Page:PoincareDynamiqueJuillet.djvu/34

 that is to say that the equations (10) only differ from equations (5) by the accentuation of the letters, it must be assumed:

l = 1.

Suppose now that we have η = ζ = 0, where ξ = V, $$\frac{d\xi}{dt}=\frac{dV}{dt}$$; the equations (5) take the form:

We can also pose:

$$\frac{dD}{dV}=f(V)=f(\xi),\quad\frac{D}{V}=\varphi(V)=\varphi(\xi).$$

If the equations of motion are not altered by the transformation, we must have:

and therefore:

But we have:

$$\frac{d\xi^{\prime}}{dt^{\prime}}=\frac{d\xi}{dt}\frac{1}{k^{3}\mu^{3}},\quad\frac{d\eta^{\prime}}{dt^{\prime}}=\frac{d\eta}{dt}\frac{1}{k^{2}\mu^{2}},$$

where:

whence, by eliminating l², we find the functional equation:

$$k^{2}\mu^{2}\frac{\varphi\left(\frac{\xi+\epsilon}{1+\xi\epsilon}\right)}{\varphi(\xi)}=\frac{f\left(\frac{\xi+\epsilon}{1+\xi\epsilon}\right)}{f(\xi)},$$