We generalize a class of magnetically charged black holes holes non-minimally coupled to two scalar fields previously found by one of us [gr-qc/9910041] to the case of multiple scalar fields. The black holes possess a novel type of primary scalar hair, which we call a contingent primary hair: although the solutions possess degrees of freedom which are not completely determined by the other charges of the theory, the charges necessarily vanish in the absence of the magnetic monopole. Only one constraint relates the black hole mass to the magnetic charge and scalar charges of the theory. A Smarr-type thermodynamic relation, and the first law of black hole mechanics for the system are obtained. We further explicitly show in the two-scalar-field case that, contrary to the case of many other hairy black holes, the black hole solutions are stable to radial perturbations.
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