206 C. F. GAUSS ON THE GENERAL THEORY OF 



ning from one pole of the earth to the other, is sufficient for 

 the foundation of the complete theory of the magnetism of the 

 earth. 



IV. Finally, it is clear that the complete theory is also de- 

 ducible from the simple knowledge of the value of Zon the whole 

 surface of the earth. In fact, if Z be developed into a series, 



Z = CP + Q' + Q' + Q" +,&c. 

 so that the general member satisfies the often-mentioned partial 

 differeiatial equation; Q" must necessarily = 0, and 

 P' = I Q', P" = ^ Q", P'" = \ Q'", &c. 



21. 



On account of the simple nature of the dependence of the 

 several forces X, Y, Z, on a single function V, and the simple 

 relation which they bear to each other, they are far better calcu- 

 lated to serve as a foundation for the theory, than the usual ex- 

 pression of the magnetic force by the three elements, total in- 

 tensity, incUnation, and declination. Or rather, the latter mode, 

 natural as it appears in itself when the question is solely that of 

 comprehending the facts, cannot directly serve for the founda- 

 tion of the theory (at least not for the first foundation) until it 

 has been translated into the other form. 



In this view it would be very desirable that a general graphical 

 representation of the horizontal intensity should be made ; partly 

 because it would be more immediately useful for theory than the 

 total intensity ; partly because, in far the greater number of cases, 

 the horizontal intensity was originally that which was actually 

 observed, the total intensity having been subsequently deduced 

 from it by means of the dip. It is the more advisable to keep 

 the elements of the horizontal force unmixed, as they can be de- 

 termined with extreme accuracy with the present instrumental 

 means ; at any rate, the observed horizontal intensity should 

 never be suppressed when publishing the deduced total intensity, 

 without at least giving the dip employed in the calculation ; so 

 that a person wishing to employ the horizontal intensity for the 

 theory may either have, or be enabled to reproduce, the original 

 observed numbers. 



Interesting as it would be to found the theory of terrestrial 

 magnetism on observations of the horizontal needle only, and thus 

 to anticipate the vertical part, or the inclination, it is at present 



