524 
atmosphere is isothermal and of constant composition, 
and that the density variations accompanying the tidal 
motion occur isothermally. 
Let p, p, T’ denote the pressure, density, and Abeaiten 
temperature of the air at height Z above the ground, 
and let the suffix zero added to these and other symbols 
distinguish the values for Z = 0, that is, the ground 
values. The equation of static equilibrium of the air is 
d(n p)/dZ = —1/H, 
where H = p/gp and the logarithm is to the natural 
base e. Clearly H is the height of the column of air of 
uniform density p required to give the pressure p (if 
the variation of g with height is ignored, though g is 
reduced by 3 per cent at 100 km). Hence H) is called 
the height of the homogeneous atmosphere; H is called 
the scale height of the atmosphere at the height Z, and 
is in general a function of Z. 
In a perfect gas 
= knT = kpT/m = RT/M, 
where 
k denotes Boltzmann’s constant (1.380 X 10—* ergs 
per degree C), 
n the number of molecules per cubic centimetre, 
m the mean molecular mass, 
R = KN, where N is Loschmidt’s (less appropriately, 
Avogadro’s) number (6.023 X 1023) = 8.313 X 107. 
M is the mean (chemical) molecular weight of the 
air (about 29); 17 = Nm. Hence 
H = kT/mg = RT/Mg. 
In the atmosphere considered by Laplace, H had the 
constant value Ho, because 7’ and m are the same at all 
heights. Hence 
p/P. = p/p = & 7". 
Laplace showed that the tidal oscillations in such an 
atmosphere could be inferred from those for a liquid 
ocean of uniform depth Ho, which is therefore in this 
case called the (tidally) equivalent depth of the atmos- 
phere. His theory, however, is not applicable to the 
actual atmosphere, in which 7 is not the same at all 
heights. Moreover, his condition that thedensity changes 
occur isothermally was doubtful. Newton had made 
the same assumption in his theory of sound waves, lead- 
ing to an erroneous value for the speed ¢ of sound, 
namely, c? = p/p = gH. Laplace had corrected this to c? 
= ygH, where y denotes the ratio of the specific heat at 
constant pressure to that at constant volume, to allow 
for the adiabatic character of such rapid density chan- 
ges. It might well seem that for such slow oscillations 
as S. and L, the density changes would be isothermal, 
but calculation [34; 36; 111, p. 36] shows that they too 
must be adiabatic, because of the long wave length; and 
this is confirmed by the determination of the lunar tidal 
variation of air temperature at Batavia (p. 519). 
It was to test his tidal theory that Laplace attempted, 
without success, to compute the lunar atmospheric 
tide Lz for Paris. He realised that the magnitude of Lp 
was very small, and as S2 is so much larger he concluded 
DYNAMICS OF THE ATMOSPHERE 
that S2 is due mainly to the sun’s thermal action, the 
tidal contribution being insignificant. He seems also 
to have thought that there was little hope of construct- 
ing a theory of the oscillations produced in the atmos- 
phere by its daily heating and cooling. 
In 1882 Kelvin gave his attention to the subject, and 
in a presidential address to the Royal Society of Edin- 
burgh [102] he quoted a table showing the 24-, 12-, and 
8-hour periodic components (S;, S2, and 83) of the solar 
daily barometric variation for thirty different stations. 
He pointed out that the 12-hour component exceeds 
the 24-hour component, especially in the higher lati- 
tudes, although in the daily variation of air temperature 
the 24-hour component is the larger. He suggested that 
the cause lies in the existence of an atmospheric free 
oscillation, of period nearer to 12 than to 24 hours, so 
that the 12-hourly thermal influence is magnified by 
resonance. His words were as follows: 
The cause of the semi-diurnal variation of barometric 
pressure cannot be the gravitational tide-generating influence 
of the sun, because if it were there would be a much larger 
lunar influence of the same kind, while in reality the lunar 
barometric tide is insensible, or nearly so. It seems, there- 
fore, certain that the semi-diurnal variation of the barometer 
is due to temperature. Now, the diurnal term, in the harmonic 
analysis of the variation of temperature, is undoubtedly much 
larger in all, or nearly all, places than the semi-diwrnal. It is 
then very remarkable that the semi-diurnal term of the baro- 
metric effect of the variation of temperature should be greater, 
and so much greater as it is, than the diurnal. The explana- 
tion probably is to be found by considering the oscillations 
of the atmosphere, as a whole, in the light of the very formulas 
which Laplace gave in his Mécanique céleste for the ocean, 
and which he showed to be also applicable to the atmosphere. 
When thermal influence is substituted for gravitational, in 
the tide-generating force reckoned for, and when the modes 
of oscillation corresponding respectively to the diurnal and 
semi-diurnal terms of the thermal influence are investigated, 
it will probably be found that the period of free oscillation 
of the former agrees much less nearly with 24 hours than 
does that of the latter with 12 hours; and that, therefore, 
with comparatively small magnitude of the tide-generating 
force, the resulting tide is greater in the semi-diurnal term 
than in the diurnal. 
The Development of the Resonance Theory. The 
periods of free atmospheric oscillation, on a plane or 
spherical earth, were investigated by Lord Rayleigh 
[90], in 1890, with conclusions somewhat in favour of 
the resonance theory; but they could not be relied 
upon, partly because he did not take the earth’s rota- 
tion into account. 
Margules [75], with the explicit object of testing the 
resonance theory, investigated in much detail the free 
and forced oscillations of the atmosphere on the basis of 
Laplace’s theory, and concluded that Kelvin’s expecta- 
tion was closely fulfilled. But Margules’ conclusions 
cannot be relied upon either, partly because he did not 
take account of the true distribution of temperature 
with height in the atmosphere, which, indeed, at the 
time he wrote, was quite inadequately known. He also 
attempted to calculate the forced oscillations due to the 
daily variation of air temperature, on various hypotheses 
