Scope mounting for an old Harrington & Richardson "Plainsman"

Recently acquired can old H&R .22 "Plainsman" that came with the original metal sights and I wanted to put a scope on the rifle. It comes with what looks like dovetail rails, but with a curved not flat top. The "dovetail" also looks quite shallow and the pair of 3/8 dovetail rings that I have do not appear to seat very tightly in the grooves. Has anyone had experience with older rifles like this and can you provide some advice on the type of mount that I should be looking for? Will post some photos later to show.

Here are a couple of photos of the rail and a standard 3/8 dovetail scope ring.Attachment 161965Attachment 161966
As you can see the dovetail on the receiver is not very deep. I suspect it may need a more specialised mount.

Gidday - you just need a ring style with "sharp" machined feet, the cast ones you have are not suited, google "Hilver 22" to get an idea of what you need

I had an old Winchester single shot scoped for our boys.
Gunsmith ( machinist) machined dovetails and used rings as suggested by tentman.

mates 22magnum had mounts that would suit that....sort of thin sheet metal that folded over top of scope with ends going down and bent inwards to enguage rails
they were common enough with skinny scopes 30ish years back...they had bolts through under scope to squeaze sides together.

Thanks for all the suggestions. I’ll take it along to a local gunsmith and see what can be done!

Thanks Tentman. I followed your suggestion and found the site. Not sure quite what I was looking for as there seemed to be a plethora of mounts on the site and it wasn't clear (to me) exactly what would suit. Was there anything in particular that you thought I should look at?

Quick update on solution that I found. I got a pair of 3/8 dovetail mounts by BKL through a supplier down in Morrinsville - Magair. All installed and very happy. Now I have stripped the whole gun down to re-blue and clean up😀