4. For larger triangles, even having sides up to 200 miles in length, and ill-conditioned in form, the same method of Legendre may be adopted without error, but for such cases the local mean radius of curvature R' used in the formula for spherical excess must be more exactly determined in accordance with the mean latitude of the triangle.

The formula for obtaining the local mean radius of curvature (R') in feet at any latitude L is

The following are the results of Colonel Clarke's computations of the dimensions of the earth.

Colonel Clarke adopts a very short value of the mètre at 32° Fahrenheit, namely 3.280 86933 feet, while even Captain Kater adopted 3.280 899; he computes that the

earth's ellipticity in the longitude of Paris is, and that the earth's mean ellipticity is, which when corrected for local attraction becomes

The mean length of a degree of latitude in the longitude of Paris hence is 364 591 feet or 69'05 miles; the mean diameter of the earth is 41 800 173 feet or 7916% miles; and the mean circumference of the earth is 23 871 miles.

The arc of the meridian subtending one second is 10136 feet; that subtending one minute is 6081.6 feet; and that subtending one degree is 365 896 feet.

It may, however, be noticed that these data, based on the scientific equivalent of the mètre at 32°, are much less than they would be if the commercial value at 62°, about 3.2818 feet, were used.

V. Formulæ occasionally required in geodetic calcula

tions.

1. If a geodetic base line (c) consist of two portions, a and b, slightly inclined to each other, forming an angle 0,

2. Reduction to Sea Level.-If the corrected length of a base line (c), situated at an elevation (1⁄2) above sea level, is to be reduced to its value (d) at sea level,

ch

d=c where is the radius of the earth.

r

3. Well-conditioned Triangles.-If one side of a triangle (a) be known, and the three angles remain to be fixed and observed, and it be desired to arrange them so that the sides b and c may be least affected by the errors

of angular observation represented by a, ẞ, y, the resulting error in the side c will be equal to

and that in the side b will correspond to this in form; hence A should be less than 90°, and B and C should be nearly equal.

4. The reduction of a slightly-oblique to a horizontal angle.

If h and be the altitudes of the two objects in seconds, and D=the observed slightly-oblique angle, then approximately the correction

5. The supplementary or satellite station.

Let A be the axis of the signal or mark observed.

"

B and C the two neighbouring stations of observation.

S the satellite station or position ofthe instrument.

m the measured horizontal distance AS.

b and c the calculated distances, AC and AB,

If BAC=A; BSC=S; ASB=B; ASC=y;

Then AS-206 264′′·8 × m.

sin B sin y)

с

b

when S lies to the right of both AB and AC.

But when S lies to the left of AC, sin y changes its sign; and when S lies to the left of AB, sin ẞ changes its sign.

6. The Distant Triangle.

Let A, B, C be the three angles, a, b, c the three sides. of the distant triangle; and P any point at which the angles B subtended by b, and a subtended by a, are observed; then the angles required are CAP=0, and CBP=&; whence PA, PB, PC can be obtained.

Let Z=360°- (a+B+C); and 4=2—0 ;.

b.sin a

Then cotg 0=cotg Z|I+ a sin B.cos Z.

Where, if Z< 90° but > 0°, cos Z is+, and cotg Z is +,

This problem becomes indeterminate only when all the points P, A, B, C fall on the circumference of a circle.

7. The determination of a distance or base line that is nearly meridional, by the difference of the two observed latitudes of its extremities.

Let L-L' be the difference of latitudes in seconds,

M be the true azimuth of the base line,

d be the reduced length of the base line in feet,

Then d=10136.sec M × (L-L').

8. The calculation of latitudes, longitudes, and azimuths.

Let L be the given latitude of any station, A and C
the colatitude.

L' be the required latitude of any station B, and
C' the colatitude.

R and Q, the local radius of curvature in feet

and mean ellipticity of the earth.

M is the azimuth of B as seen from A.

d the reduced length of the arc AB in feet.

Then L-L'=

d cos M, d2. sin2 M.tan L
+

R sin I" 2R2 sin I"

(1+Q2cos2L)

And if G the difference of longitude in seconds,

=

N=the azimuth of A as seen from B,

Then G=

d. sin M
R cos L' sin 1′′

sin (L+L')

and N 180°-M-G. sin 1" x

cos (L-LY

9. The following however are approximations frequently used when the arc AB is less than one degree. Let AB be reduced to seconds of arc, and let ʼn be the difference of azimuth.

VI. The general formulæ of spherical trigonometry.

I. Properties of any spherical triangle, having angles A, B, C, and sides a, b, c; where s=(a+b+c).

1. Any side a <πr; and any angle A < 180°.

2. a+b+c<2πR; and A+B+C<3π and >π.
3. a+b>c; a-b<c; and A+B > 180° - C.

4. If a+b=180°, then will A+B=180°.