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This process, by which we find the angle COD from the angle AOB, is called reducing an angle to the horizon.


128. It is sometimes important to know what amount of error will be introduced into one of the calculated parts of a triangle by reason of any small error which may exist in the given parts. We will here consider an example,

129. A side and the opposite angle of a spherical triangle remain constant: determine the connexion between the small variations of any other pair of elements,

Suppose C and c to remain constant.

(1) Required the connexion between the small variations of the other sides. We suppose a and b to denote the sides of one triangle which can be formed with C and c as fixed elements, and a + Sa and b + Sb to denote the sides of another such triangle; then we require the ratio of Sa to Sb when both are extremely small. We have




cos c = cos a cos b + sin a sin b cos C,

cos c = cos (a + da), cos (b + 8b) + sin (a + da) sin (b + 8b) cos C ;

cos (a + da) = cos a – sin a da, nearly,

sin (a + da) = sin a + cos a da, nearly,

with similar formulæ for cos (b +86) and sin (b + 8b). (See Plane Trigonometry, Chap. XII.) Thus

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+ (sin a + cos a da) (sin b + cos b db) cos C.

Hence by subtraction, if we neglect the product da db,

0 = da (sin a cos b - cos a sin b cos C)

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We may

this gives the ratio of Sa to 8b in terms of a, b, C. express the ratio more simply in terms of A and B; for, dividing by sin a sin b, we get from Art. 44,

cot B sin C+
sin a

бъ sin b

cot A sin C = 0;


Sa cos B+8b cos A = 0.

(2) Required the connexion between the small variations of the other angles. In this case we may by means of the polar triangle deduce from the result just found, that

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this may also be found independently as before.

(3) Required the connexion between the small variations of a side and the opposite angle (A, a).

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(4) Required the connexion between the small variations of a side and the adjacent angle (a, B).

We have

cot C sin B = cot c sin a

cos B cos a;

proceeding as before we obtain

cot C cos B&B = cot c cos a da + cos B sin a da + cos a sin B SB ;


(cot C cos B-cos a sin B) SB = (cot c cos a + cos B sin a) da;

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130. Some more examples are proposed for solution at the end of this Chapter; as they involve no difficulty they are left for the exercise of the student.


1. In a spherical triangle, if C and c remain constant while a and b receive the small increments Sa and 86 respectively, shew that

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2. If C and c remain constant, and a small change be made in a, find the consequent changes in the other parts of the triangle. Find also the change in the area.

3. Supposing A and c to remain constant, prove the following equations, connecting the small variations of pairs of the other elements:

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4. Supposing b and c to remain constant, equations connecting the small variations of elements:

SB tan C = SC tan B,
Sa = SA sin c sin B,

Sa tan CSB sin a,
SB cos a = -

prove the following pairs of the other

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5. Supposing B and C to remain constant, prove the following equations connecting the small variations of pairs of the other elements:

Sb tan c = 8c tan b,

SA = da sin b sin C,

SA cot c = 8b sin A,
Sa sin B cos c = db sin A.

6. If A and C are constant, and b be increased by a small quantity, shew that a will be increased or diminished according as c is less or greater than a quadrant.



131. The student must have perceived that many of the results obtained in Spherical Trigonometry resemble others with which he is familiar in Plane Trigonometry. We shall now pay some attention to this resemblance. We shall first shew how we may deduce formulæ in Plane Trigonometry from formulæ in Spherical Trigonometry; and we shall then investigate some theorems in Spherical Trigonometry which are interesting principally on account of their connexion with known results in Plane Geometry and Trigonometry.

132. From any formula in Spherical Trigonometry involving the elements of a triangle, one of them being a side, it is required to deduce the corresponding formula in Plane Trigonometry.

Let a, ẞ, y be the lengths of the sides of the triangle, r the radius of the sphere, so that are the circular measures


of the sides of the triangle; expand the functions of

which occur in any proposed formula in powers of

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respectively; then if we suppose r to become indefinitely great,

the limiting form of the proposed formula will be a relation in Plane Trigonometry.

For example, in Art. 106, from the formula

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sin b sin c

ß3 + y3 - a3 a* + ß* + y2 − 2a3ß2 − 2ß3y3 — 2y3a3

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and this is the expression for the cosine of the angle of a plane triangle in terms of the sides.

Again, in Art. 110, from the formula

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that is, in a plane triangle the sides are as the sines of the opposite angles.

133. To find the equation to a small circle of the sphere.

The student can easily draw the required diagram.

Let be the pole of a small circle, S a fixed point on the sphere, SX a fixed great circle of the sphere.

Let OS = a,

OSX=ẞ; then the position of O is determined by means of these angular co-ordinates a and B. Let P be any point on the circumference of the small circle, PS=0, PSX = 4, so that and are

T. S. T.


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