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 i/v 2 w, .... 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 (t/ = 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 u/i/ is small. For a 

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



p = 2p, y=-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 = -00017. 



This gives a terminal velocity u = 1280000 a 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 



av_o ^Y-9 \ 



=z -y*-y-fr&amp;gt;} 

 &quot; = *% 2 -* %r 



w = y j # = 

 dx dy 



.(xvii), 



where 



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

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



