296 MR. N. BOHR ON THE DETERMINATION OF THE 



dimensions [see (36), p. 288], and for sake of simplicity putting n = 2, which corresponds 

 to the experiments executed, 



b" b 2 b a b a 



r = a+b cos 23- cos kz+ - cos 49 cos2&z-| -- cos 45 -- cos 2kz -- ... . (36) 



6a 4 8a 8a 



and 



(37) 



The equation (37) gives the correction on the wave-length sought. 

 The equation (36) permits some further applications. 

 Putting z = 0, we get 



r = a- +6 cos 2*+ -5- -cos 43.. (38) 



4a 12 a 



(38) is the equation for the form of the orifice (supposing that the velocity, at every 

 point of the cross-section of the jet at the orifice, has the same magnitude and 

 direction), when the jet is to execute pure periodical vibrations. We see from this 

 that the opinion of P. O. PEDERSEN,* according to which a jet issuing from an orifice 

 of the form (r = + /3cos25) must be expected to execute much purer vibrations 



/ 3 R 2 \ 



than a jet from an elliptical orifice r = a + yS cos 23+ - cos 45... ), is not correct. 



\ 4 a / 



Putting 5 = 0, we get 



r = H --- \-b cos kz-\ --- cos 2kz... (39) 



8a 24 a 



(39) is the equation for the wave-profile, formed by intersecting the surface of the 

 jet by one of the two perpendicular planes of symmetry. Maximum- and minimum- 



values of r are obtained by putting in (39) respectively z = 2n-r and z = (2n-f 1) y. 



K K 



We thus get 



+ 7 Hi) alld i( 7 'mx.-nnin.) = & (40) 

 DC*/ 



These formulas will be used in the measuring of the jets. 



Calculation of the Effect of the surrounding Air. 



We have hitherto neglected the density of the air.t A sufficient approximation of 

 the small correction on the wave-length, due to the inertia of the air, is however very 

 simply obtained by the following calculation regarding infinitely small vibrations in 

 two dimensions of a cylindrical surface, separating two fluids of different density. 



* P. 0. PEDERSEN, loc. cit., p. 365. 



t Lord RAYLEIGH ('Phil. Mag.,' XXXIV., p. 177, 1892) has investigated the corresponding problem in 

 the case where the symmetry about the axis of the fluid-cylinder is maintained during the vibrations. 



