Mit GREEN, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 5 



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. ]\Ioreover, 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 in- 

 versely 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 diffiised, pdv may represent the quantity contained in this element, 

 and if afterwards we write g for the distance between dv and any 

 particle /> under consideration, and these form the quantity 



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 q and tending 

 to increase this line will always be represented by 



(1). 



I^\: 



1-n \dq) ' 



?^, being regarded as a function of three rectangular co-ordinates of 



p, one of which co-ordinates coincides with the line q, and (—7-) 



being the partial differential of V, 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 V mainly 

 depends, it will be convenient to begin with a very simple example. 



Let us therefore suppose that the body ^ is a sphere, whose centre, 

 is at the origin O of the co-ordinates, the radius being 1 ; and p is 

 such a function of x', y', %, that where we substitute for x', y', »' their 

 values in polar co-ordinates 



