THE EQUILIBRIUM OF FLUIDS. 171 



In the foregoing expressions the radius of the sphere has 

 been taken as the unit of space, but it is very easy thence to 

 deduce formulae adapted to any other unit, by recollecting that 



, ^, j~^ and yr^j, are quantities of the dimensions 0, 1, 



1 and 3 Ti respectively with regard to space: for if b re- 

 presents the sphere's radius, when we employ any other unit 



M M /- //7? fit 



we shall only have to write, 7 , -r , -r- , -W and r in the place 



b b o o o 



of r, r, B, dv l and a, and afterwards to multiply the resulting 

 expressions by such powers of Z>, as will reduce each of them to 

 their proper dimensions. 



If we here take the formula (22) of the present article as an 

 example, there will result, 



sinpjz!^ ^ 2 _ s - 



for the value of the density which would be induced in a sphere 

 A, whose radius is b } by the action of any exterior bodies what- 

 ever. 



When n>2, the value of p or of the density of the free fluid 

 here given offers no difficulties, but if n<2, we shall not be 

 able strictly to realize it, for reasons before assigned (Art. 6 

 and 7). If however n is positive, and we adopt the hypothesis 

 of two fluids, supposing that the quantities of each contained by 

 bodies in a natural state are exceedingly great, we shall easily 

 perceive by proceeding as in the last of the articles here cited, 

 that the density given by the formula (23) will be sensibly 

 correct except in the immediate vicinity of A's surface, provided 

 we extend it to the surface of a sphere whose radius is b $b 

 only, and afterwards conceive the exterior shell entirely de- 

 prived of fluid : the surface of the conducting sphere itself having 

 such a quantity condensed upon it, that its density may every 

 where be represented by 



r>, 



p = 



