Six new equations can be derived from Equations (G.24)–(G.29)
by multiplying them successively by
,
,
,
,
, and
, and then
summing the resulting equations. The righthand sides
of the new equations are

(G.30) 
The new equations can be written in a more compact form via
the introduction of Lagrange brackets, which are defined as

(G.31) 
where
,
, and
.
Thus, the new equations become

(G.32) 
for .
Note, incidentally, that
Let

(G.35) 
where and are any two orbital elements. It follows that

(G.36) 
or

(G.37) 
However, in the preceding expression, and stand for
coordinates and velocities of Keplerian orbits calculated with
treated as constants. Thus, we can write
and
, giving

(G.38) 
because

(G.39) 
where
.
Expression (G.38) reduces to

(G.40) 
where
. Hence, we conclude that
Lagrange brackets are functions of the osculating orbital
elements,
, but are not explicit functions of .
It follows that we can evaluate these brackets at any convenient point in the orbit.