**Goal **of the article is to develop a mathematical model of the movement of the cross-section center of the ellip- tical casing in the supporting elliptical bandage to increase the straightness of the geometric axis of the rotary furnace body at the support point, which depends on the sizes of the elliptical casing and bandage, their mutual location on the support, as well as the axes orientation of the ellipses of the cross-section casing relative to each other on different supports. **Significance**. The high-quality installation of the rotary furnace casing is determined by the straightness of its geometric axis, and not only at the time of control, but also during the mechanism operation. It is obvious that if this condition is observed, optimal operating conditions of the mechanism, its reliability and durability can be achieved. However, the measurement results and installation accuracy are affected by such a property of the meas- urement object as the deviation of the body shape from the circular cylindrical one, especially in the places of support. The new sub-bandages casing for installation have smaller deviations from the round-cylindrical shape than those that were in use. These deviations appear mainly as ellipticity or ovality. Since the new bandage, as a rule, does not have other distortions of the shape, except for ellipticity, to clarify the issue, the displacement of the cross-section center of the elliptical casing in the elliptical bandage should be considered. **Method**. For the cross-section of a support node with an elliptical bandage and a sub-bandage casing, using the complete elliptic integral of the second kind and its presentation in the form of a hypergeometric series, the center coordinates of the ellipse of the casing cross section in the YOZ coordinate system are determined and a detailed analysis of the factors and parameters of the system affect- ing its position is provided. **Results**. The position of the geometric axis of the body at the support point depends on the ellipticity of the sub-bandage casing and the bandage, their mutual location on the support, as well as the axes orientation of the ellipses relative to each other on different supports. Since the position of the geometric axis does not remain unchanged during the rotation of the body, the location of the scattering ellipses centers on one straight line will be optimal. **Scientific novelty**. Mathematical dependencies have been established to determine the center coordi- nates of the ellipse of the casing cross section in the YOZ coordinate system, the origin is in the center of the circle with the conventional radius of the ellipse of the rotary furnace cross section. **Practical significance. **The proposed method of estimation will further increase the straightness of the geometric axis of the rotary furnace, both at the time of control and during the operation of the mechanism.

- Bisulandu B.-J., Huchet F,
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