76 EQUILIBRIUM OF MUTUALLY ATTRACTIVE 



quired to adapt the method to the case of a fluid whose particles 

 are mutually attractive. Clairaut first supposes a mass of fluid 

 in equilibrium, and conceives an infinitesimal stratum added 

 to it, which shall produce equable pressure over the whole sur- 

 face ; the equilibrium of the original mass A will not be dis- 

 turbed, and the increased mass A + SA will be in equilibrio, 

 when the forces acting on its surface are normal to it. This 

 principle, that forces acting on a free surface must be normal 

 to it, was laid down by Huygens, and is confessedly true. By 

 a repetition of this process, the original mass can be enlarged 

 to any extent ; and the condition that the nucleus must be in 

 equilibrio becomes, Mr Ivory observes, unnecessary, by con- 

 ceiving it diminished sine limite. The mathematical condition 

 of equilibrium is, therefore, the expression of the possibility of 

 adding a stratum which shall produce equable pressure, and at 

 the free surface of which the forces shall be normal to it. 



Let us endeavour to put this symbolically. Let the force 

 at the original free surface be F\ at the point #, y, z produce 



the normal, and take a length on it = ^, &> being infinitesimal : 



thus we get a stratum producing an equal pressure o>. 



Let f (x, y , z) = c be the equation of the free surface ; then 

 F being a function of (a:, y, z), all that is requisite for the force 

 at any point of the new free surface to be normal is, that 



shall be its equation. 



Let F= J(fx)*+(f'yf+(f*)* ; then 



5, &> fx ~ G> fy . a> fz 



" = 



and f(x, y,z) = c =/(*', y', z') - [fxSx +fySy +/ 

 (where x' = x+ &) by Taylor's theorem ; 



therefore c =/(*', 3,', *') -1 



therefore c = c + Sc 0)-=,: 



