2 KEPORT— 18r2. 



■what we call the augles of declination and inclination. Indeed at any place 

 a component in any direction whatever of the magnetic force is merely pro- 

 portional to the increment which the potential there takes by a small 

 displacement in the same direction. But now the determination of 

 that potential which outside a sphere results from any magnetic actions 

 of its interior, and therefore, according to the last remark, the foretelling of 

 all magnetic phenomena produced by the same causes, become possible and 

 are facilitated by the following circumstances. In every case of the descrip- 

 tion just mentioned the mag-netic potential can be expanded into an infinite 



but converging series, proceeding by integer powers of -, the exponent of the 



r 



first one being -|-1. Among the terms of this series, that which is divided 



by r contains 3 of the Gaussian constants, 

 r^ 5 



» 



I 



2n+l „ 



In the formula for the potential, each of these constants is mtdtiplied by a 

 theoretically given trigonometrical function of u and X, and therefore, for any 

 given point, by the numerical equivalent of this function. The algebraic 

 developments which Gauss's classical work contains far the magnetic potential, 

 as well as for the observable magnetic components, relate also to the actions 

 of a sphere enclosing a finite or infinite number of any magnetic centres 

 whatsoever. Therefore these expressions can represent our terrestrial pheno- 

 mena only after the substitution, for every symbol denoting a Gaussian con- 

 stant, of that number which the individual magnetic qualities of the earth 

 require, according to observations. But then, specially, this transformation of 

 the abstract theory of the magnetic actions of a sphere into the practical 

 theory of terrestrial magnetism will amount to the determination, from a 

 sufiicient number of observed values, of 15, 24, 35, or generally n'^ + 2n 

 Gaussian constants, according as it appears that the third, the fourth, the 

 fifth, or generally the nth term in the algebraical expressions of these em- 

 pirical data is the first that is surpassed by the probable amount of their 

 inevitable errors. 



A first attempt towards the completion of the theory of terrestrial magnetism 

 was made by its illustrious author with material of which the gaps for the 

 greater part of the Antarctic Ocean, and for other vast regions, could only be 

 filled up by graphical guesswork. It led to the conclusion that a restriction 

 to four terms of the potential, and therefore the determination of 24 Gaussian 

 constants, did more than respond to the mean exactitude of the empirical data. 

 To the same effect was the computation that H. Petersen executed from 

 1846 to 1848, when commissioned for the purpose by the British Association. 

 Indeed, it being exclusively founded on 610 results of careful magnetic 

 measurements made by A. Erman on a line round the earth between 

 67° north latitude and 60° south latitude, the resulting new constants re- 

 presented these observed values fully twice as well as did the old ones, 

 and thereby, as must be avowed, up to the amount of their own probable 

 errors. But it having been shown by later experience that, just as was 

 expected, miich larger disagreement between reality and both the theoretical 

 deductions, did still exist in those parts of the earth where the one or the 



