If SB and SC have different signs the errors of b and c will be less when cos A is positive than when cos A is negative; A therefore ought to be less than a right angle. And if SB and SC are probably not very different, B and C should be nearly equal. These conditions will be satisfied by a triangle differing not much from an equilateral triangle.

If two angles only, A and B, be observed, we obtain the same expressions as before for the errors in b and c; but we have no reason for considering that SB and SC are of different signs rather than of the same sign. In this case then the supposition that A is a right angle will probably make the errors smallest.

125. The preceding article is taken from the Treatise on Trigonometry in the Encyclopædia Metropolitana. The least satisfactory part is that in which it is considered that SB and SC may be supposed nearly equal; for since 84+ SB+ 8C = 0, if we suppose SB and SC nearly equal and of opposite signs, we do in effect suppose 84 = 0 nearly; thus in observing three angles, we suppose that in one observation a certain error is made, in a second observation the same numerical error is made but with an opposite sign, and in the remaining observation no error is made.

We have hitherto proceeded on the supposition that the Earth is a sphere; it is however approximately a spheroid of small eccentricity. For the small corrections which must in consequence be introduced into the calculations we must refer to the works named in Art. 114. One of the results obtained is that the error caused by regarding the Earth as a sphere instead of a spheroid increases with the departure of the triangle from the well-conditioned or equilateral form (An Account of the Observations, &c. page 243). Under certain circumstances the spherical excess is the same on a spheroid as on a sphere (Figure of the Earth, pages 198 and 215).

127. In geodetical operations it is sometimes required to determine the horizontal angle between two points, which are at a

small angular distance from the horizon, the angle which the objects subtend being known, and also the angles of elevation or depression.

Suppose OA and OB the directions in which the two points are seen from 0; and let the angle AOB be observed. Let OZ be the direction perpendicular to the observer's horizon; describe a sphere round O as a centre, and let vertical planes through OA and OB meet the horizon in OC and OD respectively; then the angle COD is required.

Let AOB = 0, COD = 0+x, AOC = h, BOD = k; from the triangle AZB

and cos AZB = cos COD = cos (0 + x); thus

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

XI. ON SMALL VARIATIONS IN THE PARTS OF A SPHERICAL TRIANGLE, AND ON THE CONNEXION OF FORMULÆ IN PLANE AND SPHERICAL TRIGONOMETRY.

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 + 8b to denote the sides of another such triangle; then we require the ratio of da to db when both are extremely small. We have

and

also

and

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

cos c = cos (a + da) cos (b + db) + sin (a + da) sin (b + db) cos C' ;

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

COS C = (cos a

sin a da) (cos b — sin 6 db)

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

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

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

We may

this gives the ratio of da to db 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,

(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

this may also be found independently as before.

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

(4) Required the connexion between the small variations of

a side and the adjacent angle (a, 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 ;

130. Some more examples will be found among those proposed for solution at the end of this chapter; as they involve no difficulty they are left for the exercise of the student.

131. 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

B Y

of the sides of the triangle; expand the functions of

а

which occur in any proposed formula in powers of, B, 1

respectively; then if we suppose 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