INyESTIOATED BY THE METHOD OF JET VIBRATION. 347 



Similarly r. (.r) = y**. In accordance with the theory of BESKKL'S functions 



' './ 



we have 



T' -- Jt T-i-T I' -- T - W T T i- ^" T T -0 (9\ 



in l. + l. + i, * A-i - 1. l.-u-r - 1 l-i - \&)' 



X X . X 



Accordingly we have 



M (**) = -5 ; r -T ' f = j * r ' .... (3). 



or +7i 1 arl,-! ?tl. or+n 1 !_, 



x. -y-= n 



By use of the last formula (2) the values of I, and I 3 , Ac., can be calculated from 

 Io, Ii, and by substituting these in formula (3) we have ft, (x). 



In order to facilitate the use of this method for the determination of surface-tension 

 I have calculated a table of the values of /^, (x) most commonly used. This table will 

 be found at the end of this paper, and contains the values of log, p m (f) for n = 2, 3, 

 4, and 6 and for x = O'OO to x = I'OO. 



The details of the calculation of this table are given in my original paper. 



Calculation of the Vibration of a Jet. 



3. In accordance with Lord RAYLEIOH'S theory the vibration of a jet can now be 

 determined when the velocity and original cross-section is known, although, as pre- 

 viously emphasised, the theory is only available for small deviations from the circular 

 form. 



The circumference at the original cross-section is determined by 



By help of FOURIER'S series this equation can be written as 



MNB 



r = <*+ 2 b n .cos(n$ + f,) ........ (2), 



when, if necessary, the value of is changed so that b a vanishes, and the origin of 

 co-ordinates is changed so that 61 becomes zero. 



Each term in (2) can be taken alone and the resulting vibration of the jet can lie 

 calculated as the sum of all the vibrations corresponding to the different values of n 

 in (2). 



Thus it is only necessary to consider 



r = a +&,.cos(n^+e,) ......... (3). 



The wave-length X, corresponding to the vioration (3) is, according to [(2), 1], 



where 



C = /> tfl .A* 4 .V.T- w (5) 



is independent of n. 



2 Y 2 



