80 



Lord Rayleigh on 



[May 15, 



sponds to a displacement of the jet as a whole, without alteration in 

 the form of the boundary. Accordingly there is no potential energy, 

 and the frequency of vibration is zero. For n = 2 the boundary is 

 elliptical, for n=3 triangular with rounded corners, and so on. With 

 most forms of orifice the jet is subject to more than one kind of 

 vibration at the same time. Thus with a square orifice vibrations 

 would occur corresponding to «=4, n=8, n=12, &c. However, the 

 higher modes of vibrations are quite subordinate, and may usually be 

 neglected. The values of y (n 3 —7i) for various values of n are shown 

 below. 



n— 



2, 



p = 



ye 



y6x i 





n— 



3, 



P= 



v/6Xv/ 4= 



yex 2 





n= 



4, 



p = 



y6xy io= 



yex 3 



•16 



n = 



5, 



p= 



y6x y 20= 



yex 4 



•47 



71 = 



6, 



p= 



y6x y 35= 



y6x 5 



•92 



71 = 



7, 



p = 



y6x y 56= 



y6x 7 



•48 



71 = 



8, 



P= 



y6x y 84= 



y6x 9 



•17 



71 = 



9, 



p = 



y6x yi2o= 



yex io 



•95 



71 = 



12, 



V— 



yex y286= 



y6xl6 



•95 



It appears that the frequency for n=3 is just double that for n=2 ; 

 so that the wave-length for a triangular jet should be the half of that 

 of an elliptical jet of equal area, the other circumstances being the 

 same. 



For a given fluid and mode of vibration (n), the frequency varies 

 as A -1 , the thicker jet having the longer time of vibration. If v be the 

 velocity 'of the jet, \ = 2 irvp~ l . If the jet convey a given volume 

 of fluid, v oc A -1 , and thus X oc A~ l . Accordingly in the case of a 

 jet falling vertically, the increase of \ due to velocity is in great 

 measure compensated by the decrease due to diminishing area of 

 section. 



The law of variation of p for a given mode of vibration with the 

 nature of the fluid, and the area of the section, may be found by con- 

 siderations of dimensions. T is a force divided by a line, so that its 

 dimensions are 1 in mass, in length, and —2 in time. The volume 

 density p is of 1 dimension in mass, —3 in length, and in time. A 

 is of course of 2 dimensions in length, and in mass and time. Thus 

 the only combination of T, /?, A, capable of representing a frequency, 

 is TV*A-*. 



The above reasoning proceeds upon the assumption of the applica- 

 bility of the law of isochronism. In the case of large vibrations, for 

 which the law would not be true, we may still obtain a good deal of 

 information by the method of dimensions. The shape of the orifice 

 being given, let us inquire into the nature of the dependence of A, 

 upon T, p, A, and P, the pressure under which the jet escapes. The 



