\magnification=\magstep1

$$
M \quad = \quad \rho \thinspace V \quad = \quad \rho 
              ({{4}\over{3}}) \pi R^3
$$

$$
L \quad \propto \quad M \quad \propto \quad  R^3
$$


$$
L^{1/3} \quad \propto \quad R
$$


$$
{{v^2}\over{R}} \quad = \quad {{G M}\over{R^2}} \quad = \quad  
               {{G \thinspace ({{4}\over{3}}) \pi \rho R^3}\over{R^2}}
$$

$$
v \quad \propto \quad R
$$

$$
v \quad \propto \quad L^{1/3}
$$

$$
\Delta (\log{v}) \quad = \quad {{1}\over{3}} \thinspace \Delta (\log{L})
$$




\bye
