295] RESISTANCE. 533 



energy of the fluid both infinite* The steady motion here investigated 

 could therefore only be fully established by a constant force X acting on the 

 sphere through an infinite distance. 



The whole of this investigation is based on the assumption that the 

 inertia- terms udu\dx, ... in the fundamental equations (4) of Art. 286 may 

 be neglected in comparison with j/v 2 ^, .... It easily follows from (iv) above 

 that ua must be small compared with v. This condition can always be 

 realized by making u or a sufficiently small, but in the case of mobile fluids 

 like water, this restricts us to velocities or dimensions which are, from a 

 practical point of view, exceedingly minute. Thus even for a sphere of a 

 millimetre radius moving through water (i/='018), the velocity must be 

 considerably less than -18 cm. per sec.f. 



We might easily apply the formula (xiv) to find the ' terminal velocity ' of 

 a sphere falling vertically in a fluid. The force X is then the excess of the 

 gravity of the sphere over its buoyancy, viz. 



where p denotes the density of the fluid, and p the mean density of the 

 sphere. This gives 



This will only apply, as already stated, provided ua/v is small. For a 

 particle of sand descending in water, we may put (roughly) 



Po=2p, v=-018, # = 981, 



whence it appears that a must be small compared with -0114 cm. Subject to 

 this condition, the terminal velocity is u = 12000 a 2 . 



For a globule of water falling through the air, we have 

 Po = l } p = -00129, /i = ' 



This gives a terminal velocity u = 1280000 2 , subject to the condition that a 

 is small compared with '006 cm. 



2. The problem of a rotating sphere in an infinite mass of liquid is 

 solved by assuming 



C?Y_9 d\_c>. v 



.(xvii), 



where v^^Az/r 3 ( xvn i)> 



* Lord Bayleigh, Phil. Mag., May 1886. 

 t Lord Rayleigh, 1. c. ante p. 526. 



