THE practical application of variational methods, involves difficulties due to the fact that the coordinate functions usually required are often approximately linearly-dependent (ill-posed). This applies, in particular, to the system of powers of the independent variables which is in many respects very convenient. The purpose of this paper is to develop a method of carrying out calculations by Ritz's method which is as far as possible free from loss of precision. A characteristic feature connected with loss of precision is poor definition of the Ritz matrix. In [1] and [2] is shown how a coordinate system can be formulated to avoid this. Thus, if as coordinate system we take a system which is orthonormalized in the energy metric of an operator semi-convergent with the given one, the eigenvalues of the Ritz matrix will be bounded above and below by positive numbers and the numbers of definition of these matrices will be bounded above. In this case, the approximate solution will be stable with respect to perturbations of the elements of the matrix and the free terms, and also with respect to the rounding-off errors which arise in solving the system. The above-mentioned properties are exhibited, in particular, by the system of eigenfunctions of a semi-convergent operator. We shall, however, avoid choosing this as our coordinate system. The reason for this is not only analytical difficulties (for our problem this system is connected with Bessel functions), but also the poor approximative properties of such a system. Instead of this, as radial variable, we propose a system of algebraic polynomials. In this case the rate of convergence depends only on the differential properties of the solution. Nevertheless, a considerable error may still arise in this case, if the orthogonal polynomials are written in the form of power expansions — both in the process of orthogonalization and in the later calculations with a constant system. Here the poor definition of the power system is revealed in another form [3]. In avoiding this we shall everywhere base our arguments only on expansions in Chebyshev polynomials. As we have already stated, the definition number in the chosen system of coordinate functions remains bounded when the number of coordinate functions increases without limit. In practice, it is not actually very large: in our example λ max λ min ≈ 11 . The computer reserve of precision would permit the system to be solved by one of the exact methods, even for a considerably larger definition number; if, however, the system is solved by an iterative method, which is evidently more convenient in the case of high orders, the rate of convergence is very sensitive to the magnitude of the definition number. In our example the denominator of the most advantageous one-step iterative method is equal to λ max − λ min λ max + λ min ≈ 5 6 , so that to decrease the error by a factor of, for example, 10 −4, fifty iterations would be required.