Police are investigating the deaths of the animals, which happened in fields near Ravensworth Castle, close to Richmond.
Officers have branded their deaths a "terrible and cruel crime".
They were killed some time between Friday (June 22) and Saturday (June 23).
A spokesman for North Yorkshire Police said they were shot with an arrow fired from a recurve, or compound-type bow.
PC Mark Wood, Richmond rural beat manager and wildlife crime officer at the force, said: “This is a terrible and cruel crime, and I would urge anyone with any information about it to contact me.”
Anyone with information is asked call police on 101, quoting reference number 12180112952.