[From the London Matftematical Society's Proceedings. Vol. IIL] 



XL. On the Displacement in a, Case of Fluid Motion. 



IN most investigations of fluid motion, we consider the velocity at any 

 point of the fluid as defined by its magnitude and direction, as a function i.i 1 

 the coordinates of the point and of the time. We are supposed to be able to 

 take a momentary glance at the system at any time, and to observe the velo- 

 cities ; but are not required to be able to keep our eye on a particular molecule 

 during its motion. This method, therefore, properly belongs to the theory of a 

 continuous fluid alike in all its parts, in which we measure the velocity by the 

 volume which passes through unit of area rather than by the distance travci 

 by a molecule in unit of time. It is also the only method applicable to the 

 case of a fluid, the motions of the individual molecules of which are not expres- 

 sible as functions of their position, as in the motions due to heat and diffusion. 

 When similar equations occur in the theory of the conduction of heat or elec- 

 tricity, we are constrained to use this method, for we cannot even define what is 

 meant by the continued identity of a portion of heat or electricity. 



The molecular theory, as it supposes each molecule to pi'eserve its identity, 

 requires for its perfection a determination of the position of each molecule at 

 any assigned time. As it is only in certain cases that our present mathematical 

 resources can effect this, I propose to point out a very simple case, with the 

 results. 



Let a cylinder of infinite length and of radius a move with its axis parallel 

 to z, and always passing through the axis of x, with a velocity V, uniform or 

 variable, in the direction of x, through an infinite, homogeneous, incompressilile. 

 perfect fluid. Let r be the distance of any point in the fluid from the axis 



