234 Mr Oreenhill, On the rotation of a [Mar. 24, 



satisfied for an infinite elliptic cylinder, rotating as if rigid about 

 the axis of the cylinder, and here the axis of rotation is the greatest 

 axis, namely the infinite axis of the cylinder. 



For if (o denote the angular velocity of the cylinder, then the 

 equation for the pressure p is 



P-V-^(o'^x^+f)=H, A constant; 



P 

 the axis of the cylinder being the axis of z^ and the gravitation 

 potential 



F= constant — 27rp 7^ , 



'^ a+ 



when a, h are the semi-axes of the elliptic section of the cylinder. 



Hence the surfaces of equal pressure are the similar cylinders 



I -^ — o)'^ ) x^ + ( -^^ — (o'A y^ = constant, 



and these are similar to the outer surface of the cylinder, and a 

 free surface is therefore possible if 



\a + h J \a + h I 



/2 A ^^ 



or ft) = 47rp 



{a + by 



If the motion in the elliptic cylinder had been generated from 

 rest in a frictionless liquid, we should have the velocity func- 

 tion <j> for an angular velocity o) about the axis of the cylinder 

 given by 



a' - b' 



and therefore 



giving the pressure p. 



Here xy denotes a point, fixed in space, and therefore 

 dx ^ dy 



d<l) _ a^ — W (dx dy\ 



~dt~'^'^+¥[dt^'^'"dt) 



a' + h'^^ '[ 



