178, 179] ATMOSPHERE. ROTATING FLUID. 467 



179. Rotating Mass of Fluid. If a mass of fluid rotates 

 about an axis with a constant angular velocity ro, we may by the 

 principle of 104 treat the problem of motion like a statical problem, 

 provided we apply to each particle a force equal to the centrifugal 

 force. If we take the axis of rotation for the ^-axis the centrifugal 

 force may be derived from the potential, 



If an incompressible liquid rotates about a vertical axis and is under 

 the influence of gravity, we have by 11), 



23) V=gz- y0 2 + r% 



24) p = o(c-gs-\- ^(a^ + y*)). 



V * / 



Consequently the surfaces of equal pressure are paraboloids of revolution. 

 Measuring from the vertex the equation of the free surface, for 

 which p = 0, is 



O ^ \ ^ / 2 i 2\ 



The latus rectum is \> On this principle centrifugal speed indicators 

 are constructed. 



An important case which we have already treated by this method 

 in 149 is the shape of the surface of the ocean. If we seek an 

 approximation, assuming the earth to be centrobaric, the potential 

 due to the attraction of the earth and centrifugal force will be, as 

 we find either directly, or by putting K=Q in 149, 140), 



and the equation to the surface of the sea will be, U= const., which 

 may be written, writing for the constant, 



where ty is the geocentric latitude, and a is the polar radius. In the 



case of the earth ,.. = noo . . which is so small that the second 

 ym. 288.41 



term may be considered small with respect to the first, and its square 

 neglected. Accordingly putting in this term r = a the equation of 

 the surface is 



f w 2 a s 1 



1 ' 2yM ' ^J 



30 ! 



