  I consider the equations of motion which follow from d'Alembert's principle
for a general mechanical system in a space of N dimensions, constrained by a
non-holonomic constraint which is linear and homogeneous in the generalised
velocities. The variational equations of motion which follow for the same
system by assuming the validity of a specific variational action principle, in
which the non-holonomic constraint is implemented by means of the
multiplication rule in the calculus of variations are also considered. It is
shown that these two types of equations of motion are not compatible in a space
of dimension N greater than or equal to 3, if the constraint is genuinely
non-holonomic. This means that these two types of equations of motion do not
have coinciding general solutions.
