DIURNAL VARIATIONS OF TERRESTRIAL MAGNETISM. 
51 
We may suppose that the solution & is expressible in the form of a series of 
spherical harmonic functions, thus 
(51) E = K/CK 2 2 2^m n Q m n sin (nt-oi n ). 
m = 0 n = — 00 
In this equation Q m -, ‘, when n is positive, will be defined as equal to Q, a n . When 
m is numerically less than n, Q m n is zero. 
When tlie above values of E and pe are substituted on the one side, and of X and 
Y on the other, (50) becomes 
(52) 
X g, 
dVdQl 
de do 
— rr (<r + 1) VQ/ > sill (t + St — a) + 
-Q„ T dV 
0 d\ 
sin' 
cos (t 4- st — cc) 
where 
(53) 
and 
(54) 
=-C XX X 
7)1 = 0 11 = — X = — <x 
p m H R m n (s) sin {(n + s)t-a n }, 
B m n (.s) = {m(m+ 1) -ns/ sin 2 6 } Q„/*/,+ /* 
/ dQ„ 
de 
ff - 'Mi 
,s ~~ de ‘ 
By equating corresponding periodic functions of t and e on the two sides of (52), 
we may determine the values of p m n and a„. 
For the time being we will now limit the problem to the determination of the part 
of E which depends solely on local time, i.e., to the case in which a„ is independent 
of X, so that on the left-hand side of (52) the term dY/dX will be omitted. This is 
the same thing as omitting from V the part which depends on X. If we regard the 
earth as a uniformly magnetized sphere, with its magnetic axis inclined at an angle 
<p to the geographical axis, we may write 
(55) ’ V = 0 cos 6 + C tan <f> sin 0 cos X l} , 
where C is a constant (approximately equal to — f, having regard to our conventions of 
sign) while X u is the longitude measured from the meridian (68° West of Greenwich) 
which contains the earth’s north magnetic pole. The constant C has, for convenience, 
been already introduced in (51). 
Neglecting, therefore, for the present, the second term in V, (45) becomes 
(56) 2 2 2 p m n R m "(a) sin {(w+s) t-a n } 
«/ = 0 n = - ® S = — co 
{o-(o-H-2)(«r- T + 1 )Q T <r+ i + (ff 2 - 1 )(cr+ T )Q r , r . 1 } 2 g t sin {{s + r)-ct}. 
H 2 
