166 
MR. ARTHUR SCHUSTER ON THE 
and may be divided into two portions, the first of which is derivable from a potential 
S = 
In the steady state this part is balanced by a static distribution of electricity 
revolving round the earth and causing a variation in the electrostatic potential which 
is found to be too weak to affect our instruments. The second portion of the electric 
force produces electric currents ; these, neglecting electric inertia—which will be 
considered later—have pit as current function, where p is the conductivity of the 
medium. 
The comparison of (2) and (3) shows that 
n. n + 1 . It 
and by means of well known reductions Ft may be expressed in the normal form 
(2n+ 1) n . n +1 . Ft = C [n 2 (n— cr+ 1) \jj n+1 + (n + l) 2 (n + <x) . . (4). 
Here if/ n+1 and are the two spherical harmonics of degree n and type cr which 
have the same numerical factor as the current function \jj n . 
I shall confine myself to the two principal portions of the diurnal variation of 
barometric pressure which are associated with the velocity potentials 
i/n 1 = Aj sin 9 sin {(\+£) — a x } and i// 2 2 = 3A 2 sin 2 0 sin {2 (\+t) — a 2 j. 
The corresponding electric current functions are seen by (4) to be 
/TV = ^pCA^ff and pit 2 = ^-pCA,^ 2 .(5). 
It is shown by Clerk Maxwell (‘ Electricity and Magnetism,' vol. II., p. 281) 
that the magnetic forces accompanying the currents in spherical sheets which are 
derivable from a current function having the form of a surface harmonic are obtained 
from a magnetic potential which is equal to the same harmonic multiplied by a factor 
which inside the spherical shell is — 47 t (n+ 1) r"/(2n+1) a n . The thickness of the 
atmosphere being negligible compared with the radius of the earth, we may put 
r = «, and obtain thus, for the magnetic potential H due to the induced electric 
currents, 
— F = [| A 1 \fj 2 1 sin {(X + #)-<*!}+ -^A 2 xp : > 2 sin {2 (\ + £) —« 3 }] 7rpcC . . (6). 
The quantity e represents the thickness of the shell of the conducting layer, and is 
introduced because the current functions used above yield current densities, while 
Maxwell’s result applies to functions which lead directly to currents. Our 
equation (6) represents the potential of the diurnal variation of terrestrial magnetism 
calculated from an atmospheric oscillation according to our theory, and agrees in form 
