Here is the copy I have:
William Brockway’s adaptation of Dr Gaddy's technique
As posted in DoubleGunBBS
1 - Polish the barrels to 400 grit or finer (400 is usually enough). If you have tested the barrels for pattern, using acid, this area will have to be polished bright again, before proceeding.
2 - Apply a diluted solution of Laurel Mountain Forge Browner and Degreaser (see below).
3 - Hang barrels in a warm, humid location until a coating of rust forms (may take 4 hours to overnight, depending on ambient temperature and humidity). I use a steam box, powered by a percolator bottom (for the water vapor), and a light bulb (for the heat).
4 - Dip rusted barrels in a 15% solution of Radio Shack circuit board etchant and water (I keep a capped piece of 3" PVC pipe as a dipping tank). Dip for no longer than 5 seconds (important)! That's 15% RS etchant (a solution of Ferric Chloride and water) and 85% water. I am lucky enough to live where the tap water is almost chemically pure. It may be necessary for people in other parts of the country to use distilled water.
5 - Immediately after dipping, flush barrels clean with a garden hose and lots of water. I do this outside my shop in the yard. It is important to stop the etchant from working any longer than 5 or 10 seconds.
6. - Card (scrub down barrels) with medium steel wool under water. This will make the pattern begin to appear.
7 - Repeat steps 2 through 6 until the barrels are dark enough to suit you. This may take a lot of passes, maybe as many as 8 or 10.
In step 2, dilute the LMF with 2 parts water for the 1st pass only. The next couple of passes can be 5 parts water to 1 part LMF. After that, the remaining passes can be 8 - 10 parts water to 1 part LMF. I no longer use the browner diluted any more than this. The dip in Radio Shack etchant serves to remove the rust from the white striations without affecting the brown striations.
If you prefer a black and white finish, boil the barrels for 5 minutes after step 6, each pass. Dr. Gaddy also used a logwood solution dip on his black-and-white barrels, which I haven't tried, yet. Logwood is a stain used by trappers to darken steel traps.