111. To express cot B-cot A approximately.

Now we have shewn in Art. 106, that approximately

112. 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 ra.

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 Bsin (4-μ) is taken for the side of which a is the real sin (B-μ)

value

value. Let x=α

ẞ sin (4-μ)

sin (B-u)

proximately. Now approximately

sin (4-μ)

=

sin A-μ cos A

sin (B – μ) ̄ sin B – μ cos B-sin B

Also the following formulæ are true so far as terms involv

If we calculate μ from an equation corresponding to (1) of Art. 109, we have

1. If the sides of a spherical triangle AB, AC be produced to B', C', so that BB', C'C' are the semi-supplements of AB, AC respectively, prove that the arc B'C' will subtend an angle at the centre of the sphere equal to the angle between the chords of AB and AC.

2. Deduce Legendre's Theorem from the formula

3.

Four points A, B, C, D on the surface of a sphere are joined by arcs of great circles, and E, F are the middle points of the arcs AC, BD; shew that

cos AB+cos BC cos CD + cos DA

+

=

4 cos AE cos BF cos FE.

4. If a quadrilateral ABCD be inscribed in a small circle on

a sphere so that two opposite angles A and C may be at opposite

T.S. T.

G

extremities of a diameter the sum of the cosines of the sides is

constant.

5. ABC is a spherical triangle each of whose sides is a quadrant, P any point within the triangle; shew that

and

cos PA cos PB cos PC +cot BPC cot CPA cot APB=0,

tan ABP tan BCP tan CAP = 1.

6. If O be the middle point of an equilateral triangle ABC, and P any point upon the surface of the sphere,

(tan PO tan OA)3 (cos PA + cos PB + cos PC)2 =

cos2 PA+cos PB+cos PC-cos PA cos PB-cos PB cos PC-cos PC cos PA.

7. If ABC be a triangle having each side a quadrant, O the pole of the inscribed circle, P any point on the sphere, then

(cos PA + cos PB+ cos PC)2 = 3 cos3 PO.

8. From each of three points on the surface of a sphere arcs are drawn on the surface to three other points situated on a great circle of the sphere, and their cosines are a, b, c ; a', b', c' ; a′′, b′′, c". Shew that

ab"c' + a'bc" + a′′b'c = ab'c” + ab′′c + a′′be.

Prove the following approximate formula (Arts. 110, 111),

9.

10. By continuing the approximation in Art. 106 so as to include the terms involving r1, shew that approximately

11. From the preceding result shew that if A = A' + ◊ then approximately

X. GEODETICAL OPERATIONS.

114. One of the most important applications of Trigonometry, both Plane and Spherical, is to the determination of the figure and dimensions of the Earth itself, and of any portion of its surface. We shall give a brief outline of the subject, and for further information refer to Woodhouse's Trigonometry, to the article Trigonometrical Survey in the Penny Cyclopædia, and to Airy's treatise on the Figure of the Earth in the Encyclopædia Metropolitana. For practical knowledge of the subject it will be necessary to study some of the published accounts of the great surveys which have been effected in different parts of the world, as for example, the Account of the measurement of two sections of the Meridional arc of India, by Lieut. Colonel Everest, 1847; or the Account of the Observations and Calculations of the Principal Triangulations in the Ordnance Survey of Great Britain and Ireland, 1858.

115. An important part of any survey consists in the measurement of a horizontal line, which is called a base. A level plain of a few miles in length is selected and a line is measured on it with every precaution to ensure accuracy. Rods of deal, and of metal, hollow tubes of glass, and steel chains, have been used in different surveys; the temperature is carefully observed during the operations, and allowance is made for the varying lengths of the rods or chains, which arise from variations in the temperature.

116. At various points of the country suitable stations are selected and signals erected; then by supposing lines to be drawn connecting the signals, the country is divided into a series of triangles. The angles of these triangles are observed, that is, the angles which any two signals subtend at a third. For example, suppose A and B to denote the extremities of the base, and Ca