>(M) 



or PRESSURE AND EQUILIBRIUM. 107 



The variations S^, hj, ^z being considered as arbitrary 

 and independent, we may substitute, in the equation V^u 

 zizlS^s, for the coordinates Xy y, and z, three other quan- 

 tities depending on them : and then make the coefficients 

 of the variations of these quantities equal to nothing. Thus 

 if we take ^ , the radius drawn ^;.^ 

 from the origin of the co- 

 ordinates to the projection of 

 the point M on the plane of ^ 

 X and y, and let 'zsr be the 

 angle formed by f with the 

 direction of x, we shall have 

 xiz^ cos 'sr, and y^^f sin 'sr : and we may proceed to con- 

 sider u, and the values of *, as depending on f, 'zsr, and z, 



and take the variation V ^r~ ^^t~' W [This supposition 



is equivalent to taking the variation of the place of M by 

 making it move in a plane parallel to that of x and y, while 

 it remains at an equal distance from the origin of the co- 

 ordinates, the element of its motion, or its variation, being 

 f S'ar,] and the force V, so reduced to this direction, becomes 



obviously F -^j— (250). Again, if V' be the portion of F, 



"which acts in the plane of j; and y, and p be a perpendicu- 

 lar falling on its direction from the axis perpendicular to 

 X and y, passing through the origin of the coordinates, the 

 portion of F, which acts in the direction of the element 



fS'ar, will be ^ F, consequently — V'zuV -^, andp F'=: 



l^^T"' It follows, therefore, from the definition of rota- 



o'sr 



tory pressure (254), that F -r— is the rotatory pressure of 



