with uniform speed V in a straight line in inviscid liquid the total kinetic energy of 

 the system is 



%{M + H)V*, (27) 



and the body moves as if the liquid were absent and the mass of the body were 

 increased from its actual mass M to its virtual mass M + H. Here H is the added 

 or hydrodynamic mass for this particular motion, and is the coefficient of V2 V 2 in 

 the expression for the kinetic energy of the liquid. 



It is only quite recently that a physical interpretation of hydrodynamic mass 

 as an actual mass of liquid has been given by Darwin [11]. To understand Darwin's 

 interpretation consider the particular case of a circular cylinder which moves along the 

 x-axis from minus to plus infinity. Suppose that when the cylinder is at a; = — 00, 

 a wall of blue dye is used to color the particles in a plane perpendicular to the direction 

 of motion. Since the cylinder in its motion displaces a certain volume of the fluid the 

 gap left behind must be filled up and it might appear reasonable to suppose that when 

 the cylinder has attained the position x = + co , the wall of dye will have retreated 

 a certain distance to the rear of its initial position. 



Now the paths of the particles are elasticas, Fig. 1, so that a particle at A 

 when the cylinder is at x = — 00 will have drifted forward to B when the cylinder is 

 at x — + 00 . 



FIG I 



If then we consider those particles which at a given instant lie in an axial plane 

 of the cylinder, perpendicular to the direction of motion, the positions of these particles 

 when the cylinder is at * =z — 00 and at x n +00 define surfaces which, in the plane 

 of the motion, are typified by curves A,A, and B,B, Fig. 2. Thus the particles on the 

 curves A,A move forward, not backwards, to the positions B,B. The intuitive idea of 

 reflux of a wall of dyed particles is false. Darwin's discovery is that the mass of the 

 liquid enclosed between these initial and final positions is in fact the hydrodynamic 

 mass of the cylinder for this particular motion. 



The point can be established by integration since the coordinates of the points 

 on the elastica are expressible rationally in terms of Jacobian elliptic functions [12]. 

 The argument is, however, capable of general formulation independent of the particu- 

 lar shape of the cross-section of the cylinder. The argument can also be extended to 



