232 C. F. GAUSS ON THE GENERAL. THEORY OF 



the former supposition, v must first be considered as a function of 

 both r, u, and \ ; it must be differentiated according to r, and 

 then r = R must be substituted. 



But for the inner space S', to which the surface of the earth 

 belongs, v can only be developed in a series according to ascend- 

 ing powers of r. If we make 



V 

 p^ is a constant magnitude, namely, the value of-^ at the centre 



of the earth ; p', p", ;/", &c., on the other hand, are functions of 

 ti, and \, which satisfy the same partial differential equations as 

 pf, p", p'", above. 



Hence it follows, in a similar manner to Art. 20, that the 

 knowledge of the value of v at every point of the earth's surface 

 is sufficient to enable us to deduce therefrom the general expres- 

 sion for the space S' ; that w-e may arrive at the knowledge of 

 this value with the exception of a constant part, — or, which is the 

 same thing, at the knowledge of the co-efficients jo', /',/", &c., — 

 by the knowledge of the horizontal forces on the surface of the 

 earth ; but that the value of the vertical force on the surface of 

 the earth is not 



= 2p' + 3p" + 4p"' +, &c. 

 as it would be if the forces acted outwards from the interior of 

 the earth, but is 



= -y - 2p" - Sp'" -, &c. 



Now, as our numerical elements (Art. 26.), detennined under 

 the supposition of the first formula, give a very satisfactory re- 

 presentation of the phenomena generally, whereas, the pheno- 

 mena would be wholly incompatiljle with the second formula, 

 the fallacy of the hypothesis, which places the causes of terrestrial 

 magnetism in space extex'nal to the earth, may be looked upon 

 as proved. 



40. 



At the same time, this must not be looked upon as decidedly 

 disproving the possibility of a part, though comparatively a 

 veiy small part, of the terrestrial magnetic force proceeding 

 from the upper regions : a far more full and far more accurate 

 knowledge of the phenomena may in future throw light on 

 this important point of theory. If, under the supposition of 



