Theory of Terrestrial Magnetism. 405 



fore the electromagnetic potential due to the rotation of the 

 electricity on the surface of the earth is 



-g? 2awr cos X inside the earth, 

 o 



and 



47T 1 



-~- 2<rw ~2 cos 6 X outside the earth, 



where w is the angular velocity of the earth on its axis, r the 

 radial distance of any point from the earth's centre, 1 the co- 

 latitude of the place, A.ira the total quantity of electricity uni- 

 formly distributed over the surface of the earth measured elec- 

 tromagnetically, and the unit of length the earth's radius. 



These results, which we think are logical consequences of 

 the experiment performed in Professor HelmhohVs laboratory, 

 and referred to at the commencement of this paper, may now 

 be applied in various ways. 



For example, if the iron of the earth is arranged nearly in 

 a hollow sphere, of external and internal radius a 2 and a 1? 

 then, since the potential given above is a zonal harmonic, 

 we can at once apply Poisson's result; and we find that the 

 potential due to magnetization of the hollow sphere is 



47ta:(3 + Sttk) (a 3 — a 3 ) -^- awa\ 



& cos 



9 + 36™ + 32wV(aJ!--a;) ~r^~ 



for all points outside the outer surface of the sphere ; and 

 hence, for points outside the surface of the earth, the total 

 magnetic potential is 



^^ + ^){al-a^o-wal ^ Q 



L 9 + 36™ + 32wV(a;-a») + 3 J r 2 > 



where k is the coefficient of magnetization. 



Now Biot's approximation to the law of intensity of the 



force is 



\/l + 3sin 2 X, 



where \ is the latitude of the place ; and we understand that 

 this approximation is generally considered, for rough purposes, 

 as a fairly accurate one. 



Our equation for any point at a distance r from the centre 

 and having a colatitude 9 is 



