Expansion of Lagrange planetary equations
The first planet's disturbing function can be written in the form [see Equation (10.8)]

(H.1) 
where
, and is
. Thus, because
, the Lagrange planetary
equations, Equations (G.125)–(G.130), applied to the first planet, reduce to
where
The Sun is much more massive than any planet in the solar system.
It follows that the parameter is very small compared to unity.
Expansion of Equations (H.2)–(H.7)
to first order in yields
where
,
,
,
and
with
, and
. In the following, for ease of notation,
, , and are
written simply as
, , and , respectively.
According to Table 4.1, the planets in the solar system
all possess orbits whose eccentricities, , and
inclinations, (in radians), are small compared to unity, but large compared to
the ratio of any planetary mass to that of the Sun. It follows that

(H.19) 
which is our fundamental ordering of small quantities.
Assuming that
, we can perform a
secondary expansion in the small parameter . It turns out that
when the normalized disturbing function, , is expanded to second order in
it takes the general form (see Section H.3)

(H.20) 
where
is
. If we
expand the righthand sides of Equations (H.13)–(H.18) to
first order in then we obtain
Note that , , , are
, whereas and
are
.
By analogy, writing the second planet's disturbing function as [see Equation (10.9]

(H.27) 
where
, and is
, and assuming that takes the form

(H.28) 
where
is
, the Lagrange planetary equations, applied to the second planet, yield
where

(H.35) 