  We study the longitudinal spin susceptibility inside a magnetically ordered
phase, which exhibits a superconducting instability leading to a coexistence of
the two ordered phases. Inside the magnetic phase, the superconducting gap
acquires a linear term in a magnetic field applied along the direction of the
magnetic moment. We find that such a linear term generates a jump of the
longitudinal spin susceptibility when the superconducting instability occurs
via a continuous phase transition. This anomaly at the superconducting
instability is a thermodynamic signature of the microscopic coexistence of
superconductivity and magnetism, and can be a general feature associated with
the breaking of spin rotational symmetry inside the magnetic phase.
