186 
MR. ARTHUR SCHUSTER ON THE 
where 6 is the colatitude, and X + t measures the difference in longitude between the 
sun and the place at which p is required. The question is solved if we can determine 
the current function of the electric currents which are generated by the fluid moving 
through the magnetic field. The problem for constant conductivity has been treated 
in the first part of this communication and the interest of a non-uniform conducting 
power is confined to the case that the variability depends on the angular distance 
between the sun and the point considered. If oj be this angle, the effect of the sun’s 
radiation will be proportional to cos oj in the hemisphere subject to the radiation, 
i.e. for values of oj smaller than \-n. If the induction is due to the ionising power of 
the sun’s rays, the rate of recombination of ions has to be considered, but unless this 
rate is of a different order of magnitude from that observed near the surface of the 
earth, the conductivity may be considered to be proportional everywhere to the 
illuminating power. For values of u> intermediate between and n we must, then, 
give zero value to the conductivity. By means of Fourier series we may now express 
the conductivity in a series 
2 
7T 
1 2 
+ A cos &H-cos 2oj + ... 
2 3tt 
(17), 
which satisfies the condition 
p = p' 0 cos (o for 0 < oj < ~ , and p = 0 for ^ 
Confining ourselves to the first two terms and substituting the value of cos oj from (13) 
in terms of the hour-angle of the sun and its declination, we obtain 
2 
—b A sin 8 cos fl + Acos 8 sin 0 cos (A + t ) 
TT 
The conductivity has therefore the assumed form if we put 
2 
p 0 = - p 0 ; Pi = Ap'o sin 8 ; p 2 = \p\ o cos 8. 
77 
Were we to adopt the simpler form and put the conductivity proportional to 1 +cos oj, 
so that it reaches zero value only at midnight, we should have to put 
Pi — Po sin 8 ; p 2 = p 0 cos 8, 
and m every case p can be expressed in terms of a series such as (17), our investiga¬ 
tion by proper adjustment of the constants taking account of the first two terms. 
The term in cos 2oj might be taken into consideration without much difficulty should 
that become necessary. The value of p 0 can provisionally be put equal to unity and 
re-introduced at a later stage. Writing y = px/p 0 and y = p^/p,, we may therefore put 
(18). 
p = L +y' cos 6 + y sin 6 cos (A + 1) 
