SURFACE-TENSION OF WATER BY THE METHOD OF JET VIBRATION. 295 

 Proceeding in the same manner as in the second approximation, we get 



{ = A a- [l -A' "I "' < n '-i> 9 ^'- 4 ^ a+8 T 5 r 6) l cos n< cos 3 , 

 q L q 82(2n*-+l)(2n 1) J 



+ B! cos 2n cos 2qt + B 2 cos 2n3 + B 3 cos 2qt + B 4 



+ B 6 cos 3n& cos 3qt + ~B 9 cos 3n3 cos qt + E-, cos rt cos 3^, . (34) 



where the coefficients B!, B 2 , B 3 , B 4 are the same as in the second approximation, and 

 the coefficients B 6 , B 6 , B 7 are of the same order as A 3 . 



As the result of the calculation we note [putting the coefficient of cos n5 cos qt in 

 the equation (34) equal to 6], that the surface of a fluid-cylinder, executing pure 

 periodical vibrations in two dimensions, can be expressed by 



7,2 r 0-3 _ n.-.V _ n n A 



r = a + bcos n& cos qt+ - \- , , cos 2n3 cos 2qt 



a I 8(2?i/+l) 



where 



In the experiments, the jets produced (stationary waves) will execute vibrations in 

 three dimensions the cross-section will not be the same at different points of the jet. 

 If, however, the velocity of the jet c is so great that the wave-length X is great in 

 comparison with the diameter of the jet, the motion in the single cross-sections will 

 differ very little from the motion in two dimensions, and the equation (35) can, 

 therefore, also in this case give information about the form of the surface of the jet. 



The complete solution in three dimensions can be expressed by 



cos 2nS cos 2kz 



_. T,3 ,r y 

 ^ a a 



72f~ / \ 2 /\4 "H 



, - ! + !,! (^) +a 1 , 2 (~) +... 

 a \_ \A./ \A./ J 



a 

 and 



where the constant N 1; N 2 , ... arid M 1( M 2 , ... will be equal to the corresponding 



constants in the equations (35), putting in these equations t = - and q = 'In - = Tec. 



c c 



Neglecting the corrections in the terms of higher order in b/a, we get, using the 

 formula of Lord EAYLEIGH for the wave-length of infinitely small vibrations in three 



