162 



Lord Rayleigh on the Equilibrium of 



Fig. 1 represents a section by a plane through the axis 

 Oy, being the point where the axis meets the equatorial 

 plane. One of the principal curvatures of the surface at P 



is that of the meridianal curve, the radius of the other prin- 

 cipal curvature is PQ — the normal as terminated on the 

 axis. The pressure due to the curvature is thus 



*(?+& 



and the equation of equilibrium may be written 



1 1 crco 2 a 

 ^ + PQ = ~2T 



Po 

 T ' 



(1) 



where p is the pressure at points lying upon the axis, and <r 

 is the density of the fluid. 



The curvatures may most simply be expressed by means 

 of 5, the length of the arc of the curve measured say from 

 A. Thus 



1 Idy 1 d 2 y/ds 2 

 PQ ~ x ds ' p" dx/ds ' ' ' " 

 so that (1) becomes 



dy dx d 2 y o~(o 2 x 3 dx p x dx 

 dsds + *ds 2 ~ = '~2^ds + Tds' 



(2) 



or on integration 



dy o-&> 2 # 4 p x 2 



(3) 



Thus dy/ds is a function of x of known form, say X, and we 

 get for y in terms of x 



as given by Beer. 



f Xdx 

 y=±J •(i-X') ' W 



