Emitting Particles from Particels
The Aux System is used to emit particles from particles. It is useful for creating particle trails, particle traces and organic lines effects etc.
Step-by-step instructions on emitting particles from particles
Start with a new comp (Ctrl/Cmd-N), make it 640*480 at 30 fps. 5 seconds long.
Create a comp-sized solid (Ctrl/Cmd-Y), make sure to click "Make Comp Size" before OK.
Apply Effect>Trapcode>Particular to the solid.
Since we'll emit particles from the particles it easily becomes an overwhelming amount of particles so we'll start by decreasing the emission:
Set Emitter>Particles/sec to 20.
Try a RAM preview. Not too exciting.
Let's activate the Aux System:
Set Aux System>Emit to "From Main Particles"
To make the Aux Particles stronger, set Aux System>Opacity to 100.
Hit RAM preview again:

OK, now we have particles generated from the particles.
Particle Trace Effect
To get a long trace for each particle, set Aux System>Life to 5. This way the Aux Particles lives long enough to produce a long trail.
And to get a denser trace, set Aux System>Particles/sec to 20.
Try RAM preview:

Organic Lines Effect
So far, the Aux Particles have just been standing still. Let's see what happens if we put some motion to them. Many of the settings for the Aux System are in Options. Let's add some gravity, open Options and in the "Aux System" section, set Gravity to 100, and then click OK:

Notice that the Aux System has its own gravity (in Options). To change gravity for the main particles, use Physics>Gravity instead.
Particle Trail Effect
Now let's roughen up the animation a bit and make it look more like each particle is emitting a heavy trail. We'll do this by giving the Aux Particles some initial, random, velocity. Set Aux System>Velocity to 10. Hit RAM preview:
