50 
DR. S. CHAPMAN ON THE SOLAR AND LUNAR 
and 
a cc 
(46) d r — 2 , +r C r . a l+r . (sin 8 cos 0) 1 , e T = 2 ;+r C r . b l+r . (sin S cos 0)‘, 
I =0 1=0 
80 that J] , g, are power series in sin 0 and cos 
For the present we may consider an atmospheric oscillation of the general harmonic 
type, for which the velocity potential is 
(47) = K/Q/ sin (rt — a). 
The radial magnetic intensity of the earth’s field (measured positive outwards) 
will be denoted by V. The components of electric force, X and Y, measured towards 
the south and east respectively, are given by 
Y# 
r dO ' 
d& _1_ Tift 
r sin 6 dX rpe dO 
(48) 
X - 
V d!<£> 
r sin 0 d\ 
If we express X and Y in the form 
(49) 
X = 
1 dS> 1 dM 
r dO rpe sin 0 d\ 
the function will be the current function of the electric currents produced by X 
and Y.* 
In order to obtain 3ft from (49), Schuster first determined by eliminating 3ft, 
and afterwards determined 3ft by the use of £?. In my own earlier treatment of the 
problem I sought to avoid the calculation of <S by using the resistivity in place of the 
conductivity, so that 1 If* was the function which was expressed in the form (40). 
But it is better to keep p as the fundamental function, and this is easily effected, 
and 3ft directly determined, by the use of the following method. 
On eliminating £5> and multiplying both sides of the resulting equations by 
(pe/sin 6 ) 2 —this being the step which yields the improvement of method—we obtain 
the following result:— 
(50) 
(pe)*r 
sin 6 
-*h Ysin 
P e fd^ 
sin 2 0 1 d \ 2 
+ sin 0 —— sin 0 
dO 
m,) 
doi 
1 >73ft dpe 
sin 2 0 d\ dX 
djft dpe ) 
dO dO J 
* Here we neglect the effect of self-induction for the present (see, however, § 26). We define HI by 
the property that the flow across an element of length ds, measured from left to right, is 
