Note 3: canals of incompressible Jiuid 365 



NOTE 3, ART. 69. 

 On canals of incompressible fluid. 



It appears from several passages (Arts. 40, 236, 273, 276, 278, 294, 348) 

 that Cavendish considered that the weakest point in his theory was the assump- 

 tion that the condition of electric equilibrium between two conductors connected 

 by a fine wire is the same as if, instead of the wire, there were a canal of in- 

 compressible fluid defined as in Art. 69. 



It is true that the properties of the electric fluid, as defined by Cavendish 

 in Art. 3, are very different from those of an incompressible fluid. But it is 

 easy to show that the results deduced by Cavendish from the hypothesis of 

 a canal of incompressible fluid are applicable to the actual case in which the 

 bodies are connected by a fine wire. 



In what follows, when we speak of the electrified body or bodies, the canal 

 or the wire is understood not to be included unless it is specially mentioned. 



Cavendish supposes the canal to be everywhere exactly saturated with the 

 electric fluid, and that the only external force acting on the fluid in the canal 

 is that due to the electrification of the other bodies. 



Since this resultant force is not in general zero at all points of the canal, 

 the fluid in the canal cannot be in equilibrium unless it is prevented from 

 moving by some other force. Now the condition of incompressibility excludes 

 any such displacement of the fluid as would alter the quantity of fluid in a 

 given volume, and the stress by which such a displacement is resisted is called 

 isotropic (or hydrostatic) pressure. In a hypothetical case like this it is best, 

 for the sake of continuity, to suppose that negative as well as positive values 

 of the pressure are admissible. 



In the electrified bodies themselves the properties of the fluid are those 

 defined in Art. 3. The fluid is therefore incapable of sustaining pressure except 

 when its particles are close packed together, and as it cannot sustain a negative 

 pressure, the pressure must be zero in the electrified bodies, and therefore also 

 in the canal at the points where it meets these bodies. 



The condition of equilibrium of the fluid in the canal is 



dV dp 



P -j- + j = . 

 r ds ds 



where V denotes the potential of the electric forces due to the electrified bodies, 

 p the density, and p the pressure of the fluid in the canal, and s the length of 

 the canal reckoned from a fixed origin to the point under consideration. 



Since by the hypothesis of incompressibility, p is constant, 



pV + p = C, 



where C is a constant ; and if we distinguish by suffixes the symbols belonging 

 to the two ends of the canal where it meets the bodies A v and A t , 



