469 



TABLE 495.— ELEMENTS OF THE EARTH'S MAGNETIC FIELD (concluded) 



For purposes of mathematical analysis, it is convenient to recognize that the magnetic 

 intensity or field strength (like other vector fields) is derivable from a scalar function or 

 potential. If V be the potential corresponding to the geomagnetic field, we may write 



F = — grad V, 



whence any of the magnetic elements may be expressed as functions of the potential. 

 In polar coordinates (r, 0, X) with origin at the earth's center, we have 



V = a 2 Ur/a) n 7V + (a/r) D+1 TV j = V* + V\ 



where a denotes the earth's mean radius (6.37 X 10 8 cm) (see Table 827). 



m 

 Tn = 2 (<7„ m cos m\ + h n m sin tn X) p B m (6). 

 m—0 



Here is the colatitude and X the east longitude, and the affixes e and i refer to portions 

 respectively of external and internal origin. The function 



P„» {0)=\2 [ n - m \ \ V P„, m {6) when tn > 0. 

 = Pn,m (6) when m=0, 



where 



P n , m (6) = (2n) ' . , sin m 9 \ cos" 



2. , n ! (n — m) ! ( 



(n — m) (n — tn — 1) „„ n -m 



2(2» — 1) 



+ 



-I 



Magnetic surveys of portions of the earth have been made by means of observations at 

 many thousands of stations, the elements usually observed being D, H, and /. Such sur- 

 veys are repeated in part every few years in populated areas, and at intervals of one or 

 more decades in most areas, because of a substantial and usually unpredictable change in 

 the earth's field known as geomagnetic secular change. These changes are most accurately 

 measured at fixed magnetic observatories to the number of about one hundred. The U. S. 

 Coast and Geodetic Survey operates magnetic observatories at Cheltenham, Md. ; Tucson, 

 Ariz. ; Sitka, Alaska ; Honolulu, T. H. ; and San Juan, P. R. Other nations conduct 

 similar measurements. 



Magnetic surveys by airplane will no doubt be commonplace in future years. 



The part of the earth's field having external origin does not exceed a few percent, and 

 its existence has never been indicated with much certainty by the spherical harmonic 

 analyses. If the distinction between contributions of external and internal origin in the 

 first formula is disregarded, the accompanying tables give the values of the principal har- 

 monic terms at various epochs. 



The magnetic moment of the earth as given by the centered dipole approximation for 

 1922 was 8.04 X 10 2B cgs. The axis of this dipole intersects the earth's surface at points 

 called the geomagnetic (axis) poles, located in 1922 at latitude 78?S N., and longitude 

 270?0 E. ; and at latitude 78?5 S., and longitude 111° E. In comparison with these cur- 

 rently adopted values, the analysis of Vestine and Lange for 1945 shows only slight change 

 that may have taken place since 1922. 



The dipole part of the earth's field diminishes with height h approximately as (1 — 3h/a). 

 Values for 1945 have been estimated in tabulation to heights as great as h = 5000 km for 

 spherical harmonic terms up to degree six.t 



The magnetic north and south poles of popular interest are those defined by H = 0, or 

 by / = ± 90°. As H changes with time, owing to secular change, these poles must move 

 with time, except in the unlikely event that the lines of zero change of X and Y both 

 happen to pass through the poles. There are a principal north magnetic pole and a princi- 

 pal south magnetic pole, which undergo substantial change in position with time. In addi- 

 tion there are undoubtedly local (secondary) magnetic poles near each principal pole. These 

 secondary poles occur only in pairs. Of each pair, one pole has the character of a poten- 

 tial focus (like the corresponding principal pole), while the other is a "false pole" or node 

 of the equipotential lines. The secondary poles do not individually undergo large-scale 

 migration, since they are associated with localized magnetic materials in the earth's crust. 

 These occur when such materials succeed in reducing the changing value of H to zero, as 

 the principal migrates. 



The principal north and south magnetic poles are not diametrically opposite, each being 

 about 2,300 km from the antipodes of the other. 



t See bibliography, reference g, p. 501. 

 SMITHSONIAN PHYSICAL TABLES 



