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An effective means of solving boundary-value problems of the static theory of elasticity for a finite cylinder under arbitrary boundary conditions on the lateral and end surfaces is the application of the method of homogeneous solutions [3,4,6,7]. Difficulties arise in satisfying the boundary conditions of the problenl [1,7], however, because of the absence of ordinary orthogonality. In the present note we develop a method of satisfying the boundary conditions on the lateral surface of the cylinder under homogeneous boundary conditions on the end surfaces by developing the concept of generalized orthogonality [5]. We show that it is possible to generalize the method to the case of a cylinder that is piecewise-homogeneous with respect to the axial coordinate.1. We refer the elastic cylindrical body to dimensionless cylindrical coordinates a, 7, where a is the radial coordinate and 7 is a coordinate directed along the axis of the cylindrical body. By a cylindrical body we mean a finite solid or hollow cylinder and an infinite plate with a cylindrical slit, homogeneous or piecewise homogeneous with respect to the coordinate 7.In the axisymmetric case we write the equations of equilibrium relative to the components u(a, 7) and w(a, 7) of the elastic displacement vector in the absence of mass forces as follows:A + 2# 0,~0 -O,~O~w + 6~u = O,