To compute 4 C,

Here B C is the middle part, and A B and extremes; hence the equation is rad. sin B C

C are adjoining tan A B. cot C, and sin B C : cot C.

9.756034

1. In the right

118° 21′ 4′′, and 4

Answer, A C 116°

EXAMPLES FOR EXERCISE.

angled spherical triangle ABC, given A B A 23° 40′ 12′′, to find the other parts?

17′ 55′′, 4 C 100° 59′ 26′′, and B C 21° 5′ 42′′. 2. In the right angled spherical triangle ABC, given A B 53° 14′ 20′′, and 4 A 91° 25′ 53′′, to find the other parts?

Answer, A C 91° 4′9′′, 4 C 53° 15′ 8′′, and B C 91° 47′ 11′′. 3. In the right angled spherical triangle A B C, given A B 102° 50′ 25′′, and 4 A 113° 14′ 37′′, to find the other parts?

Answer, A C 84° 51′ 36′′, 4 C 101° 46′ 57′′, and B C 113° 46′ 27′′. 4. In the right angled spherical triangle A B C, given A B 48° 24′ 16′′, and B C 59° 38′ 27′′, to find the other parts ? Answer, A C 70° 23′ 42′′, 5. In the right angled

4 A 66° 20′ 40′′, and C 52° 32′ 55′′. spherical triangle ABC, given A B

151° 23' 9", and B C 16° 35′ 14′′, to find the other parts?

Answer; A C 147° 16 51", 4 C 117° 37′ 21′′, and 4 A 31° 52′ 50′′. 6. In the right angled spherical triangle ABC, given A B 73° 4′ 31′′, and A C 86° 12′ 15′′, to find the other parts?

Answer, B C 76° 51′ 20′′, ▲ A 77° 24′ 23′′, and C 73° 29′ 40′′. 7. In the right angled spherical triangle ABC, given A C 118° 32′ 12′′, and A B 47° 26′ 35′′, to find the other parts?

Answer, BC 134° 56′ 20′′, 4 A 126° 19′ 2′′, and 4 C 56° 58′ 44′′. 8. In the right angled spherical triangle A B C, given A C 91° 50′ 23′′, and A B 92° 17′ 26′′, to find the other parts?

Answer, B C 36° 33′ 29′′, 4 A 36° 34′ 50′′, and C 91° 22′ 00′′. 9. In the right angled spherical triangle A B C, given A B 138° 25′ 34′′, and A C 49° 27′ 16′′, to find the other parts?

Answer, B C 150° 20′ 8′′, ▲ A 139° 21′ 36′′, and C 119° 9′ 34′′. 10. In the right angled spherical triangle A B C, given A C 68° 14′ 20′′, and ▲ C 70° 21′ 15′′, to find the other parts ?

Answer, B C 40° 6′ 19′′, A B 61° 0′ 22′′, and 4 A 43° 55′ 2′′. 11. In the right angled spherical triangle A B C, given A C 118° 25′ 21′′, and ▲ € 53° 27′ 46′′, to find the other parts?

Answer, B C 132° 16′ 22′′, A B 44° 57′ 38′′, and ▲ A 44° 57′ 38′′. 12. In the right angled spherical triangle A B C, given A C 53° 25′ 31′′, and ≤ A 124° 26′ 7′′, to find the other parts ?

Answer, BC 138° 31' 13", A B 142° 41' 19", and 4 C 130° 59′ 38′′.

13. In the right angled spherical triangle A B C, given A C 102° 15′ 27′′, and B C 49° 13′ 18′′, to find the other parts?

Answer, A B 108° 58′ 9′′, 4 A 50° 47′ 47", and C 104° 35′ 21′′. 14. In the right angled spherical triangle A B C, given 4 C 38° 14′ 3′′, and ▲ A 59° 20′ 7′′, to find the other parts ?

Answer, B C 34° 30' 11", A B 24° 3' 2", and A C 41° 11' 17". 15. In the right angled spherical triangle A B C, given 4 C 171° 4′ and 4 A 92° 6', to find the other parts ?

Answer, A C 76° 30′ 37′′, A B 171° 18′ 56′′, and B C 103° 38′ 57′′. 16. In the right angled spherical triangle A B C, given C 90° 18′ 18′′, and ▲ A 93° 17′ 20′′, to find the other parts?

Answer, A C 89° 58′ 56′′, A B 90° 18′ 20′′, and B C 93° 17′ 20′′. 17. In the right angled spherical triangle A B C, given A B 40° 18′ 23′′, and A C 100° 3′ 7", to find the other parts?

Answer, 4 A 98° 38′ 53′′, 4 C 41° 4' 6", and B C 103° 13′ 59′′. 18. In the right angled spherical triangle A B C, given A C 61° 3′ 22′′, and 4 A 49° 28′ 12′′, to find the other parts?

Answer, A B 49° 36′ 6′′, 4 C 60° 29′ 19′′, and B C 41° 41′ 32′′. 19. In the right angled spherical triangle A B C, given A B 29° 12′ 50′′, and 4 C 37° 26'21", to find the other parts?

Answer, ambiguous, 4 A 65° 27′ 58′′ or its supplement, A C 53° 24′ 13′′ or its supplement, B C 46° 55′ 2′′ or its supplement.

20. In the right angled spherical triangle A B C, given A B 54° 21′ 35′′, and 4 C 61° 2′ 15′′, to find the other parts?

Answer, ambiguous, B C 129° 28′ 28′′ or its supplement, A C 111° 44′ 34′′ or its supplement, and 4 A 123° 47′ 44′′ or its supple

ment.

21. In the right angled spherical triangle A B C, given A B 100° 10′ 3′′, and C 90° 14′ 20′′, to find the other parts?

Answer, ambiguous, A C 100° 9′ 55′′ or its supplement, B C 1° 19′ 53′′ or its supplement, and 4 A 1° 21' 8" or its supplement. 22. In the right angled spherical triangle A B C, given A B 121° 26' 25'', and 4 C 111° 14′ 37′′, to find the other parts?

Answer, ambiguous, 4 A 136° 0′ 3′′ or its supplement, A C 66° 15′ 38′′ or its supplement, and B C 140° 30′ 56′′ or its supple

ment.

APPLICATION OF THE FORMULA FOR RIGHT ANGLED SPHERICAL TRIANGLES TO THE SOLUTION OF CASES RELATIVE TO QUADRANTAL TRIANGLES.

A quadrantal triangle is a spherical triangle one of whose sides is a quadrant.

Let A B C or ABC be a quadrantal triangle, A C being the quadrantal side, on C B or C B produced, let CD be taken

equal to a quadrant, and let A D be an arc of a great circle passing through A and D. Then the angles C A D, CDA (or B D A) and B' D A will be right angles, and A D will be the measure of the angle C. The angles D A B, D A B' will respectively be complements of CA B, CA B, and C B, C B' will also be respectively complements of B D, B' D. Hence the different parts of the quadrantal triangles B' A C, BAC may be determined from the corresponding parts of the right angled triangles A D B, A D B'.

EXAMPLE I.

In the triangle A B′ C, given A C 90°, the angle C A B′ 112° 2′ 9′′, and A B 67° 3′ 14′′, to find the other parts?

Let CD be a quadrant, then as CAD will be a right angle, DA B' will be 22° 2′ 9′′. Hence D B' will be acute; and as A B' is acute, the angle B' and A D, the measure of the angle C, will also bẹ acute.

To compute A D, or the measure of 4 C. Equation, rad. cos D A B' = cot A B' tan A D.

9.626715 10.000000

9.967057

10.340342

To compute D B', the complement of D C.

Equation, rad. sin D B' sin D A B'. sin A B'.

To find the angle B'.

Equation, rad. cos A B' = cot DA B'. cot B'.

Or cot D A B 22° 2′ 9′′....

10.392809

10.000000

9.590915

9.198106

In the triangle A B' C, given A C 90°; A B' 79° 18′ 40′′, and C B'

123° 16' 3", to find the other parts ?

Let C D be a quadrant, then D B', the complement of CD, is 33° 16' 3".

To find the angle B'.

Equation, rad. cos 4 B' = cot A B' tan D B'.

To find A D, or the measure of the angle C.

Equation, rad. cos A B′ = cos D B'. cos D A.
Or cos D B' 33° 16′ 3′′....

9.922268

10.000000

9.268288

9.346020

To find the angle D A B'.

Equation, rad. sin DB' = sin A B'. sin D A B'.

1. In the quadrantal triangle A B C (see the last figure) A C being the quadrantal side, given A B 67° 3′ and ▲ A 49° 18', to find the other parts?

Answer, C 60° 48′ 54′′, B C 53° 5′ 46′′, and 4 B 108° 32′ 27′′. 2. Given A 118° 40′ 36′′, and B C 113° 2′ 28′′, to find the other parts?

Answer, A B 54° 38′ 57′′, ≤ C 51° 2′ 35′′, and ▲ B 72° 26′ 21′′: 3. Given C 69° 13′ 46′′, ▲ A 72° 12′ 4′′, to find the other parts? Answer, A B 70° 8′ 39′′, B C 73° 17′ 29′′, and ▲ B 96° 13′ 23′′. 4. Given B C 86° 14′ 40′′, and ▲ A 37° 12′ 20′′, to find the other parts?

Answer, A B 4° 43′ 2′′, ▲ B 142° 42′ 2′′, and ▲ C 2° 51′ 23′′. 5. Given C 60° 41′ 30′′, and B C 78° 12′ 19′′, to find the other parts?

Answer, A B 61° 22′ 7′′, ▲ A 76° 31′ 59′′, and 4 B 96° 32′ 45′′. 6. Given B C 118° 32′ 16′′, and A B 67° 48′ 40′′, to find the other parts?

Answer, 4 C 64° 32′ 21′′, 4 A 121° 3′ 40′′, and B 77° 11′ 6′′,

7. Given BC 58° 3′ 42′′, and A B 61° 4′ 19′′, to find the other

parts?

Answer, C 55° 15′ 0', 4 B 110° 9' 10", and 4 A 52° 48′46′′. 8. Given 4 B 104° 41′ 17′′, and B C 73° 21′ 6′′, to find the other

parts?
Answer,
9. Given 4

parts?
Answer,
10. Given

parts?

A 67° 56′ 13′′, 4 C 47° 32′ 39", and A B 49° 42′ 18′′.
A 21° 39′ 48′′, and ▲ C 53o 26′ 45′′, to find the other

B 123° 36′ 32′′, B C 26° 18′ 40′′, and A B 74° 41′ 35′′.
B 94° 29′ 54′′, and B C 56° 31′ 26′′, to find the other

and

Answer, A 56° 15′ 28′′, ≤ C 81° 53′ 0′′, and A B 83° 14′ 11′′. 11. Given B C 78° 38′ 1′′, C 93° 18′32′′, to find the other parts?

Answer, A 78° 26' 54", 4 B 89° 20′ 16", and A B 93° 14' 30". 12. Given A B 96° 32′ 18′′, and B C 85° 32′ 4′′, to find the other parts?

Answer, 4A 85° 30′ 19′′, 4 C 96° 33′ 29′′, and 4 B 89° 29′ 14′′.

APPLICATION OF TRIGONOMETRICAL FORMULE TO THE NUMERICAL COMPUTATION OF THE DIFFERENT PARTS OF OBLIQUE ANGLED SPHERICAL TRIANGLES.

ALL the cases of oblique angled spherical triangles may be solved by the formulæ for right angled ones, except when the three sides are given to find the angles, or the three angles given to find the sides. For the solution of these two cases, we have the following rules.

To find any angle of a spherical triangle when the three sides are given.

RULE 1. From half the sum of the three sides subtract the side opposite to the required angle, then add together the log cosecants of the other two sides, (rejecting 10 from each of their indexes) and the log sines of the half sum, and remainder; half the sum of these four logarithms will be the log cosine of half the required angle.

RULE 2. From half the sum of the three sides, subtract each of the sides containing the required angle, then add together the log cosecants of these two sides, (rejecting 10 from each of their indexes) and the log sines of the two remainders; half the sum of these four logarithms will be the log sine of half the required angle.

Remark. When the required angle is large, the first of these rules may be used in preference, and the second when the angle is small.