13th Texas Cavalry Burnett's Regiment
I have not been able to determine who this soldier may be. He may be a William or a Windsor but they are already accounted for in other units. He is listed on the muster rolls for this company and regiment. The census rolls do not list anyone else who's name begins with the letter "W", so he may be using his middle name.
Here is a link to a short history of the 13th Texas Cavalry.