of Viscous Liquid under Capillary Force. 153 



We shall see that this corresponds to the maximum insta- 

 bility, and it occurs when the wave-length of the varicosity 

 is very large in comparison with the diameter of the cylinder. 

 The following table gives the values of (34) for specified 

 values of x : — 



00 

 0-2 

 0-4 



0-6 



(34). 



-3-0000 

 -3-0000 

 -3*0004 

 -3-0023 



1-0 

 2-0 

 4-0 

 6-0 



(34). 



-3-0188 

 -3-2160 

 -4-458 

 -6-247 



It will be seen that the numerical value of (34) is least when 

 x = 0, which is also the value of x for which the numerator of 

 (31) is greatest. On both accounts, therefore, in is greatest 

 when x or ka = 0. But over the whole range of the insta- 

 bility from Jca = to &a = l, (34) differs but little from —3, 

 so that we may take as approximately applicable 



T(l-W) 

 6fj,a 



(36) 



The result of the investigation is to show that when vis- 

 cosity is paramount long threads do not tend to divide 

 themselves into drops at mutual distances comparable with 

 the diameter of the cylinder, but rather to give way by 

 attenuation at few and distant places. This is, I think, in 

 agreement with the observed behaviour of highly viscous 

 threads of glass or treacle when supported only at the ter- 

 minals. A separation into numerous drops, or a varicosity 

 pointing to such a resolution, may thus be taken as evidence 

 that the fluidity has been sufficient to bring inertia into play. 



The application of (31) to the case of stability (ka>l) is 

 of less interest, but it may be worth while to refer to the 

 extreme case where the wave-length of the varicosity is 

 very small in comparison with the diameter. We then fall 

 upon the particular case of a plane surface disturbed by waves 

 of length X. The result, applicable when the viscosity is so 

 great that inertia may be left out of account, is the limit of 

 (31) when a, or x, is infinite, while k remains constant, or 



Tk 

 in=^r--7- Lim x{ J 2 (ix) fiiiix) + 1 } . 



JifJb 



By means of the expressions appropriate when the argument 

 is large, the limit in question may be proved to be — 1 ; so 

 that 



Tk 



