1879.] the Capillary Phenomena of Jets. 93 



The velocity potential (0) of the motion of the fluid, satisfies the 

 equation 



dr* r dr r 2 c?# 2 dz 2 



or, if in order to correspond with (12) we assume that the variable 

 part is proportional to cos n6 cos kz, 



^+lf-fe ! +4=0 .... (18). 



The solution of (18) under the condition that there is no introduc- 

 tion or abstraction of fluid along the axis of symmetry is — 



0=jS»J«(i&r)cos nO cos kz .... (19), 



in which i— V( — 1), and Z n is the symbol of the Bessel's function of 

 the nth order, so that 



K J 2»r(»+i)l 



1+ + 



2.2n + 2 2 .4. 2» + 2. 2w+4 



+ ^ + (20). 



2.4.6.2w + 2.2»+4.2»+6 J 



The constant /3« is to be found from the condition that the radial 

 velocity when r=a coincides with that implied in (12). Thus 



ik(3 n J n '(ika) = ^L (21). 



dt 



The kinetic energy of the motion is, by Green's theorem, 



2P 0—1 a d6 dz = Wpz . ika . J n (ika) J n '(ika) . /3 W 2 ; 



J J L drjr=a 



so that, by (21), if K denote the kinetic energy per unit length, 



K=j7r / >a 2 3 »( iha ) /dccnV ] % > (22). 



ika . J n ' {ika) \ dt J 



When n=0, we must take, instead of (22), 



K=Wpa*^^ a ±—(^) 2 . . . (23). 

 ika . J ' (ika) \dt J 



The most general value of K is to be found by simple summation, 

 with respect to n and k, from the particular values expressed in (22) 

 and (23). Since the expressions for P and K involve only the squares, 



and not the products, of the quantities <x, — , it follows that the 



dt 



