THE EQUILIBRIUM OF FLUIDS. 123 



distributes itself in the latter case, has long since been deter- 

 mined experimentally by Coulomb. We have thus had the 

 advantage of comparing our theoretical results with those of a 

 very accurate observer, and the differences between them are not 

 greater than may be supposed due to the unavoidable errors of 

 experiment, and to that which would necessarily be produced by 

 employing plates of a finite thickness, whilst the theory supposes 

 this thickness infinitely small. Moreover, the errors are all of 

 the same kind with regard to sign, as would arise from the 

 latter cause. 



1. If we conceive a fluid analogous to the electric fluid, but 

 of which the law of the repulsion of the particles instead of 

 being inversely as the square of the distance is inversely as some 

 power n of the distance, and suppose p to represent the density 

 of this fluid, so that dv being an element of the volume of a 

 body A through which it is diffused, pdv may represent the 

 quantity contained in this element, and if afterwards we write g 

 for the distance between dv and any particle p under considera- 

 tion, and these form the quantity 



v _pdv, 

 " 



the integral extending over the whole volume of A, it is well 

 known that the force with which a particle p of this fluid situate 

 in any point of space is impelled in the direction of any line $ 

 and tending to increase this line will always be represented by 



1 



1 n \dq 



V being regarded as a function of three rectangular co-ordinates 

 of p, one of which co-ordinates coincides with the line ^, and 



being the partial differential of F, relative to this last co- 

 ordinate. 



In order now to make known the principal artifices on which 

 the success of our general method for determining the function F 

 mainly depends, it will be convenient to begin with a very 

 simple example. 



