413 



caused to press upward against the water In order to accelerate It; the water 

 loads the sphere, In fact, with an equivalent mass equal to half the mass M 

 of the displaced water, and an upward force F = Ma/2 must act on the water 

 where o is the acceleration of the sphere. This force imparts upward momen- 

 tum jFdt to the water. 



Yet if the total amount of upward momentum is calculated from the 

 usual formulas, for the water lying within any given distance r of the center 

 of the globe, the result is zero. The water around the sides of the gas 

 globe moves downward as that above and below it moves upward, and, as re- 

 gards the water inside of any spherical surface concentric with the gas globe, 

 the downward momentum on the sides Just cancels the upward momentiam above and 

 below the sphere. The question thus arises, what has become of the upward 

 momentum imparted to the water by the upward force F? 



The paradox Is redoubled by the following consideration. Since 

 the sphere moves upward, water must on the whole move downward. The total 

 momentum in the water must, therefore, be directed downward, not upward. It 

 is easily shown that this downward momentum is, in fact, of magnitude ZjFdt, 



The solution of the paradox is found upon careful consideration to 

 lie in the occurrence of a decrease in the pressure near the bottom. As a 

 result of the motion of the sphere, the upward force of the bottom on the 

 water is decreased by 3F. One-third of this decrease compensates for the 

 lifting effect of the sphere caused by its upward motion and thereby absorbs 

 the upward momentum given by it to the water; the remaining two-thirds of the 

 decrease allows part of the downward force due to gravity to develop in the 

 water downward momentum of magnitude 2jFdt. 



These considerations are based, of course, upon the assumption of 

 incompressible water. If the action is so rapid, or the body of water so 

 large, that non-compressive theory is not adequate to describe the motion 

 throughout, then part or all of the upward momentum given to the water will 

 remain in it, although perhaps not in the neighborhood of the sphere. 



The motion of the water around a moving spherical gas globe of 

 fixed radius is exactly the same as around a sphere of equal size moving at 

 the same rate, hence the same considerations apply to the motion of the gas 

 globe. The statements made lA this report have been carefully worded so as 

 to be correct as they stand; caution must be used if the references to mo- 

 mentum are altered or extended. 



