470 Prof. L. Natanson on Douhle-Be fraction 



however, confine ourselves to an examination of that part of 

 the problem which one may hope to solve by purely hydro- 

 dynamical means ; we shall not attempt to enter into con- 

 siderations derived from optical theory, which would be 

 necessary to render the theory of this phenomenon complete. 

 § 1. Thanks to the investigations of which we have given 

 a brief account, the particular case discussed by Stokes and 

 realized in the experiments of Kundt has become of the 

 greatest importance ; it is therefore the case which will 

 claim our attention. Imagine a cylinder kept in rotation 

 about its axis. At a sufficiently small distance from its 

 surface imagine a fixed cylindrical wall having the same 

 axis. The annular space bounded by the surface of the 

 cylinder and the wall is filled with the liquid which it is 

 desired to investigate. Let a = internal, and b = external 

 radius of annular space, and r = distance from the axis of any 

 arbitrarily chosen point M in the interior of the liquid. Let 

 q be the velocity, and h the angular velocity of M. Let the 

 axis of z be chosen along the axis of the cylinder, the axes 

 of x and y being arbitrarily chosen in a plane normal to the 

 axis. Let % be the angle between the direction of r and the 

 #-axis, and s = velocity of rotation of cylinder; this rotation 

 will be assumed to be uniform. The motion communicated 

 to the molecules of the liquid will evidently be along circles 

 whose planes are normal to the axis of z. We shall assume 

 that the velocity q of a molecule depends solely on its distance 

 r from this axis : 



q = q(r); g(b)=0 ; q(a)=s; . . . . (1) 



it is independent of the time t, so that the motion may be 

 called steady. In practice, the actual motion of a liquid caTi 

 only approximately fulfil the conditions of the ideal case 

 which we shall be content to discuss here. 

 From what has been said, it follows that 



»-*-*?< m 



and 



S=+Jd, (3) 



the angle ® being reckoned from the instant t = Q. The 

 sign + has been introduced in order to express the fact that 

 the motion of the cylinder, and hence also that of the liquid 

 molecules, may take place in two opposite directions rela- 

 tively to the co-ordinate axes. 



In our calculations, it will sometimes be convenient to 



