THE EQUILIBRIUM OF FLUIDS. 153 



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

 is equal to zero, we shall have 



dV , dV , , dV , ,, 7 



= -j- dx + - T - dy + -^- dz = d V\ 

 dx dy dz 



and therefore, by integration, 



V const. 



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

 nates are x, y> z, we immediately deduce, by the method ex- 

 plained in the fifth article, 



^- 2 A 

 m ( ^ 



sin 

 P = 



&'n 



seeing that in the present case the general expansion of F there 

 given reduces itself to 



V F (0) . 



If moreover Q serve to designate the total quantity of free 

 fluid in the sphere, we shall have, by substituting for 



. /n-2 \ . A , TT 



sin r TT its value 



ON .. N , 



r( n "" 2 )r( 4 "" n ) 





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 for- 



mulae may readily be adapted to any other unit by writing 



i 



- in the place of r', and recollecting that the quantities p, F, 

 a 



and Q, are of the dimensions 0, 4 w, and 3 respectively, with 

 regard to space ; a being the number which represents the radius 

 of the sphere when we employ the new unit. In this way we 

 obtain 



