(38.) Solution of triangles by series.

[1] Given the sides a, b, and the angle C of a plane triangle :

[2] Given the sides a, b, and the angle C of a spherical tri

log, cos c=log. (cosa.cos 16) + tana.tan 1b.cos C

log,c=log, a — § (2)

? } (A + B) =

m

90° — † C + S„(−1)−1(
C + Sm(−1)m-1 (tan 3b)m sin m C

cot

a

m sin m C

m

m

'tan bm cos m C

m

m

tan

∞

cosm C

0

log.cosc=log. (cosa.cos 1b)+(-1)-1(tana.tan)

m

(39.) Correction for a compound base. posed of two straight lines a and b, 180°-a", the correction is

If the base be cominclined at an angle

ab x2

0,00000000001175.

a+b

(40.) Reduction of an oblique to a horizontal angle.

Let a, ẞ, be the angles of elevation of two objects;
their angular distance;

C, the horizontal angle: then

[1] C−y={(a + ß)2 tan Ly− (a — ß)2 cot1y}.

[2] (sin¦ C)2 = sin § (a+a— ß). sin § (a — a — ß).

cos a. cos B

;

log SinC={log Sin (a + a-ẞ) + log Sin (a-a-B)

(41.) Reduction of a spherical to a plane triangle.

[1]

Let C be the spherical angle,

C- the corresponding plane angle; then

1

α= {(a + b) tan C - (a —b)2 cot C}.
16

[2] Legendre's Theorem:

00=

area of triangle

3r2

= {(A + B + C − 180°).

The value of a in seconds is

area of triangle

3 x 0,000004848

If the area be found in feet, and a degree on the Earth's

surface=365155 feet, then log10 divisor = 9,803894.

Reduction of a plane to a spherical triangle.

Let

γ be the plane angle contained by the sides a, ß;

C the corresponding spherical angle; then

(L. 293-313; C. ch. xx; W. ch. xii; Enc. Met. V. 1. p. 698,9.)

FORMULE FOR THE CONSTRUCTION OF TABLES.

(42.) cos 30° = (1 + 1) ]* = c1 ;

cos 1 30° = 1 (1 + c,) ]* = c2 ;

cos + 30° = ↓ (1 + c2) |3

] = c3 ;

= $1

sin 15° = (1+1) ↓ (1 − 1 ) 1 — s1 ;

sin 15° =

sin 15° =

&c. &c.

By continuing this process we obtain sin

15°
210

and thence

which equation may be solved by approximation.

sin (a + b) = sin a {sin a ― sin (a - b)} — 4 sin a. (sin 6)2: Putting b=1°, we obtain Delambre's formula,

sin (a+1°)=sin a+ {sin a― sin (a — 1o)} — 4 sin a. (sin 30′)2. sin (60° + a) = sin a + sin (60o — a).

tan (45° + a) = 2 tan 2a + tan (45° — a).

cosec a=cot a + tana.

(43.) Sines of arcs expressed by surds.

(W. ch. iv ; C. ch. vi.)