February 21, 1901] 



NA TURE 



409 



approaches for about four-sevenths of the arc from its eastern 

 end closely to that of the Clarke spheroid ; whereas, for the 

 remaining three-sevenths, or for the region across the Rocky 

 Mountains to the Pacific, the curvature comes more nearly to 

 that of the Besselian spheroid. In the published paper two 

 tables are given containing the results needed for combination 

 with any other arc and, in conclusion, some preliminary rough 

 combinations of American arcs are presented ; all of which 

 point to a reference spheroid of larger dimensions than those of 

 the Besselian and are in favour of continuing the use of Clarke 

 for reference. 



The second arc under consideration extends from Calais, Me. , 

 in the north-east and opposite the Canadian boundary, to the 

 Gulf of Mexico, and terminates at New Orleans, La. It iS 

 known as the Eastern Oblique Arc of the United States. Its 

 length is 2612 km., or 1623 statute miles ; its difference of lati- 

 tude is 15° i', and of longitude 22° 47'. The general direction 

 is therefore favourable, and the length ample to secure fair 

 results for an o.sculating spheroid. In the main the triangula- 

 tion follows the Appalachian chain of mountains ; in Western 

 North Carolina and Eastern Tennessee it bifurcates, leaving an 

 oval space between the two branches. The length of sides 



of the vertical and for variation of pole according to Dr. 

 Chandler's and Dr. Albrecht's researches. The same scrutiny 

 as before had been extended to the deflections of the vertical, 

 both regional and local. Partly on account of avoiding un- 

 necessary labour, but principally on account of the crowding 

 together of astronomical stations in certain very limited localities, 

 and all of them, therefore, partaking of the deflections charac- 

 teristic of this area, the total number of astronomical stations 

 admitted into the final equations for the determination of the 

 best spheroid were thirty-six for latitude, fourteen for longitude, 

 and thirty-four for azimuth, or eighty-four conditions in all. 



These eighty-four differences between the astronomical and 

 geodetic results constitute the data needed for a new deter- 

 mination of a spheroid ; next the functional relations between 

 the positions of these stations upon the reference spheroid to 

 the earth's equatorial radius and to the compression of the polar 

 axis had to be established. 



The final normal equations contain, therefore, four unknown 

 quantities, viz. the correction to the meridional deflection of the 

 vertical at the initial or reference station of the oblique arc ; 

 second, the corrections to the deflection of the vertical, in the 

 plane of the prime vertical, at the same place ; third, correc- 



PRINCIPAL ARCS OF THE MERIDIAN. THE PARALLEL AND OBLIQUE ARCS 



„. ,„t ..r.-itmCMCNT or THC cartms nouRr .., 



MAP ON 



MOUWEIDE'S EQUIVALENT 



OR H0MAL06RAPH1C PROJECTION 



PROJtCTEO 



depends upon six base lines, and in general the development is 

 closely accommodated to the hypsometric and other natural 

 conditions along the course. It includes among its stations the 

 two highest points in the eastern part of the United States, 

 viz. Mount Washington, N. H., rising to about 1920 metres, or 

 6300 feet, and Mount Mitchell, N.C., rising to about 2038 

 metres, or 6687 feet. 



The adjustment of the wh9le triangulation is effected precisely 

 as explained in the use of the arc of the parallel ; the small re- 

 duction to the sea-level of the observed horizontal directions, on 

 account of the altitudes sighted, was only applied when exceeding 

 0'05". The principal labour of adjustment was demanded by the 

 necessity of bringing into accord the measured lengths of the 

 Fire Island, the Massachusetts and the Epping base lines, and 

 fulfilling the geometrical conditions of the intervening net of 

 triangles. This demanded the satisfying of fifty-seven condi- 

 tions and involved the simultaneous solution of an equal 

 number of normal equations and the working out of 131 

 corrections of observed directions. Of astronomical measures we 

 have seventy-one latitute stations, seventeen longitude stations, 

 and fifty-six azimuth stations, tolerably well distributed over the 

 whole extent of the arc. The latitudes, as were those of the 

 arc of parallel, were corrected for height of station or curvature 



NO. 1634, VOL. 63] 



tion to the equatorial radius of the reference spheroid ; and, , 

 last, the correction to its compression. 



In the combination of conditional equations arising from 

 observations of a diff"erent nature, the question of their relative 

 weights must be considered. In the present case, four as- 

 sumptions were made and the consequent normal equations 

 solved, viz. for equal weights, for weights one-half, one- third 

 and one-fourth to the azimuth equations, the latter being, 

 necessarily inferior to the equations derived from latitudes and 

 longitudes. A comparison of these four results showed that it 

 was of small consequence which of these hypotheses was finally 

 adopted, since the corrections to the equatorial radius of the 

 reference spheroid were practically the same for any of these 

 hypotheses, and nearly the same could be said of the resulting , 

 compressions. The weight one-third to each of the azimuth 

 equations was finally decided upon, and the resulting dimen- 

 sions of an osculating spheroid were found to be : — 

 Equatorial radius, 6,378,157 + 90 metres ; compression, 

 i/304'5 + I "9. The equatorial radius, therefore, difl'ers but 

 49 metres from Clarke's value of 1866 adopted on the Survey, 

 while the Besselian value is apparently too small by 809 metres. 

 On the other hand, the compression or the ratio of the difference, 

 of the equatorial and polar semi-axes to the former is in favour 



