Article so far as terms involving 4.

T. S. T.

G

a3

12r*

113. To find an approximate value of the error in the length of a side of a spherical triangle when calculated by Legendre's Theorem.

Suppose the side ẞ known and the side a required; let 3μ denote the spherical excess which is adopted. Then the approximate ẞ sin (A - μ) is taken for the side of which a is the real sin (B-)

value

value. Let x= a

ẞ sin (4-μ); we have then to find x apsin (B – μ)

proximately. Now approximately

sin 4 {1+μ (cot B — cot 4) + μa cot B (cot B — cot A

sin B

t 4)}

Also the following formulæ are true so far as terms involving 2: