318 MAGNETISM 



constant, it may be omitted, for in determining the forces we 

 differentiate V. 



If we suppose a thin shell round the Mirfnrr of the earth to be 

 free from magnetism, within that shell the potential will be the 

 of the potentials due to external and internal magn-ti>m. Then 



The intensity in any direction .v is -- T-' It I is drawn in tin- 



geographical meridian towards the North, f/.v = }{dl. 

 component of the hori/ontnl intensity in that direction 1>< \ It 

 ,9 is drawn towards tli phical West, <2f K I/t tin- 



component in that direction In- V. It .v i-. drawn \erticall\ down- 

 wards, ds = dr. Let the vertical intensity = /. 



_ \ l dV 



~ IT 3T 



v 



~ 



7 - dV 



-777' 



Gauss began by assuming that the earth's field is entin 1\ due 

 to inside magnetisation, and that the series converges so rapidly 

 that the first four terms are sufficient to express V. So that 



YD H R 3 H '* 



= Bj 



Bj contains three coiManK H 2 ti\e. H 3 >. ven. H 4 nine, giving 

 twenty-four in all. It mu>t he noted that though these constants 

 depend on the internal distrihution. they tell us nothing as to that 

 distribution, for a given distribution of i-xtcriml potential may be 

 produced by an infinite number of different arrangements of mag- 

 netism. To determine the constants it would be sufficient to 

 eight stations at which X. Y, and Z were known, if these were known 

 exactly. But, of course, there are errors in the values, not only of 

 observation, but also of reduction to the same instant of time. Gauss 

 took twelve stations in -i \cn different latitudes, and computed the 

 values of the twenty-four constants best satisfying the values oi \ . Y . 

 and Z at the stations. He then calculated the values oi \. \ . /, for 

 a larger number of known stations all over the earth where X. ^ . / 

 had been observed. The differences between observation and 



