238 Mr GreenhUl, On the rotation of a [Mar. 24, 



In the general case, therefore, when the axes of the cylinder are 

 revolving with angular velocity eu + «', the co-ordinates x, y of a 

 particle initially at a point whose co-ordinates are ma cos ^, mh sin ^, 

 are given at time thy 



X = ma cos [ ^ , ^ «< — ^j cos (&)+&)')< 



+ w6 sin f -2 — p «* ~^) sin (to + qj') t 



= ^m (a 4- 6) cos f 0)'^ + ^-— , 2 &)i + ^ j 



+ ^m(a— 5) cos ( &)i + 2 , ;^2 tof ^ 9) , 

 and 

 y = wa cos f -2 r2 ct)^ — ^ ) sin (eo + tu') t 



— TTib sin [ -a — 7-3- toi — ^ j cos (ft> + 0)') i 



= I w (a + 5) sin f w'i + - ^ ,s wf + ^ j 



+ ^ m (a — 6) sin {(o't + 2 , 12 ft>f - j . 



The particles of the liquid therefore describe pericycloids, which 

 ' 2 1,2 



(1) when — = -2 — T2 ^re epicycloids ; (2) when a) + w' = are 

 ellipses ; (3) when to = are circles ; (4) when 



, , . 2a5 

 a^ + 6'' 



are circles ; the particular case considered by Kirchoff. 



In the Gottingen Transactions for 1859 and 1860 Lejeune 

 Dirichlet and Riemann have attacked the general problem of the 

 motion of a mass of liquid in the shape of an ellipsoid, and dis- 

 cussed the particular cases where a free surface is possible, the 

 liquid being under the action of the mutual gravitation of the 

 particles. 



If we take the liquid, supposed frictionless, filling the ellipsoid 

 222 



-2 + ^ -h -2 = 1, originally at rest, and if at any instant the com- 



ponent angular velocities about the axes be Wj, w^, Wg ; then the 

 velocity function 



