190 RESEARCHES OX FUNGI 



APPENDIX 



THE MOTION OF A SPHERE IN A VISCOUS MEDIUM 

 Contributed by Dr. GuY I3arlow. 



(The notation is the same as that employid in Chapters X V. and X V II.) 



As shown by Stokes, the resisting force on a sphere of radius a when moving 

 Avith velocity v is given by 



Y = &irtJ.av (1) 



Since the force is directly proportional to the velocity, it is evident that the 

 component of this force in any direction is also directly proportional to the 

 component of the velocity in that direction. The motion of the sphere when 

 projected under gravity can therefore be regarded as com2)ounded of tlie 

 inde})endent horizontal and vertical motions, and these may be conveniently 

 investigated separately. 



1. Fall f7'om rest under gravity. 



The equation of motion is — 



dv 

 m ^ = riuf - btr/mav, 



where ni is the mass of the sphere and v its velocity downwards at time t. 

 The density o- of the medium is here neglected. 

 This equation may be written 



dv 



dt^^-'' (2) 



Avhere 



c = — . 

 m 



When the steady terminal state is reached, -^ — o, v = V, hence from (2), 



V-^ (3) 



c 



for the teniiiniil velocity 



Substituting value of c and putting m = ^d^p we obtain Stokes' expression 



Equation (2) may now be written — 



dv 



dt = '^''-''^- 



Integration with initial condition r = o when t = o gives 



f = V(l-e-") (5) 



