251-252] VIBRATIONS OF A CYLINDRICAL JET. 459 



For s=l, we have <r = 0; to our order of approximation the 

 section remains circular, being merely displaced, so that the 

 equilibrium is neutral. For all other integral values of s, o- 2 is 

 positive, so that the equilibrium is thoroughly stable for two- 

 dimensional deformations. This is evident a priori, since the 

 circle is the form of least perimeter, and therefore least potential 

 energy, for given sectional area. 



In the case of a jet issuing from an orifice in the shape of an 

 ellipse, an equilateral triangle, or a square, prominence is given to 

 the disturbance of the type s = 2, 3, or 4, respectively. The motion 

 being steady, the jet exhibits a system of stationary waves, whose 

 length is equal to the velocity of the jet multiplied by the period 



(27T/0-). 



252. Abandoning now the restriction to two dimensions, 

 we assume that 



< = facoskz . cos(<rt + e) ..................... (9), 



where the axis of z coincides with that of the cylinder, and fa is a 

 function of the remaining coordinates a?, y. Substituting in the 

 equation of continuity, V 2 ^> = 0, we get 



(?,-*)$, = <) ..................... (10), 



where Vi 2 = d*/dx? + d z /dy 2 . If we put x = r cos 6, y r sin 0, this 

 may be written 



dr* r dr 



This equation is of the form considered in Art. 187, except for the 

 sign of & ; the solutions which are finite for r = are therefore of 

 the type 



fc= /,(&) jlrf ..................... (12), 



olil^ 



where 



z s 



Hence, writing 



<f> = BI 8 (Ar) cos s6 cos Icz . cos (at + e) ......... (14), 



we have, by (4), 



g = - B kal * (A?a) cos s0 cos kz . sin (at + e) ......... (15). 



(7(1 



