450 
summation with respect to the index 7; we then have 
the normal differential system 
a 
dx? J 
The general integral of the nonhomogeneous linear 
system (86) is composed additively of a particular 
integral of this system and the general integral of the 
homogeneous differential system 
L(Ac*) = —L™ (0 (86) 
Lee) =0, @=i1,2,8) (87) 
in which 
= “t+ (yi eee tacos Nye dis 
+ yo" + auf?) © +0 G,7 = 1, 2, 3) 
with f* = 2w;;, the index series (k, 7, 7) bemg derived 
from the series (1, 2, 3) by a rotating permutation. 
The f* terms are nothing but the Coriolis parameters 
associated with the coordinate system 122°. Let us 
note that the elongations Az? of an inertial oscillation 
(Ail = 0) of the air particle under consideration satisfy 
the differential system (87). The characteristic equation 
ye — By +e —o =0 (88) 
of this system is obtained in the same way that the 
characteristic equation (49) is obtained from the differ- 
ential system (48). In equation (88) we have set 
B= You + Yul f € = 507 + off. 
Since equation (88) is cubic with respect to 5, it 
follows that in the atmosphere a particle set unto any 
type of motion can undergo, in general, three inertial 
oscillations of different periods (7). 
Let us now attempt to determine the three roots, 
y,, of equation (88). In order to do this let us sid 
stitute for the coordinate system a2 the system 
ayz whose z-axis coincides with the zenith direction at 
the point P; let us then adopt the approximation 
allowed (p. 438) and, finally, let us introduce the 
geostrophic flow u(uz, uy , 0) associated with the arbi- 
trary motion (83); then, by definition, we have 
and 
L260 | jas Sums ” Sas 
5 = Foy? CO a en uz =0. (89) 
We will admit moreover that 
6 a ig =O. (90) 
fn (89) and (90) we have set =a2,y =2,2=2, 
and f = f = 2w sing and f = f' & 0. To the approxi- 
mation to which we have just consented, the equation 
of motion (83, 7 = 3) in the vertical, z = 2’, has been 
replaced by the equation of hydrostatic balance (90). 
This simplification is amply justified in the atmosphere. 
Let us note that in the system xyz, it is possible to 
consider the second derivatives of the geopotential ® as 
DYNAMICS OF THE ATMOSPHERE 
negligible. Substituting (89) and (90) into the o;; ex- 
pression, we obtain 
0 fu 0 fu g 008 
Ora =i) = 22) eee, 
f Ae) fos, (= 6 0% 
he O (Uy 0 (uz g 00 
r=liouli= io2(¥) -o2(%) 2% 
F l= jo 2 i. g 60 
dz \® dz \O 0 dz 
with the symmetry relations 
0 [Ux 0 [wy\ _ oO [Uz g 00 | 
Bie ee 2-0. foo (us wean 7! 
and 
(91) 
in which we have set, (35), 
5. 8 _ SS) a Orcs aS a 
dx Ox 00/dz) dz’ sy dy d0/dz ) dz 
From the relation (91) we are able to deduce the exist- 
ence of a stream function for geostrophic flow within 
the isentropic surfaces (6 = const). 
It is easy to show that, (36), 
OUz OUy 
—, 2.2 _ Uz Oly 
Cage ri (% bys Oy = 
See (Sy 4 = =| 10 to 10 sec™® 
and 
2 (u;/O, w/8) ~ ,2,2 
Ca 
eS i Ei at (a = ll & 10 * see’, 
in which 6u,/6% — 6u,z/dy represents the vertical com- 
ponent of vorticity determined in the isentropic sur- 
faces (© = const) by the geostrophic current 
u(uz, u,, 0). Finally, we obtain 
ae a i Uy = i us) =~ pe Ry Ome sec, 
€ = veya fe = 10°” sec * 
with 
2 2 
Vs + Vi 
Ill 
B 
in which 
— Oty _ Uz ay —8 —2 
nas|r+(% me) | 2 10 sec. 
The expressions y? and vg defined above generalize the 
corresponding expressions (37) and (40), respectively, 
which are valid in the case of a particle in geostrophic 
motion. 
