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
* 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.