Certain Problems of Two-Dimensional Physics. Ill 

 Figures of Equilibrium of Rotating Fluid. 

 11. If the gravitational potential of a cylinder take the 

 form* C — \<d 2 {(x — k) 2 +y 2 } on the surface tj=0, i. e. if it is 

 possible (in the case of constant density) to have 



V,= -*/>{(*- /Lf+^ + ^p-arKX^ + X^ + Y^-f-Y/) 



+ 1 {!■(*) -F(t)}, 

 V =-i ft , 2 (X 1 2 + X 2 2 + Y 1 2 +Y 2 2 )+i<»^(X 1 + X 2 ) 



+ «>»£ (XY'-X'Y>*,+ \ t {P(«)-F(t)}, 



then the cylinder is a possible form of equilibrium of liquid 

 rotating, under the influence of its own attraction, with 

 angular velocity co. 



In particular, the hypotrochoids of equation (2), § 4, are 

 possible figures of equilibrium if k = and 



co 2 n — 1 j 1 fr 2 | 



Thus for any given positive integral value of ??, and for 

 values of e varying from to 1, the hypotrochoids 



X\J n{n — e 2 ) = c(ii cos rj + 6 cos ????), 



y^/n(n — e 2 ) = c{n sin 77 — 6 sin nrj), 



[with 6)7277/3 = (n - e 2 )/(n + 1) ] 



form a linear series of figures of equilibrium (unstable if 

 n > 1) passing out of the circle of radius c. The case of 

 7i—l is that of the elliptic cylinders, which are stable if 

 e < J, the bifurcating ellipse being that for which e = \. 

 The case of n > 1 is that of the hypotrochoid which passes, 

 as e increases to unity, into the ?i + l-cusped hypocycloid, 

 after which fluid escapes at the cusps, as is easily verified in 

 any particular case. 

 For example, take 



x ±L7j = aeZ +lT1 + be-t !i + lr >Uce- 2 tt+ l1 >\ 

 and therefore 

 i (Xi 2 + X 2 2 + Yi 2 + Y 2 2 ) = a 2 + b 2 + c 2 + 2bc cosh f cos v 

 + 2ab cosh 2£ cos 2?? + 2ca cosh 3£ cos 3*7, 

 ^ f (XY' -X'Y)d7] = (a 2 -b 2 -2c 2 )Z-3bc sinh f cost;- icasinh 3f cos 3t7- 



* By taking the surface-condition in this form we are really making 

 use of Poincare°s theorem that there is a plane of symmetry (y = 0) 

 through the axis. 



Phil. Mag. S. 6. Vol. 30. No. 180. Dec. 1915. 3 E 



