118, no] The Tidal Problem 119 



and the pressure at any point will be given by 



,....(347). 



V U / 



The function 



...(348) 



p \g? o* c* 



is a linear function of # 2 , y 2 and z*. It can accordingly be made a spherical 

 harmonic by assigning a suitable value to & ', and on operating on the func- 

 tion with V 2 , the requisite value for 6' is found to be given by 



2 1 fa. I ON ,/l 1 IN (349) 



abc ^Trpabc \a b 



Now during the motion of the spheroid, the pressure over the surface 

 will be uniform at each instant*, so that the function (348) must be con- 

 stant over the surface at each instant. If we assign to 8' the value given 

 by equation (349), the function (348) is a spherical harmonic, so that being 

 constant over the surface of the ellipsoid, it must also be constant throughout 

 its volume, and hence the coefficients of # 2 , y 2 and z* must vanish separately. 

 Equating these to zero, we obtain the system of equations 



(350), 



2jrpabca Trpabc a 2 " 



c" A '/^-+l.-Ai-Ti (351), 



%7rpabc c 2 -rrpabc c 2 ' 



On adding corresponding sides of these three equations, we again obtain 

 equation (349) which determines 6', so that the three equations (350) to 

 (352) contain within themselves the necessary and sufficient condition that 

 the pressure shall remain uniform (or zero) over the boundary throughout 

 the motion. The equations are equations expressing d, b and c in terms 

 of the configuration at any instant ; they may accordingly be regarded as 

 equations of motion for the ellipsoid (343). Naturally they reduce to the 

 statical equations (78) to (80) when the ellipsoid is at rest. 



Clearly the relation between 6, as given by equation (81), and & ', given by 

 equation (349), is 



"-i^ + n+ll (353) 



by equation (344). Thus 6' becomes identical with 6 when there is no 

 motion, or when the figure is instantaneously at rest, so that d = b = c 



* It will be seen that the method we follow is that of Dirichlet, Gott. AMand. 8 (1860), p. 3, 

 or Coll. Works, n. p. 263. See also Lamb's Hydrodynamics (4th Edition), p. 689. 



