'298 MR. N. BOHR ON THE DETERMINATION OF THE 



Before proceeding to the experimental part of the investigation, we will yet 

 consider a question which may be of interest for the discussion that follows. 



In the jets produced in the experiments the velocity must be supposed to be greater 

 in the middle of the jet than closer to the surface. 



We can, however, in the following manner, get an idea of the rate at which the 

 velocity-differences will be extinguished by the viscosity of the liquid. For this 

 purpose we will consider a circular fluid-cylinder, in which each part moves with a 

 velocity parallel to the axis of the cylinder, and in which the velocities of the different 

 parts are only functions of the distance from the axis and the time. 



Using the axis of the cylinder as Z-axis, we have with the same notation as above, 



a = 0, fi = 0, and w=f(r,t). 

 From the two first equations it follows that -^- = 0, arid, as further, p = const, for 



r 



r = a, we get p = const. 



Supposing w = (f> (r) e~ ft , the equation of motion 



Dw dp . 3 2 </> 1 3(4 pe , 



p,Vw-p -=- - = -f- gives _r + -- + L< = 0, 

 Dt Sz or rtir p, 



of which the solution, subject to the condition to be imposed, when r = 0, is 



<f> = cJ (kr), in which P = - . 



P- 



The dynamical surface-condition gives ( ) = ; therefrom it follows that k must 



be a root of the equation 



J' v (ka) = 0, (k = 0, V* = irl'2197, k. 2 a = 7r2'2330, k 3 a = 7r3'2383,...). 

 The general expression for iv is consequently 



_*."( 



w = Sr,,J &> e " . 



We see that the term of the expression for w containing J (k^r) decreases much 

 more slowly than the terms with higher index. 



For the jets in question the term mentioned will furthermore be predominant 

 already at the orifice, as dw/dr must be supposed to have the same sign between 

 and a. 



The velocity in a jet-piece must therefore be expressed with a high degree of 

 approximation by 



T n \ -. U/V1-2197Y 

 w = c +f'iJ (AI?") e , where e = " I -1 , 



p \ a I 

 t being the time which the jet-piece has taken to move from the orifice. 



