have been applied to the distance obtained and used on this section, one being that for reduction to sea level, amounting to -8 feet, and the other that for reduction on account of tape, which, after comparison with the United States Coast and Geodetic Survey standard, was found to be +12 feet. Thus, theoretically, the monument was placed 4 feet too far north, but the error, compared with that which might result from unavoidable station error, is so small that it may be considered as negligible.

It might be well to explain what is meant by "station error." It is the error which is more or less inherent in all astronomic determinations, being caused by the deviation of the plumb bob, on account of varying local attractions, from the true vertical. Its existence may be detected by direct measurements between two astronomic stations. Assuming an astronomic location to be made at A, a connection by triangulation with another station, B, and the position of B thus deduced from that of A, then if the astronomic position of B is determined, the discrepancy between the astronomic position and the position determined by triangulation from A represents the combined station error. Efforts have been made to deduce a law, based on the contour and density of adjacent land forms, which shall define the amount of attraction affecting the plumb bob which might be expected in any particular case, but nothing satisfactory has been accomplished, owing principally to the fact that it is impossible to give an accurate value to the various components which constitute the mass of any given section of the earth's crust.

The following table gives the results obtained directly from the stadia, and which were used in reducing the random line to the true line, both in distance and in azimuth. These figures do not, however, represent the degree of precision obtained by the stadia in measuring the distances, as they are unaffected by any corrections.

Table showing discrepancy between computed and measured distances, also discrepancy in azimuth.

The distances measured by the stadia should be corrected before any satisfactory comparison can be made with the computed distances for the purpose of testing the accuracy of the stadia work.

Two conditions will be considered. The first is based upon the

supposition that the rods were graduated arbitrarily into regular divisions, which were sufficiently small to admit of reading distances approximately to a foot, and that the stadia wires were set at some fixed interval. The fact that the rods were actually graduated so as to be read, as nearly as might be, to feet, tenths, and hundredths, and that the wire interval was approximate in the ratio of 1 to 100, does not enter at all into the theory of this condition. In fact the rod might be graduated in any systematic manner and the wires placed at any fixed interval without affecting the results after the proper corrections had been applied. An analogous case would be one in which certain distances were measured with a chain or tape the exact length of which was not known at the time the measurement was made but which was afterwards determined and the correction applied. The tape might prove to be 99 feet in length, but this would not affect the ultimate results after proper allowance had been made. The fourth section of the boundary line furnished a means by which the relation between the distances determined by the stadia and careful measurements with a steel tape, in other words, the stadia factor, might be ascertained.

Table of comparisons of stadia and steel-tape measurements from transit station 949 (post 8) to the international boundary (post 0), section 4.

An inspection of the foregoing table will develop the fact that the ratio existing between the results from the steel-tape and the stadia measurements is fairly constant. The steel-tape measurements were carefully made with a 100-foot steel tape, under conditions approximating those under which the other portions of the line were measured independently by the stadia, except that the surface was probably not so broken and the change of elevation not so great, thus favoring the accuracy of the tape measurement. The steel tape was afterwards compared with the United States Coast and Geodetic Survey standard, and the values in the table are affected by a small correc

tion necessary to reduce them to the standard. Thus the comparison may be considered as determining the stadia factor, or the amount by which each stadia distance should be corrected in order to reduce it to a true measurement.

It appears from the table that in a distance of 33,219 feet the stadia measurement was 33,103 feet, or a difference of 116 feet, which corresponds to 1 in 286; this correction to be added to the stadia results.

It might be argued that it would be preferable to obtain the stadia factor by direct comparison with the computed lengths of the lines in sections 1, 2, and 3; and so it would if the rods were to be used for other purposes, but for the present purpose, which is to make a comparison between the computed distances and the direct stadia measurements, the other method is adopted.

Table showing discrepancies between stadia distances corrected for stadia factor and computed distances.

A combination of the figures in the foregoing table indicates that in a total distance of 340,200 feet, or about 65 miles, there was an apparent error of 133 feet, or 1 in about 2,565. It would not, of course, be proper to claim any such accuracy for the stadia work, and in fact the figures themselves do not justify such a claim, as the range between them is too great to admit of any general deduction from the three combined results. For the purposes of this discussion it would have been better if the line could have been divided into more numerous sections and if more than one comparison for the determination of the stadia factor could have been made.

A second condition is represented by the supposition that the rods were originally accurately subdivided so as to read to feet, tenths, and hundredths and that the stadia wires were at a fixed interval of 1 to 100. A correction inherent to this condition, from the theory of stadia measurement, is that of f+c, f corresponding to the distance from the plane of the cross wires to the objective, and c being the distance from the center of the instrument to the objective. In order to obtain a correct distance the measurement should, then, be from a point which is f+c, or, in the instrument used, 0.95 feet, ahead of the center of the transit. In other words, this amount should be added to the distance obtained at each transit station.

Table showing discrepancies between stadia distances affected by "f+c" correction and computed distances.

The mean discrepancy for the whole distance is 175 feet, or 1 in 2,133. This, it will be observed, approximates that obtained after application of the stadia-factor correction.

The results from the application of the f+c correction serve to indicate that the rod was graduated approximately correctly, as intended, and the stadia wires were fixed so as to read distances on the rod approximately in the proportion of 1 to 100. As the stadia-factor correction was systematic, the results obtained for distances along the boundary line by a direct adjustment of the original stadia distances to fit the triangulation exactly correspond to those which would have been obtained if the stadia-factor correction had been first applied and the remaining discrepancy afterwards adjusted to the triangulation.

The only check obtained with reference to vertical distances or measurements of height is that furnished by the portion of the line between the Northern Pacific and Great Northern railroads. Along the former road is a line of precise levels run by the United States Geological Survey from a mean sea-level connection at Tacoma, Washington. The stadia line was connected with a bench mark of the precise levels, and this bench mark is the datum upon which the elevation of the stadia line is based. The profile of the Great Northern Railway adjusted to the United States Geological Survey precise levels furnished an elevation at Leonia to which the stadia line was joined. The resulting discrepancy was 73 feet, which is greater than might reasonably be expected, but it is possible that some of the error may be in the railroad profile.

The details relating to the profile of the boundary line between these two railroads are as follows: Sum of horizontal components, 201,188 feet; sum of vertical components, 42,700 feet; discrepancy in closure, 73 feet; discrepancy referred to horizontal components equals 1 in 2,756; discrepancy referred to vertical components equals 1 in 585.

Extensive and careful experiments with the stadia were made by the