Co-ordinate conditions arising from the requirement of yielding a metric tensor which is best approximation to a Minkowski metric tensor are obtained and are applied to several spaces. Another set of co-ordinate conditions are derived by considering the co-ordinates to be comoving co-ordinates of a fictitious fluid and finding those co-ordinate systems having minimum total intrinsic rotation. It is found that none of the commonly-used co-ordinate conditions fall into either of the two categories considered. A variational principle is stated which leads to the harmonic co-ordinate conditions, but the associated integral seems to have no fundamental physical or geometrical significance.