\magnification=\magstep1

$$
d \quad = \quad \ell \sqrt{N}
$$

$$
\tau \quad = \quad \rho \kappa s
$$

$$
{{dI}\over{I}} \quad = \quad - \rho \kappa \thinspace ds
               \quad = \quad - d \tau
$$

$$
I(\tau) \quad = \quad I_0 \thinspace e^{- \tau}
$$

$$
dI \quad = \quad - \rho \kappa \thinspace ds \ I
$$


$$
-{{1}\over{\kappa \rho}} \thinspace {{dI}\over{ds}} 
    \quad = \quad
    I
$$

$$
dI \quad = \quad - \rho \kappa \thinspace ds \ I  \ + \  \rho j \thinspace ds 
$$


$$
{{1}\over{\kappa \rho}} \thinspace {{dI}\over{ds}} 
    \quad = \quad
    {{j}\over{\kappa}} \ - \ I \quad = \quad S \ - \ I
$$

$$
I(s) \quad = \quad 
 \int_{s=0}^{s=s_f} {{1}\over{\kappa \rho}} \thinspace {{dI}\over{ds}} 
    \quad = \quad
 \int
    {{j}\over{\kappa}} \ - \ I \quad = \quad S \ - \ I
$$

$$
I(s) \quad = \quad I_0 \thinspace e^{ - \rho \kappa s} \ + \  
                S \thinspace (1 \ - \ e^{ - \rho \kappa s})
$$


$$
{{I(\theta)}\over{I(\theta = 0)}} \quad = \quad 
     a \ + \ b \thinspace \cos(\theta) 
$$




\bye
