103-ios] Rotating Cylinders 103 



Then it can be verified* that the potentials of the cylinder, assumed com- 

 posed of homogeneous matter of density p, are given by 



(?) d+r<l> (rj) drj - ?J + cons ....... (291), 



e Jo ) 



(292). 



105. Now suppose that a cylinder of matter of density p has for its 

 equation x* + i/ 2 = a 2 or 



^=a 2 



when there is no rotation, and that under a rotation a), this gives place to a 

 boundary of equation 



^a' + a^f + ^ + a^p-M'H .................. (293), 



or, in polar coordinates, 



r 2 = a 2 + 2a 1 rcos0 + 2a 2 r 2 cos20 + .................. (294). 



The condition that the surface (294) can be a figure of equilibrium is 

 that, at every point of the boundary, 



F; + JwV 2 = acons ......................... (295). 



Since the value of r 2 at the boundary is given by equation (294), this 

 condition readily transforms into 



Vi + Trp r 2 - Trp ( 1 - ^ J (a 2 + 2a t r cos 6 + 2a 2 ?^ 2 cos 20 +...) + cons. = 



...... (296). 



This expression must vanish at every point of the boundary ; it is readily 

 seen to be harmonic, and so must vanish at every point inside the boundary. 



From equation (291) the potential "Pi must be of the form 



f) - 77] 



acons (297), 



where 6,, 6 2 , ... are functions of a,, a 2 , .... Hence equation (296) assumes the 

 form 



26 n (" + 77*0 - 1 - [a 2 + ^^ 



This must be satisfied at every point inside the boundary ; equating co- 

 efficients we obtain * 



* Phil. Trans. 200 A (1902), p. 67. 



