Page:PoincareDynamiqueJuillet.djvu/44

 or, since t = -r,

so that our 4 invariants (5) become:

and our 4 invariants (7):

In the second of these expressions I wrote r1 instead of r, because r is multiplied by ξ - ξ1 and I neglect the square of ξ.

On the other hand, 's law would us give for these 4 invariants (7)

So if we denote the 2nd and 3rd invariants (7) by A and B, and the 3 first invariants (7) by M, N, P, we will satisfy 's law up to terms of order of the square velocities, by:

This solution is not unique. Indeed, let C be the fourth invariant (5), C - 1 is of the order of the square of ξ, and it is equal to (A - B)².

So we could add to the 2ds members of each of equations (8) a term consisting of C - 1 multiplied by an arbitrary function of A, B, C, and a term of the form of (A - B)² also multiplied by a function of A, B, C.

At first sight, the solution (8) seems the most straightforward, it may nevertheless be adopted and in effect – since M, N, P are functions of X1, Y1, Z1 and $$T_{1}=\sum X_{1}\xi$$ – we can draw from these three equations (8) the values of X1, Y1, Z1, but in some cases these values become imaginary.

To avoid this, we will operate in another way. Let:

This is justified by the analogy with the notation

which appears in the substitution of.

In this case, and because of the condition, -r = t, the invariants (5) become: