536 XI. HYDRODYNAMICS. 



163) U (r , if>, <p) = const. 



r , ty, (p, denoting the radius, latitude , and longitude. If F denote 

 the potential of the disturbing body, we have according to the equi- 

 librium theory, for the disturbed surface, 



164) U(r,i}>,<p) + V= const. 

 and subtracting equation 163) from this, we have 



165) U (r, 1>,g>)-U (r , ^ 9) + F = const, = C. 



But if we put h = r r , h is the height of the tide, and being 

 small with respect to the radius, we may put 



166) U(r,^ri-U(r Q ,^,ri = h, 

 giving 



167) v-c=-hj?- 



But g = ?-*-> as in 149, so that we obtain for the height of 

 the tide 



168) h = r . 



We may determine the constant in 168) by the consideration 

 that the total volume of the water is constant. If dS is the area 

 of an element of the earth's surface, the total volume of the tide 

 above the surface of equilibrium must vanish, giving 



169) 0=hdS, VdS = cdS, V=C, 



where F is the mean value of the disturbing potential over the earth's 

 surface. Now we have found in 150, equation 154), the value of 

 the potential of the tide - generating forces, 



170) F-g(3cosZ-l), 



where Z is the zenith-distance of the heavenly body at the point in 

 question. If we refer other points on the earth's surface to polar 

 coordinates with respect to this point and any plane through it, 

 with coordinates Z, <&, we have 



2rt TI 



C CvdS = Cd f@ cos 2 Z - 1) sin Z dZ = 0, 







so that the mean of F vanishes. Accordingly we have 

 171) fc-.(3oo.Z-l). 



