454-457] Terrestrial Magnetism 391 



cooling and heating either of the earth's mass, or of the atmosphere as 

 suggested by Faraday. Schuster*, who has analysed the daily-varying 

 terms in the potential, and Balfour Stewart have suggested that the cause 

 of this variation is to be found in the field produced by electricity induced 

 in the upper strata of the atmosphere, as they move across the earth's 

 magnetic field, a suggestion which has received a large amount of experi- 

 mental confirmation f. In addition to this field produced by external 

 sources, Schuster finds that there is a smaller field, roughly proportional 

 to the former, having its source inside the earth. This he attributes to 

 the magnetic action of electric currents induced in the earth by the 

 atmospheric currents already mentioned. 



456. The non-periodic part of the earth's field, since it is found to arise 

 entirely from magnetism inside the earth, has a potential of the form 



& 81 n =* m ^ n (P(sin /) , x ] 



n = 4 + ;J+ = 2 ^ J-- w -^_^ (A n>m cosm\ + B n>m smm\)\, 



n-l rn = ( T ) 



in which the values of the coefficients may be obtained in the manner 

 already explained. 



This method of analysing the earth's field is due to Gauss, who calculated 

 the coefficients, with such accuracy as was then possible, for the year 1830. 

 The most complete analysis of the field which now exists has been calcu- 

 lated by Neumayer for the year 1885, using observations of the field at 

 1800 points on the earth's surface. 



The first few coefficients obtained by Neumayer are as follows : 



^,= -0057, 



2)1 = -0130, 2)2 =--0126, 



A =-'0244 1 = *0396, A 3>2 = -'0279, 4,,, = --0033, 



3} ! = -0074, 3) 2 = - -0004, ,, , = - -0055, 

 =- '0306, 2 =' 



- '0344 



4 , 2 =--0071, 4)S = -0051, 4)4 =--0010. 



457. The simplest approximation is of course obtained by ignoring all 

 harmonics beyond the first. This gives as the magnetic potential 



fl = - JA.o % (sin I) + % l (sin I) (A lfl cos X + B lfl sin X)l 

 = - j'3157 sin I + cos / ('0248 cos X - '0603 sin X)i . 



* Eoy. Soc. Phil. Trans., 1889, p. 467. 



t See, for instance, a paper by van Bemmelen, Konink. Akad. Wetenschappen (Amsterdam), 

 Versl. 12, p. 313, in which it is shewn that the field of daily variation may be regarded roughly 

 as revolving around the pole of the Aurora Borealis (80'5N., 80 W.). 



