We show that the existence of a new massive sterile neutrino can be manifested employing a novel experimental method of neutrino oscillations, namely, neutrino oscillometry. With a judicious monochromatic neutrino source the relevant oscillation length is expected to be shorter than 1.5 m. Thus the needed measurements can be implemented with a gaseous spherical time projection chamber of modest dimensions having a very good energy and position resolution. The best candidates for oscillometry are discussed. The expected sensitivity to the mixing angle θ ₁₄ has been estimated: sin ^-2(2θ ₁₄)=0.05 (99%) with only two months of data handling with Cr51.