32G Mr, H. S. Allen on the Motion of 



some power of the velocity, the index having a value between 

 1"7 and 2, depending on the material of the pipe. 



3. Lata of Resistance to the Movement of a Sphere ivhen the 

 Motion is Sinuous. 



When the velocity of the sphere moving uniformly in an 

 infinite fluid becomes so great that eddying motion is set up, 

 we have no means of determining theoretically the way in 

 which the resistance varies. In the parallel case of the flow 

 of liquids through pipes the empirical results show that the 

 resistance may be taken proportional to some power of the 

 velocity. If we assume that the resistance, R, to the motion 

 of a solid sphere may in like manner be taken proportional to 

 V", so that it may be represented by a single term., 



R=ka x pVv z Y n , 



we may employ the principle of dynamical similarity to deter- 

 mine the form of the indices x, y., z. 



The corresponding " dimensional " equation is 



ML T /MV/L 5 



t^)W(r)*- 



Since tlhe quantities M, T, and L must occur to the same 

 degree on both sides of this equation, we must have 



yielding 



Thus 



B,=kayv 2 - n V u 



= kpv 2 ~ n (aY) . 



When the sphere has attained its terminal velocity, the 

 effective weight is exactly balanced by the fluid resistance. 

 If V now represent this terminal velocity, 



k P i> 2 - n {aY) n = §7rg{<T-p)a 2 ; 



and therefore V is proportional to 



3 — x 2 -l 

 an /J/» 



