Hi nobbyv,
Welcome to the forum.
The small bowl is the condensation bowl. It is to prevent "black rain" from falling down on your smoked foods. Black rain is caused by condensation forming around the vent inside the smoker. When it builds up it will drip on the food. There is a wire rack that attaches to the top of the side rails. That wire rack has a circular slot. The bowl goes in that slot.
The oval shaped object is the smoke diffuser it sets on top of the porcelain drip tray. The end with no slots goes over the large hole opening. When you look at the drip tray you will see a about three raised knobs, near the large opening. The end with the slots should be place over those knobs.
If you click on the below link to the manual and scroll to page ten. The item labeled #3 is the smoke diffuser; the item labeled #7 is the condensation bowl.
Bradley P10 ManualHappy Smoking
