NOTES TO BOOK XII. 73 



ferentials of which the elements of the magnetic force at 

 any point will be expressed. This function Fis well known 

 in physical astronomy, and is obtained by summing all the 

 elements of magnetic force in each particle, each multiplied 

 by the reciprocal of its distance ; or as we may express it, 

 by taking the sum of each element and its proximity 

 jointly. Hence it has been proposed (Quart. Rev. No. 131, 

 p. 283,) to term this function the ' integral proximity 1 of 

 the attracting mass 1 . By using the most refined mathe- 

 matical artifices for deducing the values of V and its dif- 

 ferentials in converging series, he is able to derive the 

 coefficients of these series from the observed magnetic ele- 

 ments at certain places, and hence, to calculate them for all 

 places. The comparison of the calculation with the observed 

 results is, of course, the test of the truth of the theory. 



The degree of convergence of the series depends upon 

 the unknown distribution of magnetism within the earth. 

 " If we could venture to assume," says M. Gauss, " that 

 the members have a sensible influence only as far as the 

 fourth order, complete observations from eight points 

 would be sufficient, theoretically considered, for the deter- 

 mination of the coefficients." And under certain limita- 

 tions, making this assumption, as the best we can do at 

 present, M. Gauss obtains from eight places, 24 coeffi- 

 cients (each place supplying three elements), and hence 

 calculates the magnetic elements (intensity, variation and 

 dip) at 91 places in all parts of the earth. He finds his 

 calculations approach the observed values with a degree 

 of exactness which appears to be quite convincing as to 

 the general truth of his results ; especially taking into 

 account how entirely unlimited is his original hypothesis. 

 1 See the Addition at the end of this Note. 



