34 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 



1 dV 



.-^- = the force directed parallel to the axis Z. 



1 — » d% 



But since, in consequence of the equilibrium, each of these forces is 

 equal to zero, we shall have 



„ dV J . dV , , dV . .-, 

 = -5— dx + -7— dy + -J- d% = dV\ 

 dx dy d% 



and therefore, by integration, 



F = const. 



Having thus the value of V at the point p, whose co-ordinates are 

 X, y, %, we immediately deduce, by the method explained in the fifth 



article, 



/w-2 \ 



sm 



P = 



2' 



^.(l-r'*) 



seeing that in the present case the general expansion of K there given 

 reduces itself to 



If moreover Q serve to designate the total quantity of free fluid in 

 the sphere, we shall have, by substituting for 



sin f TT j its value 



rl^)r[^y 



\ 2 / rrz-ij >i.Ji/i '2\""S~ ^"^ 



sm 



Q = /pe/«; = ^5 i F/liW^dril-r") 



See Legendre. Exer. de Cal. Int. Quatrieme Partie. 



In the preceding values, as in the article cited, the radius of the 

 sphere is taken for the unit of space ; but the same formula may 



readily be adapted to any other unit by writing — in the place of r', 



and recollecting that the quantities p, V, and Q, are of the dimensions 

 0, 4 — «, and 3 respectively, with regard to space; a being the number 



