36 ROTATING LIQUID. 



from the axis. Let m be the mass of the particle at P, w 

 the angular velocity of the liquid, and j the distance MP, 

 and denote the centrifugal force by P; then (Anal. Mechs., 

 Art. 198) we have, for the centrifugal force on the parti- 

 cle Wi, 



P = mtfy. (1) 



The particle is also urged vertically downwards by its 

 own weight mg, due to the force of gravity ; hence the par- 

 ticle is in equilibrium under the action of gravity mg, of 

 the centrifugal force muPy, and of the reaction of the sur- 

 face of the" liquid which is normal, and therefore the result- 

 ant of mg and muPy must be normal to the surface. 



Let PF and PG represent the centrifugal force and force 

 of gravity, respectively ; then, completing the parallelogram 

 of forces, the resultan t of these PR, must be normal to the 

 surface at P. Let this normal meet the axis in N; since 

 the triangles, GPR and MNP, are similar, we have 



NM : MP :: PG : GR (= PF); 



or NM : y : : mg : muPy ; 



- (2) 



But NM is the subnormal of the curve, AVD ; therefore 



the subnormal NM ^= = a constant, 

 W 2 



which is a property of the parabola. Hence the curve 



AVD, is a parabola whose latus rectum is , and therefore 



tj* 



the surface is a paraboloid of revolution. 



Sen. It will be seen that this result is independent of 

 the form of the containing vessel. The axis of rotation, in 

 fact, may be within or without the fluid, but in any case it 

 will be the axis of the surface of the paraboloid. 



