along with , the gas law and an energy equation. Having obtained the equilibrium we then set

where the perturbed quantities, denoted by a subscript `1' are smaller than the equilibrium quantities, denoted by a subscript `0'. Note that the equilibrium quantities are independent of time, . Next we substitute these expressions into the MHD equations and neglect any products of small terms. Thus, the mass continuity equation becomes

Hence, the linearised mass continuuity equation is

In a similar manner () - () in the ideal MHD limit (i.e. ) reduce to

and

Therefore, is constant in time. However, if it is zero at then it is zero for all time.

The linearised equations may now be combined into one equation in the
following manner. Take the time derivative of (4.7), the
linearised equation of motion, to get

and use (4.6), (4.8) and (4.9) to eliminate the perturbed density, pressure and magnetic field and get

(4.11) is the linearised equation of motion and forms the basis of the study of MHD waves and MHD instabilities.