Bodies communicating by a canal 43 



I am not able to answer this Problem accurately, except when the 

 repulsion is inversely as the simple or some lower power of the distance; 

 but I think we may be certain of the following circumstances. 



35] CASE i. Let the repulsion be inversely as some power of the 

 distance between the square and the cube, and let AD be overcharged. 



First, It is certain that the density of the fluid must be everywhere 

 the same, at the same distance from the planes A a and Dd. Secondly, 

 There can be no space as EC, of any sensible breadth, in which the matter 

 will not be overcharged. And thirdly, The fluid close to the planes Aa 

 and Dd will be pressed close together. Whence, I think, we may conclude, 

 that the density of the fluid will increase gradually from the middle of 

 the space to the outside, where it will be pressed close together. Whether 

 the matter exactly in the middle will be overcharged, or only saturated, 

 I cannot tell. 



36] CASE 2. Let the repulsion be inversely as some power of the dis- 

 tance between the square and the simple power, and let AD be overcharged. 



There will be two spaces AB and DC, in which the fluid will be pressed 

 close together, and the quantity of redundant fluid in each of those spaces 

 will be more than half the redundant fluid in A D; so that the space BC, 

 taken all together, will be undercharged; but I cannot tell in what 

 manner the fluid will be disposed in that space. The demonstrations of 

 these two cases are exactly similar to those of the two cases of Prob. 2. 



37] CASE 3. If the repulsion is inversely as the simple or some lower 

 power of the distance, and AD is overcharged, all the fluid will be collected 

 in the spaces AB and CD, and BC will be intirely deprived of fluid. If 

 AD contains just fluid enough to saturate it, and the repulsion is inversely 

 as the distance, the fluid will remain in equilibrio, in whatever manner it 

 is disposed; provided its density is everywhere the same at the same 

 distance from the planes Aa and Dd: but if the repulsion is inversely as 

 some less power than the simple one, the fluid will be in equilibrio, whether 

 it is either spread uniformly, or whether it is all collected in that plane 

 which is in the middle between A a and Dd, or whether it is all collected 

 in the spaces AB and CD; but not, I believe, if it is disposed in any other 

 manner. 



The demonstration depends upon this circumstance; namely, that if 

 the repulsion is inversely as the distance, two spaces AB and CD, repel 

 a particle placed either between them, or on the outside of them, with 

 the same force as if all the matter of those spaces was collected in the middle 

 plane between them. 



It is needless mentioning the three cases in which AD is undercharged, 

 as the reader will easily supply the place. 



