\magnification=\magstep2


$$
\sigma \quad = \quad \sqrt{  { \sum_{i=0}^{N} {(v_i - \bar{v})}^2 } \over { N }}
$$

$$
({\rm luminosity})^A \thinspace ({\rm size})^B 
    \thinspace ({\rm velocity\ dispersion})^C \quad = \quad {\rm constant}
$$

$$
2*({\rm kinetic\ energy }) \thinspace + ({\rm potential\ energy}) \quad = \quad  0 
$$

$$
2 K + U \quad = \quad 0
$$

$$
K \quad = \quad -{{1}\over{2}} U
$$

$$
U \quad = \quad -{{G M m} \over {R}}
$$

$$
K \quad = \quad {{1}\over{2}} m v^2
$$

$$
v^2 \quad = \quad {{G M}\over{R}}
$$

$$
{\rm if\ \ } M \propto L \thinspace  \quad \quad 
     \rightarrow \quad 
     v^2 \quad \propto \quad L 
$$

$$
v^2 R \quad = \quad 2 G M
$$

$$
\sigma^2 \quad = \quad K \thinspace ({{M}\over{R}})
$$

$$
\sigma^2 \quad \propto \quad {{L}\over{R}}
$$

$$
L \quad = \quad \mu \thinspace (\pi R^2) \quad \rightarrow 
       \quad R \quad \propto \quad L^{1/2}
$$

$$
L \quad \propto \quad \sigma^4 
$$

$$
({\rm luminosity})^A \quad = \quad 
         {   { {\rm constant} } \over 
     { \thinspace ({\rm size})^B \thinspace ({\rm velocity\ dispersion})^C } }
$$




\bye 
