[Methods for approximate and exact complanation of spheroidal-cell nuclei as an aid for statistical karyometric...

After short mathematical explanations on exact formulas for surface area calculation of the three-axis ellipsoid and the spheroid, approximation formulas, which for the most part are modified formulas of the spherical surface area, are developed. By application of of Jacobj's linear method and Lange's curvimeter-planimeter method, these approximation formulas are used to calculate the surface areas with identical semi-axis and diameter values being inserted, and the results are compared with each other and the true values. If the design procedure is extended to other axial rations, numeric and per cent surveys result in the form of tables. The approximation formulas, whose results differ only a little from the true value and which require only a minimum of computing effort are selected. For the three-axis ellipsoid as well as for the spheroid, the true surface area values can be determined from the approximate value, with help of tabulated percentages and by interpolation of intermediate values are required. Interpolation formulas have been formulated, for that purpose, in addition. In the last part an exact formula for the nucleus surface area, if it has the shape of a spheroid, has been developed. According to the formula there exists a dependence of nucleus volume and axial ratio as the basis of a mixed cyclometric function. A further detailed table contains these surface area values, arranged in groups of equal axial ratios. In addition two other formulas have been evolved for exact computing of the nucleus surface area. For one it is dependent on the two diameters and the mixed cyclometric function which is based on the axial ratio; for the other it depends on the large diameter, the axial ratio and the mixed cyclometric function. Numerical examples have been worked out for all representations. For u = n = const, the surface values from terms of an arithmetical second-order procession. In case of equal axial ratios a recurrence formula can be worked out, according to which the succeeding term can be calculated from its predecessor.