SURFACE-TENSION OF WATER BY THE METHOD OF JET VIBRATION. 289 

 From (35) and (36) we get, with the same approximation as used hitherto, 



p ac 

 With the same approximation it will be permissible, by use of (13), to put in (37) 



cp\ 1 ' 2 ,, .,fa 2 k 

 -\ =l-i- 2 



choosing the sign for iad so that the real component is positive [see (30)]. 

 The equation (37) now becomes 





2n(n'-l) 



. v .o. . (38) 



2 \ pea 2 Ay 4 \pca 2 & 



Now solving (38) with regard to b, and setting /; = k + ie, we get 



--* fl - M (- 1 )Y ^ Y /3 3*(*-l)V 2 ^ VI 



"T~ "Ipca^y 4 "v^o^y J 



and 



n-l 



_ 

 " 



, - ^ , , 



2 - 



As all the equations used are linear, it will be seen that the physical meaning of 

 the above calculation is the proof of the existence of a real fluid-motion corresponding 

 to a surface of the form 



r = a + be~" cos kz cos n$, 



where k and e are expressed by the equations (39) and (40). 



The correction, which on account of the effect of the viscosity is to be introduced in 

 the expression for the surface-tension, can be found by (36) and (39) ; we get 



3/3 , 3n(n-l)V ' 

 - i '- 



\ 2 ~| 



. . (41) 

 / J 



- - y 

 \pca 2 kj 2 \pca 2 k 



Calculation of the Influence of the Finite Wave-amplitudes. 



We will now calculate the correction of the wave-length due to the magnitude of 

 the wave-amplitudes. The method of approximation which will be used is on the 

 principle indicated by G. G. STOKES.* 



The following calculation will only treat of the vibrations in two dimensions of a 

 fluid-cylinder without viscosity. The problem in three dimensions could be treated 



* G. G. STOKES, ' Camb. Trans.,' VIII., p. 441, 1847. 

 VOL. COIX. A. 2 P 



