4 



BETWEEN THE UNITED STATES AND MEXICO. 253 



II. Discussion of lines of equal magnetic declination, dip, and Jiorizontal intensify. 



In this discussion, the observations made in previous years in connexion with the Mexican 

 boundary surveys, and published in the fifth volume, new series, of the Memoirs of the 

 Academy, have been combined with those communicated in the present paper. Observations 

 made under the direction of Professor A. D. Bache, Superintendent of the Coast Survey, at 

 stations Dollar Point, East Base, and Jupiter, near Galveston, Texas, and near San Diego, 

 Monterey, and San Francisco, and affording important co-ordinates for the curvature of the 

 lines, have also been introduced ; likewise, an observation of the declination at the Great Salt 

 Lake, by Captain Howard Stansbury, U. S. Topographical Engineers, published in his report 



of the survey of that region. 



The method employed in determining the lines ' of equal declination and dip was partly 

 graphical and partly analytical, being the same pursued by Professor Bache and Mr. J. E. Hilgard 

 in their discussion of the Coast Survey magnetic observations. — (Coast Survey Report for 1S55, 

 Appendix, p. 47.) The stations were projected on a map, and their positions referred to a right 

 line graphically assumed as axis of co-ordinates, the origin being chosen about the mean posi- 

 tion of the stations, and the direction so as nearly to divide the positive and negative ordinates 



equally. 



The co-ordinates being read off on any convenient linear scale, conditional equations are 

 formed for each station or group of stations, and the whole scheme is solved by the method of 

 least squares. The conditional equations representing an interpolation by second differences 



are of the form, 



V is the observed declination (or dip) ; 



Fq, the assumed declination at the origin ; t?, the correction to be applied ; 



Xand y, co-ordinates of position. 



x^ y, z, py qy coefficients to be determined. 



The solution of a considerable number of such equations involves a great deal of labor, which 

 the results amply repay, however. The process being well known, there is no occasion to give 

 the steps of the calculations in this place. After determining the coefficients, the co-ordinates 

 of points in the lines sought were computed, the lines projected on the map, and the latitudes 

 and longitudes of points read off and tabulated. 



In the absence of any data to determine the secular changes, the results are doubtless liable 

 to an uncertainty from that source. We know, however, that the changes are small, and, for 

 the limited period over which the observations extend, they may be considered as merged in the 

 local errors, and the average date of 1852 as belonging to the resulting lines on the map. 



The following tables give a general resume of the observations, and the corresponding values 

 in the computed system ; also the residuals, the distribution of which is the best evidence of the 

 successful representation of the general facts involved. 



The geoo-raphical position of points in the lines of equal declination and dip — isogonic and 

 isoclinal lines — are also given in tables from which they may be readily projected in any map. 



