-0-$4h-. vets 
2 1 
=p -hvwt hy. +9¥p 
— pp + 
Here the singly underlined terms vanish because they 
form together the undisturbed equation. The doubly 
underlined terms can be omitted because they are 
small of higher order. In a similar manner the other 
perturbation equations can be obtained so that the 
complete system becomes 
0 
at WWW + VW + 20 X v 
Se 
See erg NES 
Oon 
at + div (gv) + div (Qx) = 0, (42) 
og 
oe + V-VG + ¥-VQ 
(43) 
ap 
= ¥ (e + V-Vp + v-vP) = 0, 
F(x, t) + fo, ) = 0, (44) 
at V-Vf + v-VF = 0, 
(45) 
of / / 
at WMS + v-VF = 0, 
P(r, t) — P(t, ) + pl, t) — pit, ) = 0, (46) 
0 
a, (p — p') + V-V(p — p’) 
+ v-V(P — P’) =0, 
miMh (47) 
0 / y/ / 
=) Se SD = 7) 
+ v'-V(P — P’) = 0. 
In equations (44) and (46) the undisturbed terms alone 
are no longer zero in spite of (87) and (89), because 
these equations hold only if the values at the un- 
disturbed boundary are substituted for r, while in (44) 
and (46) the values at the disturbed boundary have to 
be inserted into fF, P, and P’. On the other hand, it 
is permissible to replace in the perturbation quantities 
contained in the equations (44)—(47) the values at the 
disturbed position of the boundary by the values at 
the undisturbed position because this substitution pro- 
duces only an error of higher order. Consider, for 
instance, the perturbation pressure p. For the moment 
let r be the undisturbed position of the boundary, 
DYNAMICS OF THE ATMOSPHERE 
rt + Ar the disturbed position where Ar must be small 
of the same order as the other perturbation quantities. 
Then 
p(t + Ar, t) = p(t, t) + (Ar)-Vp(, 4), 
where the second term on the right-hand side is smaller 
than the preceding by one order of magnitude. 
V. Bjerknes has pointed out that the perturbation 
equations (41)—(47) can be obtained from the equa- 
tions of the undisturbed motion (34)—(40) by forming 
the variation of these equations and substituting the 
perturbation quantities wherever variations of the un- 
disturbed quantities appear. 
The Lagrangian System of Perturbation Equations. 
In the formulation of the perturbation equations in 
the Lagrangian form it will be assumed that at the 
time ¢ = ft the position and the parameters of state 
for the undisturbed particle are the same as in the 
undisturbed case. This assumption implies that the 
perturbation is induced suddenly at the time 4. It 
has the advantage that the three parameters a, b, 
and c, which identify each particle, are not only the 
components of the radius vector Ro for the undisturbed 
motion at the time # , but also for the disturbed mo- 
tion since f , the deviation of the disturbed from the 
undisturbed position at f, vanishes. The equations 
for the undisturbed motion in the Lagrangian system 
are now obtained by substituting for the appropriate 
quantities with asterisks in (22), (23), (14), (30), (32), 
and (33) the corresponding quantities characterizing 
the undisturbed motion which are denoted by capital 
letters. Thus, 
we op + 2Q X lealll = — — V9, (48) 
R R oR 
Oe ee elas) nee 
Q = Qa, b, ¢, P), (50) 
F(a, b, c) = 0, F(a’, b’, c’) = 0, (1) 
F(R, t) = 0) = [F’(R’, 24) = O, (52) 
P(R, 1) = P'(R,t) for R=R’. (58) 
The physical equation (50) has here been written some- 
what differently from (14) by making use of the fact 
that the parameters A, B, and C, --- may vary from 
particle to particle and are therefore functions of a, 
b, and ec. 
The perturbation equations are now obtained by 
making in these same six previously mentioned equa- 
tions the substitutions indicated by (1), (8), and (4) 
and taking into account also that the surface of dis- 
continuity will in general change its position because 
of the perturbation motion. The resulting equations 
can be simplified by the same two considerations as 
stated on page 407 for the Hulerian equations. It is 
to be noted that the force potential 6 depends on the 
position in space. Hence, while in the undisturbed mo- 
tion it is given by ®(R), it will be given by ®(R + r) 
for the same particle after the beginning of the perturba- 
