48 



Proceedings of the Royal Irish Academy. 



These, however, vanish on integration by virtue of 



J,{mR) = 0. 



Hence, we find 



(60) 



i-i 



Also q. p. V= a^, whence we have ^ being small compared with I 



Si i-i 



It is obvious that this formula will also give the kinetic energy of an 

 infinite fluid due to the motion of two thin circular disks which are 

 in motion towards one another with equal and opposite velocities, their 

 distance being supposed small compared with their common radius. 



If we suppose ^ comparable with R, but both small compared 

 with I, the expression for the kinetic energy of the fluid will have 

 the same form in a, but a will now be given in terms of V by 



^ 2aRr l-2K,{x,)Y,(x) . ^ ^x 



Jo .sm^-^.. (61) 



I. — Case where dish fits not quite tightly an enclosing cylinder radius R\ 

 i.e. R' = R + JB, where B is supposed small compared with R. 



Let V, as before, be the potential corresponding to the vertical 

 velocity of any point in the fluid. Then the expression of v for the 

 internal cylindrical space above and below the disk will be the same as 

 before, but that for the outer cylindrical sheath R^ R' will now be 



v' = A,,Y,{nr) + B',Klnr), 



where now the conditions of motion give 



for 



r = R', i.e. A,,Z {nR') + B'^K, {nR') = 0, 

 60 that v' may be written 



