The distressed ewe went into labour after being chased by a greyhound-type dog wearing a red jacket in the Reeth Road area, west of the town on Saturday.
The dog’s owner left the scene.
A vet attended, but both delivered lambs died.
North Yorkshire Police’s Rural Taskforce has called for dog owners to behave responsibly and always keep their dogs on leads near livestock, following the incident.
Ewes are at particular risk during the lambing season which runs from around January to April.
Livestock worrying and livestock attacks are a growing problem in rural communities, North Yorkshire Police reports.
North Yorkshire is one of five police forces across the country that takes part in a working group that is trying to tackle the problem.
Speaking earlier this year at the launch of a report highlighting the scale of dog attacks on livestock, North Yorkshire Police’s Chief Constable Dave Jones said: “We need dog owners to take responsibility for their animals - not just by putting their dogs on a lead when out walking, but by preventing them from escaping from home and causing damage to livestock.
“We need livestock owners to report incidents so that we can gather intelligence and launch investigations. Above all, we need the powers to tackle this problem effectively and an overhaul of the outdated and sometimes ineffective rules surrounding livestock worrying.”
Anyone with information about the Richmond incident is asked to call North Yorkshire Police on 101, quoting reference number NYP-31032018-0284.