Accordingly the capacity C of the ellipsoid is given by the equation _ 1 1 / dS -- = -------- | ---------------------------------------------------- (5). C 4[pi]abc _/ [root](x² + y² + z²)[root](x²/a^4 + y²/b^4 + z²/c^4) It has been shown by Professor Chrystal that the above integral may also be presented in the form,[7] _ 1 /[oo] d[lambda] -- = ½ | ----------------------------------------------------- (6). C _/0 [root]{(a² + [lambda])(b² + [lambda])(c² + [lambda])} The above expressions for the capacity of an ellipsoid of three unequal axes are in general elliptic integrals, but they can be evaluated for the reduced cases when the ellipsoid is one of revolution, and hence in the limit either takes the form of a long rod or of a circular disk. Thus if the ellipsoid is one of revolution, and ds is an element of arc which sweeps out the element of surface dS, we have /dx\ /py\ 2[pi]b² dS = 2[pi]yds = 2[pi]ydx / ( -- ) = 2[pi]ydx / ( -- ) = ------- dx. \ds/ \ b/ p Hence, since [sigma] = Qp/4[pi]ab², [sigma]dS = Qdx/2a. Accordingly the distribution of electricity is such that equal parallel slices of the ellipsoid of revolution taken normal to the axis of revolution carry equal charges on their curved surface. The capacity C of the ellipsoid of revolution is therefore given by the expression _ 1 1 / dx -- = -- | --------------- (7). C 2a _/ [root](x² + y²) If the ellipsoid is one of revolution round the major axis a (prolate) and of eccentricity e, then the above formula reduces to 1 1 /1 + e\ -- = --- log_[epsilon]( ----- ) (8). C1 2ae \1 - e/ Whereas if it is an ellipsoid of revolution round the minor axis b (oblate), we have 1 sin^-1 ae -- = ---------- (9). C² ae In each case we have C = a when e = 0, and the ellipsoid thus becomes a sphere. In the extreme case when e = 1, the prolate ellipsoid becomes a long thin rod, and then the capacity is given by C1 = a/log_[epsilon] 2a/b (10), which is identical with the formula (2) already obtained. In the other extreme case the oblate spheroid becomes a circular disk when e = 1, and then the capacity C2 = 2a/[pi]. This last result shows that the capacity of a thin disk is 2/[pi] = 1/1.571 of that of a sphere of the same radius. Cavendish (_Elec. Res._ pp. 137 and 347) determined in 1773 experimentally that the capacity of a sphere was 1.541 times that of a disk of the same radius, a truly remarkable result for that date.