III. Right-angled Spherical Triangles, right-angled at A.

NOTE. In the following Formulæ, as already observed, the rules of the signs in Algebra require to be attended to. The quantities marked * are of the same affection, and those marked are ambiguous, in which case the quantities required may be either those found directly, or their supplements.

tan a tan b sec C, and cot a cot b cos C 29

cos B' cos b sin C, and sec B = sec b cosec C31

= cos a sec b, and sec c ́ sec a cos b 32 sin* B cosec a sin* b

33

cos C cot a tan b, and sec C = tan a cot b 34

COS C

Cos a

= cos b cos c, and sec a sec b sec c 35

opposite to

B

tan B tan b cosec c

36

the oblique

angles.

C

tan C cosec b tan c

37

cos bcos B cosec C, and sec b= sec B sin C39

C

cos c cosec B cos C, and sec c = sin B sec C40

VI. The three angles A, B, and

C,

2 K

a

sina =

{

cos } a = { cosec Bcosec Ccos (K∞ B) cos (K∞ C)}163 ={

|tan } a={-cos K cos (K ́» A) sec (K B) sec (KC) } } 64

V.-Particular Solutions.

I. To reduce the length of an inclined base to its horizontal

measure.

Let B be the length of the base on the inclined plane, b that reduced to the horizontal plane, and the inclination, then

b = B cos

(1)

But as is generally a small angle, and need not be known with extreme precision, it is better to calculate the excess of B above b, and supposing to be given in minutes,

0 2

B-b=B (1-cos 4) = 2 B sin =B 2 sin 2 l′ =

or B-6 0.00000004231 02 B

Logarithmically,

Log. (B—b) = const. log. 2.626422+2 log. ◊ + log B

sin 2 l'

2 B,

2

(2)

(3)

II. In measuring a base, it sometimes happens that all its parts do not lie in the same straight line, but are inclined to one another at very obtuse angles. In this case there are given the two sides and contained angle to find the third side.

1. When the contained angle is very obtuse.

Let a and b be the given sides, and C the contained angle in the given triangle ABC. Then put C 180° —, in which case is very small, and is the defect of C from two right angles.

=

b(a—b)

sin A+ sin 3 A
6

R"

a {1+ a+b22

6(a+b) 2 (R")

(1)

2

2. Supposing that the side b is very small in comparison of a,

b

3a3

b2 2a2

3. Log. c=log. a— -M cos C-M cos 2 C-M

a

(2)

&c. (3)

3 a3

in which M is the logarithmic modulus, or 0.4342945.

63

III. Having given the hypotenuse of a right-angled spherical triangle, and one of the oblique angles, to find the value of the side adjacent to this angle expressed in series.

Let a be the hypotenuse, c one of the sides about the right angle, and B the contained angle, then

a-c=tan2B sin 2a-tan4 B sin 4a+tan6 B sin 6a-&c.(1) Let tan c tan a, and

IV. Having given a and b, the two sides of a spherical triangle, little different from a quadrant, together with the side c, it is required to find C from the three given sides a, b, c.

By hypothesis a = 90°- b=90°-ß; and as a and ẞ are very small, the angle C is measured by an arc nearly equal to c. Let Cc+x, and

V. In some cases of trigonometrical surveying, one side of the spherical triangle is very small in comparison of the other two, and therefore to obtain the requisite accuracy the following series must be employed.

180°

BA+c sin A cot b+c2 sin A cos A (1+2 cot b.)
+c sin A cos2 A cot B (3+4 cotb.)
c3 sin A cot B (1+2 cot 2 b)

a=b-c cos A+ c2 cot b sin A+ c3 cos A sin 2A (3+cot 2b.) (3)

Spheroidal and Geodesic Formulæ.

I.

The following formulæ, given by Mr Ivory in the Philosophical Magazine of 1828, for determining the relations of arcs and angles upon the surface of the terrestrial spheroid, are introduced here on account of their simplicity and general utility.

Let A denote the length of any arc on the earth's surface in fathoms or feet.

▲, the length of a degree of longitude on the equator in the same measure = 60856 fathoms, or 365136 feet.

دا

the length of the chord between any two points on the earth's surface.

E, the excess of A above y, or of the arc above its chord.

R, the mean radius of the earth = 20887680 feet.

r, the length of an arc in degrees equal to the radius.

d, the chord of the elliptical meridian between two places near to each other.

the ellipticity of the meridian = 0.00324.

a, the radius of the equator.

a (1), the polar semiaxis.

x', the latitude of any place nearest the equator.

λ, that of another more distant.

m, the azimuth at one of these places, or the angle between the meridian and the arc passing through the other.

m', that at the other place.

w, the difference of longitude between them.
M, the logarithmic modulus = 0.4342945.

β.

(7) Log sin = 4 log sin+sin_M (sin" + sin x')

2

2

cos à cos λ'

*(sino