DIURNAL VARIATION OF TERRESTRIAL MAGNETISM. 
165 
2. The velocity potential of a horizontal irrotational motion of the earth’s 
atmosphere considered as an infinitely thin shell is necessarily expressible as a series 
of spherical harmonics from which we may select for consideration the one of degree 
wand type <r, writing it xfj/ sin {cr (X + t) — a} or i ft/ according to convenience. The 
longitude X is measured from any selected meridian towards the east, while t is the 
time of the standard meridian in angular measure. As in the greater part of the 
investigation X and t occur in the combination X + t only, we may frequently omit t 
without detriment to the clearness, noting, however, that if the differentiation with 
respect to t is replaced by a differentiation with respect to X we must apply the factor 
2 tt/N, where N is the number of seconds in the day. 
I consider in the first instance the electric currents which are induced in air moving 
horizontally under the influence of the earth’s vertical magnetic force. Assuming the 
earth to be a uniformly magnetised sphere, its potential may be resolved into the 
zonal harmonic of the first degree and the tesseral harmonic of the first type and 
degree. The angle between the magnetic axis and the geographical axis not being 
great, the zonal harmonic constitutes by far the largest part, and forms the first 
subject of our investigation. As far as this part is concerned, we may put the vertical 
force equal to C cos 9, where 9 is the colatitude and C, measured upwards, has a 
numerical value differing little from — 
The components of electric force, X and Y, measured towards the south and east 
respectively, are 
Xa = C cot 9 
dxjj 
dX’ 
Ya = — C cos 6 
dxp 
die 
(i), 
and these equations may be written in the form 
n . n+ 1. Xa = C- ^ +C . (n .n+l.\]j cos 6 —sin 6^ 
d6dX sin 9dX\ dO 
n.n + l .Ya = C . 
sm 6dX 2 dd 
n . n+ 1. xfj cos 9 —sin 9-^ 
d9)J 
( 2 ), 
where n is the degree of the harmonic. 
The identity between (1) and (2) is obvious as regards the first of the equations, 
and the reduction of the second is obtained with the help of the fundamental equation 
sin 9-^- sin 9 ^ ^ +n.n+ 1. sin 2 9 . x/j = 0, 
ad d9 dX 
\{t being a zonal harmonic of degree n. 
The components of electric force are by (2) reduced to the form 
_ _ c/S c/Pt 
a d9 a sin 9 dX ’ 
Y = - 
d$ 
a sin 9 dX a d9 
(3). 
