pigmented colorants: dependence on media and time - Cornell ...

∆i − =(K + S) idx
202
∆j − =(K + S) jdx (A.1)
On the other hand, the gains in each flux come from scattering alone. Assuming
a single scattering event in the layer dx, the gains are:
∆i + = Sj dx
∆j + = Si dx (A.2)
Then the total loss in each direction is the loss minus the gain. Note that the
upward-moving quantity is negated so that we can measure both changes in the
same coordinate system.
di =∆i − − ∆i +
=(K + S) idx− Sj dx
dj = − ∆j − − ∆j +
=(K + S) jdx− Si dx (A.3)
Letting the constant a = 1+ K
yields the two differential equations:
S
di
Sdx
− dj
Sdx
= ai − j
= aj − i (A.4)
Adding these two equations together and rearranging the terms leads to: