The following method has been used to determine the relative frequencies with which nebulae of a given actual ellipticity, oriented at random, will be observed as having various apparent ellipticities. In Figure 8, let the co-ordinate axes OX and OY coincide with the major and minor axes, a and b, of a meridian section of an ellipsoid of revolution. Let OO′ be the line of sight to the observer, making an angle i with OX, and let OR be perpendicular to OO′. Let PP′ be a tangent to the ellipse, parallel to and at a distance from OO′. Let x_{0} and y_{0} be the intercepts of the tangent on the X- and Y-axis, respectively. The apparent ellipticity is determined by b_{x}, which, for various values of the angle i, ranges from b to a. The problem is to determine the relative areas on the surface of a sphere whose center is O, within which the radius OY must pass in order that the values of b_{1}, and hence of the apparent ellipticity, e_{1} may fall within certain designated limits. This requires that the angle i be expressed in terms of b_{1}. [Illustration: Fig. 8] From the equation of the tangent, PP′, $y = -x \tan{i} + \sqrt{a^2 \tan^2{i} + b^2}$ $y_{0} = \sqrt{a^2 \tan^2{i} + b^2}$ Since $b_{1} = y_{0} cos(i)$ $b_{1}^2 = a^2 \sin^2{i} + b^2 \cos^2{i}.$ Let a = 1, then $\cos^2{i} = \frac{1 - b_{1}^2}{1 - b^2},$ where $b_{1} = 1 - e_{1},\;b = i - e.$ From these equations, the values of i can be determined for all possible values of e_{1}. The limits for the observed classes E0 to E7 were chosen midway between the consecutive tenths, E0 ranging from e = 0 to e = 0.05; E1, from e = 0.05 to e = 0.15; E7, from e = 0.65 to e = 0.75. The relative frequencies of the various observed classes are then proportional to the differences in sin i corresponding to the two limiting values of e_{1}. These frequencies must be calculated separately for nebulae of different actual ellipticities. The results are given in Table X, where the actual ellipticities, listed in the first column, are followed across the table by the percentages which, on the assumption of random orientation, will be observed as having the various apparent ellipticities. The bottom row will be seen to show the percentages of apparent ellipticities observed in an assembly of nebulae in which the numbers for each actual ellipticity are equal and all are oriented at random.