152 Dr. C. V. Burton on the Thermally 
This corresponds to absolute zero of temperature at the 
poles ; we are more concerned, however, with the diurnal 
changes, which probably furnish a better representation of 
actual conditions than do the mean temperatures for different 
latitudes deduced from our assumptions. From (18) and 
p-p = A sin 6 J" sin (It—u) J. , when 0< /,f -&> O; * 
d H A sin 
K20) 
, when 7r</^— &)<2tt I 
Also 
~=^-w = Ah sin 6 cos (kt — ©), when < kt — co < tt ; 
^2 H > . (21) 
^=0, when 7r</^— ft)<27r j 
It is now proposed to examine the disturbances of pressure 
arising from the temperature-disturbances (21) imagined as 
affecting a spherical sheet of air whose mean temperature is 
uniform. 
Expanding d 2 H/c^- in a Fourier series we find 
™\2TT 1 
-^ =\Ak sin 6 cos (let — ay) — — Ak sin 6 sin 2 (kt — co) 
+ terms of higher frequency. (22) 
The right hand of this equation must now be expanded in 
such a form that 0, cd appear only in a series of surface har- 
monics. It can be shown that (22) is equivalent to 
^2 = ±Ak sin cos (kt -co) -Akl ^sin 2 0sin 2 (kt—e>) 
+ ^ sin 2 6 (7 cos 2 6- 1) sin 2 (kt-&) 
6435 -^ 
+ 3543Q4 sin2 e (33 cos 4 0-18 cos 2 0+1) sin 2 (kt-co)+... I 
+ terms of higher frequency ; (23) 
in which the first term, considered as a function of 0, <o is a 
surface harmonic of degree 1, and the terms given within the 
bracket of degrees 2, 4, 6 respectively. For convenience let 
