APPENDIX II.

ON UNLIMITED SPHERICAL TRIANGLES AND THEIR SOLUTION.

Page

The various triangles formed with the same three points on the sphere,

Demonstration of spherical formulas of unlimited application,

Proof that there may be 128 triangles, each the first term of an infinite

series formed by three great circles on the surface of the sphere,

179

Ambiguity in the solution of the general spherical triangle,

181

Formulas required for the solution of the general spherical triangle. Gauss

Theorem,

182

Mode of using the Gauss equations, .

184

Proof that the parts of the triangle may be interchanged in the Gauss

equations,

ib.

Auxiliary angles,

186

Solution of the several cases of the general spherical triangle, with the use

of checks,

Formulas to be used in place of Napier's rules, when the required part

expressed by its sine is very small, or expressed by its cosine is very

large,

Rules for deciding upon the ambiguity of the result in the use of the sine

proportion,

ib.

196

Modes of determining the effect of small errors of observation in the data

upon the quæsita, and under what circumstances these will be

minima,

197

187

193

194

Great circle courses from New York to Liverpool or Havre,

From the Chesapeake to Bordeaux, .

228

229

Description and use of the theodolite and its adjustments,

238

239

Survey of cross sections and computation of excavation and embankment,

Computation of the contents of fields,

255

256

Mode of determining the apparent right ascension and declination of a star

from the British Catalogue,

269

Of the corrections to be applied to the observed altitudes of heavenly

bodies, . :

270

DETERMINATION OF THE LATITUDE AT SEA BY MERIDIAN ALTITUDE,

279

Example and exercises,

281

Determination of the time of the moon's meridian passage,

282

Method of finding the local time by an altitude of the sun or other

Description of Troughton's Circle and mode of observing with it,

ib.

Use of one of Bowditch's tables, and tables of versed sines,
Example of circum-meridian altitudes with repeating circle,
Method of Littrow by three altitudes near the meridian,
Latitude by an altitude of the pole star out of the meridian,
THE MURAL CIRCLE,

Description and use of the ghost apparatus for adjustment,

66 "the reading microscopes,

Method of taking the readings,

Reduction to the meridian of an observation with the mural,
Meridian circle,

PART V I.

GEODESY.

Page

Mode of computing the spherical excess,

323

Ditto between the latitude of 450 and 250,

324

Form and example of the above used on the U. S. Coast Survey,

DETERMINATION OF LATITUDES, LONGITUDES, and Azimuths of the STATIONS,

Demonstration of formulas for the spherical figure of the earth, .

Necessary modifications for the spheroidal figure,

Specimens of table from which co-efficients of the formula for difference of

325

ib.

ib.

327

latitude are taken,.

328

Example of geodetic determination of latitude, .

329

Indication of cases in which the third and fourth terms of the formula may

be omitted,

ib.

Number of places to which the logs. of the different terms must be carried,

Demonstration of the formula for difference of longitude,

330

ib.

Form and rationale of subsidiary table for its correction when the geodetic
line exceeds 16 miles,

Demonstration of the formula for difference of azimuth,

Rule for the direction in which azimuths are to be estimated,

Form used on the U. S. Coast Survey and example of geodetic determina-

tion of latitude, longitude, and azimuth,

PROJECTION OF MAPS,

Use of blank maps by plane table parties,

331

332

333

ib.

334

ib.

By difference in time of culmination of the moon and a star,

Example,

ib.

350

Rule when the meridian is distant from that of Greenwich,.

Formula for interpolation,