Thank you, Dave and Gully!!!
Those suggestions allowed me to break through my conceptual barrier. I followed your instructions and created a tube but my ends were faces, not hemispheres. I drew a half circle arc across the diameter of one end then selected the tube face and clicked on the Follow Me Tool. To my surprise, I lost the tube but got 2 hemispheres, one at each end where the tube had been. I then bisected the hemisphere giving a quarter of a circle. I deleted the line across the diameter and, lo and behold, there was a groove end as pretty as you please.
So many ways to skin a cat!
Thank you, both, for the kind assistance.