ASTRONOMY AND GEODESY.

The observations made in California were computed in the field by myself, Lieutenant Whipple, and James Nooney. They were all subsequently revised by Professor Hubbard, and the new element of corresponding observations introduced into the computations for longitude. The observations on the Rio Bravo were computed in the field by myself and Assistants J. H. Clark and J. O'Donoghue, and revised in the office by Captain George Thom, corps of topographical engineers, and Assistant C. N. Thom.

The observations on the parallels 31° 47′ and 31° 20' were all computed in the field by Principal Assistant J. H. Clark, Lieutenant Turnbull, corps of topographical engineers, and Assistant Hugh Campbell, and revised in the office by the same.

The observations at the point where the line strikes the Rio Colorado were made and computed by Lieutenant Michler and Assistant J. O'Donoghue.

A.

Determination of the line forming the boundary between the United States and the republic of Mexico, from the Initial Point on the Pacific ocean to the point where the "Gila river empties into the Colorado." By Brevet Major William H. Emory, Chief Astronomer.

This portion of the boundary consists of a straight line from a point on the Pacific ocean, one marine league south of the port of San Diego, to the junction of the Gila and Colorado. The most obvious way of determining the direction of this line was to connect the two points by triangulation, and in this way ascertain their relative positions on the face of the earth, and compute the azimuth of the line joining them. But the character of the intervening country made it impossible to pursue this mode of operating when the time and means at the disposal of the joint commission were considered. Triangulation is the surest, but it is the slowest and most expensive method of surveying, even in old-settled countries, where the stations to be selected are easily accessible in wagons. In the country under consideration obstacles presented themselves almost insurmountable. The whole distance, about 148 miles, may be divided into two nearly equal parts, differing in character, but both equally unfavorable to geodetic operations. The first, rising in steppes from the sea, and covered with spinose vegetation, attains, in abrupt ascents, the height of five or six thousand feet in the short space of thirty miles. From this point, for about thirty miles, the country is occupied by a succession of parallel ridges, striking the boundary nearly at right angles, and separated by deep, and sometimes impassable chasms. It then falls abruptly to near the level of the sea. The remainder of the line stretches across the desert of shifting sand at the head of the Gulf of California, destitute, for the most part, of both water and vegetation.

The following is the order in which are arranged the subjects embraced in the determination of the line:

1. The longitude of Camp Riley, near the Initial Point.

2. The longitude of Camp near the junction of the Gila and Colorado.

3. The latitude of Camp Riley, near the Initial Point.

4. The latitude of Camp near the junction of the Gila and Colorado.

5. Transfer of the latitude and longitude of Camp Riley, by triangulation, to the Initial Point.

6. Azimuth of straight line from Initial Point, on Pacific, to junction of Gila and Colorado.

Numbers 1, 3, 5, and 6 are by myself. Numbers 2 and 4 are by Lieutenant Whipple. The tracing of the line on the ground was partly by myself and Lieutenant Whipple, but chiefly by Captain E. L. F. Hardcastle, corps topographical engineers.

The computation of the azimuth of the line was made in the field. In this computation the earth was supposed to be a spheroid of revolution of the following dimensions, which are those determined by Bessel from all the measurements up to that time, (1849,) and the elements given by him were converted into English measure by adopting the following value of the metre, viz:

I am indebted to Professor Airy for the observations at Greenwich for 1849; and for the recomputation of the longitude and the application of the correction due to the corresponding observations on moon and moon culminating stars, I am indebted to the assistance of Professor J. S. Hubbard, of the National Observatory.

I. LONGITUDE OF CAMP RILEY, NEAR THE INITIAL POINT.

The observations with the transit instrument have been reduced in the following manner: The equatorial intervals of the transit wires having been determined as accurately as possible, the imperfect transits were corrected, by applying to the mean of the observed wires the mean of their equatorial intervals, multiplied by the secant of the stars' declination.

For circum-polar stars, each wire was reduced separately, and the mean of the results taken. In the case of the moon, allowance was made for its motion by the method and tables of Bessel. (Tabulæ Regiomontanæ, pp. LII and 537.)

Denoting by a the constant of correction for azimuth of the instrument, by b the constant for level, and by c that for collimation, and by d the star's right ascension, & its declination and z its zenith distance, and by t the chronometer time of its transit, and by At the correction of the chronometer at the time t, we have the known formula

or

a. cos. ❤,

a = t + s t + m + n. tan. d + c. sec. d,

a = t + At +m+ (n + c.) tan. + c. (sec. tan. d)

which last form has been employed in the reductions.

In the observations at Camp Riley, co for nearly the whole series, and is small enough at all times to have no effect in the last term of the formula; in the other series, one or two cases occur where it has been necessary to take this last term into account.

Where, as in the present case, only the right ascension of the body is wanted, the quantities A t and m being constant for the evening, may be combined together, and then the last term of

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one equation, always small, and vanishing at no great distance from the equator, being introduced when necessary, and the requisite correction for the chronometer rate being applied, it is evident that but two equations are necessary for the determination of the unknown quantities. One is generally furnished by a circum-polar star; the other by the mean of the equations, corresponding to all the stars near the moon's path, in order the more completely to remove all chance of constant error from the desired result. The equations being solved, furnish the quantities given below, and which have been applied to the observations.

The first column contains the date; the second the name of the object observed; the third shows the position of the instrument, (lamp east or west.) Next follow columns 4, 5, 6, 7, 8, 9, 10, the seconds of observed transit, and column 11, the mean of the transit over as many wires as have been observed. Column 12 contains the correction to be applied to this mean for an imperfect transit; column 13 the correction for instrumental error, or the quantity (n + c) tan. d + c. (sec. d -tan. o,) the last term of which is generally equal to o; and column 14 gives the correction of chronometer, and the constant term of instrumental correction, or the quantity ▲ t + m.

In cases where a mean-time chronometer has been used, this column includes also the reduction of mean to sidereal time, the quantities in all the preceding columns being in mean time. In column 15 is given the sum of the quantities; in columns 11, 12, 13, and 14 are the observed right ascension of the object; and the last column shows the tabular right ascension, taken in the order of preference from the Nautical Almanac, the Greenwich Twelve-year Catalogue, or the Catalogue of the British Association.

The next step was to deduce the required corrections of the assumed longitude of the place, by comparing the observed AR. of the moon with that corresponding to the assumed longitude already determined very approximately by computations in the field. For this purpose, the tabular AR. was interpolated from the moon culminating list of the Nautical Almanac, using fourth differences, and it was found that the assumed longitude corresponded perfectly to the results from the uncorrected tabular place of the moon. But the extracts from the observations at Greenwich, given below, show a correction of the latter to be necessary; and this being applied, the corresponding correction of the assumed longitude was determined and also applied.

* The value of At adopted for the mean-time chronometer being the complement of the time value, it becomes necessary to change the sign of (n+c.)

t