OF FLUIDS ON THE MOTION OF PENDULUMS. [53] 



is much larger than n, which refers to air, although /a' is larger than p*. Hence the results 

 will be nearly the same as if we had taken /m = °° , or regarded the sphere as solid. 



If, however, instead of a globule of liquid descending in a gas we have a very small bubble 

 ascending in a liquid, we must not treat the bubble as a solid sphere. We may in this case 

 also neglect the motion of the fluid forming the sphere, but we have now arrived at the other 

 extreme case of the general problem, and the two equations of condition which have to be 

 satisfied at the surface of the sphere are that R = and T e = when r = a, instead of R = and 

 = 0, when r = a. 



The equation of condition T e = which applies to a bubble, as well as the fourth of equa- 

 tions (128), will not be the true equations, if forces arising from internal friction exist in the 

 superficial film of a fluid which are of a different order of magnitude from those which exist 

 throughout the mass. At the end of the memoir already referred to, Coulomb states that in 

 very slow motions the resistance of bodies not completely immersed in a liquid is much greater 

 than that of bodies wholly immersed, and promises to communicate a second memoir in con- 

 tinuation of the first. This memoir, so far as I can find out, has never appeared. Should the 

 existence of such forces in the superficial film of a liquid be made out, the results deduced 

 from the theory of internal friction will be modified in a manner analogous to that in which the 

 results deduced from the common principles of hydrostatics are modified by capillary attraction. 

 It may be remarked that we have nothing to do with forces of this kind in considering the 

 motion of pendulums in air, or even in considering the oscillations of a sphere in water, except 

 as regards the very minute fraction of the whole effect which relates to the resistance experienced 

 by the suspending wire in the immediate neighbourhood of the free surface. 



It may readily be seen that the effect of a set of forces in the superficial film of a liquid 

 offering a peculiar resistance to the relative motion of the particles would be, to make the re- 

 sistance of a gas to a descending globule agree still more clearly with the result obtained by 

 regarding the globule as solid, while the resistance experienced by an ascending bubble would 

 be materially increased, and made to approach to that which would belong to a solid sphere of 

 the same size without mass, or more strictly, with a mass only equal to that of the gas forming 

 the bubble. Possibly the determination of the velocity of ascent of very small bubbles may 

 turn out to be a good mode of measuring the amount of friction in the superficial film of a 

 liquid, if it be true that forces of this kind have any existence. But any investigation relating 

 to such a subject would at present be premature. 



45. Let us now attempt to determine the uniform motion of a fluid about an infinite 

 cylinder. Employing the notation of Section III, and reducing the problem to one of steady 

 motion as in Art. 39, we obtain the same equations of condition (116), (117), as in the case of 

 the sphere. Assuming ^ = sin 0F(r), and substituting in the equation obtained from (69) by 



transforming to polar co-ordinates and leaving out the terms which involve — , we get 



U t 



Id 1 Id 1\» „ 



{^ + rTr-?) F ^ = ° <*« 



The integral of this equation may readily be obtained by multiplying the last term of the 



