258 THE EQUILIBRIUM OF EEVOLVING LIQUID curvatures of the surface at P is that of the meridianal curve, the radius other principal curvature is PQ—the normal as terminated on the axis, pressure due to the curvature is thus and the equation of equilibrium may be written p + pQ=~2r~+r°' ........................... where p0 is the pressure at points lying upon the axis, and a- is the dens the fluid. The curvatures may most simply be expressed by means of s, the 1( of the arc of the curve measured say from A. Thus _L -1 ^y 1 _ d*y/ds2 PQ so ds ' p doe/ds ' so that (1) becomes dy dx dzy _ aco^x* dx p$x dx dsds+a)'d^~~2T"ds+Tfo' or on integration tdy _ g-afx* Thus dy/ds is a function of as of known form, say X, and we get for y in 1 of x f Xdx * y=±J^(i-X2)' ........................... as given by Beer. If, as in fig. 1, the curve meets the axis, (3) must be satisfied by c dyjds = 0. The constant accordingly disappears, and we have the ] simplified form At the point A on the equator dyjds = 1. If OA = a, whence eliminating p0 and writing "" ST ' we get ds a? ^ ~~e of revolution about the axis of rotation.