SURFACE-TENSION OF WATER BY THE METHOD OF JET VIBRATION. 283 



(^\ 2 

 a ^L + C q = 0. One degree ot 

 Gv 



freedom. Small oscillations. Free motion I and the equation of motion for the 



dissipative system consists in the addition of a frictional term ( a -% + b-^ + cq = 



\ dt 2 tit 



In the present pi-oblem as in all fluid-problems in which a velocity-potential exists 

 for the conservative system, but not for the dissipative one the coefficient of inertia a 

 will not be the same for the two systems, since a in the dissipative system is dependent 

 on the coefficient of viscosity. 



As will be seen from what follows, the correction on the wave-length will not be 

 proportional to S 2 but to 8 :V3 . 



In order to find the variation of the wave-length due to the viscosity, the problem 

 "must be treated in greater detail. Such an investigation is given by Lord RAY LEIGH* 

 for the vibrations of a cylinder of viscous fluid under capillary force, in the case where 

 the original symmetry about the axis of the cylinder is maintained. In the develop- 

 ment to be found in the paper cited above, the assumption mentioned (the symmetry) 

 is, however, from the outset used in such a manner -that the calculation cannot be 

 extended to treat the more general vibrations which will be mentioned here. The 

 result of our development does not include the problem investigated by Lord RAYLEIGH, 

 as, in order to simplify the calculation, special precautions relative to the limiting case 

 (n = 0) are not taken. 



The general equations of motion of an incompressible viscous fluid, unaffected by 

 extraneous forces, are 



and 



in which u, v, w are the components of the velocity, p the pressure, p the density, 

 ju, the coefficient of viscosity, and 



v = + J*L + f*l *L ~ ?. M A r jl w l 



In the problem in question the motion will be steady. Putting w = c + w, and 

 supposing that u, v, and w have the form f(x, y) e^ z , and that u, v, and o> are so small 

 that products of them and quantities of the same order of magnitude can be neglected 

 in the calculations, we get from the equations (1) 



- 19 P. /ai 



T7 */i ' 1 Ir i T7 '/ r \ 



* Lord RAYLEIGH, 'Phil. Mag.,' XXXIV., p. 145, 1892. 

 2 o 2 



