530 Mr. H. S. Allen on the Motion of 



proportional to the weight. It was then found that save for 

 the largest sphere the points lay almost exactly on a straight 

 line (fig 4)*. 



The slope of the straight line on which the observed points 

 fall determines the value of n. The straight line drawn in 

 the diagram is that line passing through the point VI., for 

 which n = 2. Hence it appears that the resistance is propor- 

 tional to the square of the velocity. 



Referring to § 3 we obtain 



U = kpa 2 V 2 , 



indicating that the resistance to the steady motion of a sphere 

 with velocity V is independent of the viscosity of the fluid. 



It should be borne in mind that this does not imply that 

 equal spheres moving with the same velocity in two liquids, 

 one of great the other of small viscosity, necessarily ex- 

 perience the same resistance. For in the more viscous liquid 

 the sphere will require a greater velocity before the regime 

 indicated by the above law can be entered upon. 



The exceptional case of the largest sphere presents some 

 difficulty. The observed values of log aV fall short of those 

 required by the assumption of a resistance proportional to 

 the square of the velocity. It is scarcely possible to suppose 

 that any higher power than the square could be involved, for 

 this would necessitate a resistance decreasing with increasing 

 viseosity. 



Perhaps the simplest explanation is to be found in the 

 supposition that the influence of the walls of the fall-tube 

 and vessel has become appreciable in the case of this sphere, 

 which possesses the greatest diameter and the greatest ter- 

 minal velocity. 



In this connexion it is interesting to recall a remark of 

 Sir I. Newton in a discussion of experiments on the resist- 

 ance experienced by a pendulum oscillating in water. " I 

 found (which will perhaps seem strange) that the resistance 

 in the water was augmented in more than a duplicate ratio 

 of the velocity. In searching after the cause I thought upon 

 this, that the vessel was too narrow for the magnitude of the 

 pendulous globe, and by its narrowness obstructed the motion 

 of the water as it yielded to the oscillating globe " (Mathe- 

 matical Principles of Natural Philosophy, Book II. Sect. vi.). 



Assuming that n = 2 we can determine the value of the 



* In the diagram (fig. 4) observations in the small vessel are indicated 

 by crosses, those in the large vessel by points. The latter are the more 

 reliable (see § 8). 



