EXPRESSION OF POTENTIAL. 415 



439. EXPRESSION OF POTENTIAL. Whatever may be the mag- 

 netisation of the Earth, the external potential may be represented, as 

 we have seen (369), by the expression 



which, for a point on the surface, reduces to 



We deduce from this, as the components of the Earth's magnetism, 

 1 3V JA SA 



K . . . 



a ou uu ou 



=-i_^=-!-r '+^2+ i 



a sin ull sin u [_ W <>/ " J ' 





The coefficients A 15 A 2 , A 8 , are functions of the two angles 

 / and u. A n is expressed (368) by 2n+i terms in sines and co- 

 sines. Hence, if we wish to represent the condition of the Earth by 

 a series of this form, we must determine three numerical coefficients 

 for A 1? five for A 2 , seven for A 3 , etc. 



Gauss found that, in the then existing condition of magnetic 

 determinations, it was useless to push the development beyond the 

 fourth term, so that there are still twenty-four numerical coefficients 

 to calculate. 



Every point of the surface gives three equations by the values of 

 the components X, Y, Z ; hence, if we know these three elements at 

 any eight places in the earth, we have a complete solution of the 

 problem. In order to avoid errors arising from neglected terms, and 

 from inexact observations, Gauss applied the method of least squares 

 to the data for eighty-four points, taken on twelve equidistant meri- 

 dians, and seven parallels. The results thus obtained were then 

 applied to ninety-nine other points. 



