Rachel had no middle name. Died 18 May 1899 Failford,Tarbolton. Children all surname gibson William Davidson b1/3/1851,Alexander b10/2/1853,George James b abt 1856,Janet Bell b2/4/1858,James b25/6/1860, Margaret b30/12/1862, Thomas Imrie b17/9/1865,Mary Ann b21/9/1867, Agnes Jane b8/4/1870, Rachel H b1874 (believe the H to be Helen but will try and confirm soonest).
William's middle name is interesting after you mentioning the connection that Edward Davidson (the visitor)has to the family. James b1860 married Janet daughter of Rachel's sister Mary Ann Imrie 30/12/1887.