Page:PoincareDynamiqueJuillet.djvu/42

 Q, then the invariants are functions of mutual distances of five points

and among these functions we must retain only those that are homogeneous of degree 0, on the one hand in relation to

(variables that can then be replaced by X1, Y1, Z1, T1, ξ, η, ζ, 1), on the other hand in relation to

(variables that can be replaced later by ξ1, η1, ζ1, 1).

Thus we find in addition to the four invariants (5), four new distinct invariants, which are:

The last invariant is always zero, according to the definition of T1.

This granted, what are the requirements?

1° The left-hand side of relation (1), which defines the velocity of propagation must be a function of the four invariants (5)

One can obviously make a lot of hypotheses, we only look at two:

A) It may be

where t = ±r, and since t must be negative, t = -r. This means that the propagation velocity is equal to that of light. At first it seems that this hypothesis should be rejected without consideration. has indeed shown that this propagation is either instantaneous, or much faster than light. But had considered the hypothesis of finite speed of propagation, ceteris non mutatis; here, however, this hypothesis is complicated by many others, and it may happen that there is a more or less perfect compensation, as the applications of the  transformation gave us already so many examples.

B) It may be

The propagation velocity is much faster than that of light, but in some cases t may be negative, which, as we have said, seems hardly acceptable. We will add this to hypothesis (A).

2° The four invariants (7) must be functions of the invariants (5).

3° When the two bodies are in absolute rest, X1, Y1, Z1 must have the value