Like Pat said, size the bow with the TBIII first and give it 24 hours to cure up. Give it a light once over with coarse sand paper before putting on the rawhide.
I use tacks to stretch my damp rawhide strips on a board as I rub the TB glue into it. I also use a rubber spatula to lay a very thin layer over the bow limb before laying on the rawhide.
Start at the handle. Use string or rubber banding to hold the rawhide down in the middle of the handle and start laying the rawhide down a little at a time using your fingers to smooth out any air bubbles that might be trapped. When you get to the tip, put gentle stretching pressure on the rawhide with one hand and with the othe rubbing the rawhide in long strokes from the handle toward the tips. This action will stretch the rawhide and work out any excess glue and/or air bubbles. When you are satisfied that the rawhide is stretched and bubble free, use a tack or pushpin to hold the rawhide in place. Push the tack in thru the rawhide into the belly of the bow right about where the nocks are. Small hole, no damage to the bow, really.
Now start at the handle of the bow and using your thumbs, strecth the rawhide down over the sides. This squishes out some more excess glue. Have a wet washcloth handy and a bowl of warm water. Wipe up any excess glue. When you have 6 inches of limb done, wrap with Ace bandage and go do the next 6 inches. Getting the glue off will help get the Ace bandage off later, trust me, and don't ask how I know this.
Once the limb has dried for about an hour, take off the Ace bandage. Take a fresh bowl of warm water and a clean washcloth and wipe down the whole rawhide side of the limb. You can use plenty of water at this point because the PVA glue that makes up TB II and III will not re-wet at this point. But you are plumping up the rawhide again, you want it good and wet so that it swells up and gets rid of the fabric pattern of the wrapped Ace bandage. Wipe down with a dry paper towel and let it cure overnite. Repeat on second limb next day.
I've been meaning to take photographs of this process and do an article for PA because I have gotten some really nice comments on my rawhide backings.