40 Mr GREEN, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 



/(or', y\ «')=/'(") = ^„ + P,r'^ + ^.r'^ + &c. = if„ (l - ^']", 

 and p = (1 - O ^ ./(x', y', %') = ^ sin {—^ ^) . («= - 1)"... 



(«^ - /'')-Ml - 0~- 



In the value of p just exhibited, the radius of the sphere is taken 

 as the unit of space, but the same formula may easily be adapted to 



any other unit by writing j and y- in the place of a and / respectively, 



and recollecting at the same time that in consequence of the equation 



•dv.p . rdaP' 



const, 



= r^r^j!^ + ji 



s -^ g 



before given, ^ , is a quantity of the dimension — 1 with regard to 



space: h being the number which represents the radius of the sphere 

 when we employ the new unit. Hence we obtain for a sphere whose 

 radius is bg, acted upon by an exterior concentric spherical surface 

 of which the radius is a, 



2P'a.sin {—-■"] 2-n ^ 



(/3) p = -^ ff-b') ' {a'-r")-' {b'-r") ^ ; 



7r 



P' being the density of the fluid on the exterior surface. 



If now we conceive a conducting sphere A whose radius is a, and 

 determine P' so that all the fluid of one kind, viz. that which is re- 

 dundant in this sphere, may be condensed on its surface, and afterwards 

 find b the radius of the interior sphere S from the condition that it 

 shall just contain all the fluid of the opposite kind, it is evident that 

 each of the fluids will be in equilibrium within A, and therefore the 

 problem originally proposed is thus accurately solved. The reason for 

 supposing all the fluid of one name to be completely abstracted from 

 S, is that our formulas may represent the state of permanent equilibrium, 

 for the tendency of the forces acting within the void shell included 

 between the surfaces A and B, is to abstract continually the fluid of 

 the same name as that on ^'s surface from the sphere S. 



