46
5 The Wave Equation and the Third Axiom
We now turn to the accepted conclusion of the equivalence of Eq. (53) with Eq. (55)
and will prove this using the definitions (54) and (56). In doing this, we accept the
validity of Eq. (55). The complete area of space occupied by masses is divided up
into the areas G K , k = 1, 2, . . . , N ; see Eq. (16). Thus,
d
dt
m a
d
dt
x a
=
n
b =a
f ba + f a
and without loss of generality x a ∈ G 1 . Addition of all ‘particles’ in the area G 1
results in
a∈G 1
d
dt
m a
d
dt
x a
=
n
b =a
a∈G 1
f ba +
a∈G 1
f a ,
so including the definitions (54) and (56)
d
dt
M 1
d
dt
X 1
=
⎛
⎝
(b =a)∈G 1
+
b∈G 2
+ · · · +
b∈G N
⎞
⎠
a∈G 1
f ba + F 1
=
b∈G 1
a∈G 1 (a =b)
f ba +
b∈G 2
a∈G 1
f ba + · · · +
b∈G N
a∈G 1
f ba + F 1
and once again using the definitions (54) and (56),
d
dt
M 1
d
dt
X 1
= F 11 + F 21 + · · · + F N 1 + F 1 .
Here is
F 11 =
b∈G 1
a∈G 1
f ba = 0
because of the Third Axiom for f ab according to (55). Because G 1 was an arbitrary
area, we receive the Newtonian equation (53) for the arbitrarily considered mass M A
d
dt
M A
d
dt
X A
=
N
B =A
F B A + F A ,
and once again (under observation of the reaction axiom for forces according to (55),
f ab = −f ba ) the reaction axiom is also valid without loss of generality for the forces
F AB , for example for F 21 ,