52 
DR. S. CHAPMAN ON THE SOLAR AND LUNAR 
Hence it appears that for all values of n, 
(57) 
and on equating the factors of corresponding periodic terms on the two sides of the 
equation (56), we find that 
(58) 
2er+ 1 
1 <T (cr + 2) (er — T+ l) Q T <r + l + (o- 2 —l) (f + r) Qb-li g s = 2 2 pf^f ( s> ) 
oo co 
"V 
m = 0 n = — r -D 
where 
(59) s' = s + r—n. 
There are an infinity of equations of type (58), one for each positive and negative 
integral value of s. Both sides of (58) may be expressed as the sum of a series of 
spherical harmonics of type Q,/ s+T , where (cf 44, 45) r may take all integral values. 
By equating the factors of corresponding harmonics on the two sides, a doubly infinite 
set of equations is obtained, from which the doubly infinite set of constants pf may 
be determined. 
If the atmospheric conductivity is uniform, f s and g s are zero except when s = 0, so 
that only the central equation of the set (58) appears, and on the right-hand side 
R to " (s') vanishes except when s' = 0, i.e,, when n = r. Also R ((t T (0) = m(m+ 1) Q m T , 
and g 0 = 1. Thus, comparing the two sides of the equation, the only two values of 
m for which pf is not zero are a± 1. Consequently, a term Qj in the velocity 
potential of the atmospheric oscillation produces harmonics of types Qh+i ctncl Qb-i, 
and no others , in the electric current function Tv when the conductivity is uniform. 
If also a- = t, as in the atmospheric oscillations Q/ and Q 2 2 , Qb_j is zero and only 
the single harmonic Q T T+1 will appeal’ in 3ft. In this case the value of p\ + i is found 
to be 
(60) t/(t + l) (2r +1). 
If the atmospheric conductivity, or pe, is of the form K(l+a 1 cos w), s in (58) can 
take five values ( — 2 to +2), while s' on the right can take three values ( — 1 to + 1 ). 
Hence n may range from t —3 to r + 3, by (59), while m may range from <r — 3 to <x + 3. 
Only job+i andpb-i contain a w as before, and they also contain only even powers of 
a x ; p T (r± i, p a ±\ ±l contain a x to the first and odd powers, and so on. The coefficients 
p m n have been worked out by Schuster, for this law of conductivity, to the fourth 
power of a x , for the two harmonics Qfi and Q 2 2 in the velocity potential. The more 
important of these coefficients, which will be required in the subsequent discussion, 
may be obtained from Table O by writing a 2 — 0. 
For more complicated forms of pe the calculation of the values of p m n , although 
straightforward and simple in principle, becomes increasingly laborious. In my 
