290 ME. N. BOHE ON THE DETEEMINATION OF THE 



in a corresponding manner ; but the calculations would in this case be very extensive, 

 and, with regard to the present investigation, it would not be of any practical 

 importance. Using jets the diameter of which is small in proportion to the wave- 

 length, the motion will differ so little from the motion in two dimensions that the 

 small correction of the wave-length due to the finite values of the amplitudes can be 

 considered the same in both cases. 



In the present problem the existence of a velocity-potential < can be supposed. 

 Using polar co-ordinates and calling the radial and tangential velocity respectively a 

 and |8, we get 



Considering the fluid as incompressible, we get 



" Zr r r 8S ~ \3r 2 r Sr r 2 



The solution of (l), subject to the condition that the velocity shall be finite for 

 r = 0, can be written (n being a positive integer) 



)sin(^ + e, 7 ) ........ (2) 



The equation of the surface can be written 



The surface-conditions are, using the same notations as above and calling the radius 

 of curvature of the surface R, 



From (3) we get 



1 ^b ^ w y ^b , ^y \ _ A 



r\ . f> ~f\ n f\ n I ^\ / 



and 



P 1 g^ 2 I i a-v. / Ijl 2o 



=0. . (5) 



Considering only small vibrations of the surface about the position of equilibrium 

 r = a, is a small quantity which we will consider as being of the first order. From 

 (2), (4), and (5) it can be seen that <f> must also be of the first order, when F (t) is 

 defined in such a manner that < does not contain terms independent of r or 5. 



