ON THE CONDITIONS OF FLUID-MOTION. 199 



XXIII. On the Conditions of the Motion of a Portion of 

 Fluid in the Manner of a Rigid Body. By R. F. 

 GWYTHER, M.A. 



Bead November 2nd, i88o. 



The condition that a portion of fluid may comport itself 

 as a rigid body, or that fluid may remain in a state of re- 

 lative rest upon or within a moving solid^ has not, as far as 

 I am aware, been mathematically investigated. We know, 

 however, that in certain cases, as on the surface of the 

 earth, the condition can be realized, or that any deviation 

 has not been discovered by undirected investigation. 



In the case considered the velocity at points in the fluid 

 must consist of a common linear velocity and a common 

 angular velocity about some axis, moving or fixed. There- 

 fore, using the quaternion notation, the velocity must be 

 of the form 



o- = S + VeT, 



where 2 is the common linear velocity, e the vector-axis of 

 instantaneous Totation, and r the vector of any point in 

 the fluid. 



The equation of motion is 



I^<o- + -iVj^ = a, ^^(l) 



a denoting the force acting on the element of the fluid, p 

 and q having the usual meanings. Under the condition 

 stipulated no force due to viscosity is called into action. 

 If q be a function oi p only, we may write 



