597 



surface of the earth, then the horizontal component directed 

 towards the east or west would follow of itself; and not fess re- 

 markable was the consequence deducible from this, that the know- 

 ledge of the value of the potential function which the horizontal 

 component, as above stated, would furnish for all points of the 

 earth's surface, would also give its value for all points of external 

 space. But of all the conclusions which this memoir contained, 

 those which excited the most sanguine hopes of the ultimate and 

 complete solution of the great problem of the earth's magnetism, 

 were the successive theorems in which he showed that the com- 

 ponents of the magnetic force for any point of the earth's surface 

 may be represented by combining, with given functions of the lati- 

 tude and longitude of that point, certain constant coefficients of 

 which not more than twenty-four were likely to be required which 

 were deducible from a sufficient number of the observed values of 

 those components in different and assigned localities. 



The calculation of these coefficients, a work of no ordinary 

 labour, was effected by the author of this theory ; and the results 

 which they afforded were compared with their values, as given by 

 observation, at ninety-one stations. The discrepancies between 

 observation and theory, which were shown by these results, were 

 not more considerable than might have been expected from the 

 inadequate extent to which the calculations had been carried, and 

 from the necessarily imperfect character of the data which were 

 made subservient to them. The calculation of the same coefficients 

 was renewed, and greatly extended, by Petersen, under the direc- 

 tion of the younger Erman, at the request and expense of the 

 British Association, and the results are published amongst their 

 Reports for 1847. Some of these results would seem, however, 

 rather to indicate defects in the theory than errors of the obser- 

 vations, and it seems highly desirable, with a view of further testing 

 its correctness and applicability, that the calculations should be 

 resumed with the aid of more accurate and multiplied observations. 



In a subsequent memoir Gauss enters upon a discussion, which is 

 at once elementary and profound, of the general properties of the 

 same potential function which plays so important a part in the 

 " Allgemeine Theorie des Erdmagnetismus," ending with a series of 

 propositions on the relations of this function for a distribution of 



