506 MR. IVORY ON THE EQUILIBRIUM OF A MASS OP 



wherefore, p denoting the intensity of pressure, we obtain 



, rdm , rdm , rdm 



J ~r J ~r J ~T 



dp=^—jJ-dx+ ^/ dy-\-—^-dz-\'2{ydy^zdz)', 

 and by integrating;, 



/'=/x+(i^^ + ^0-C: . (1.) 



and from this the equation of the surface of the fluid is derived by making p = 0, viz. 

 = [f^f] + -1 (2,2 + ^2) _ c, . (2.) 



the brackets signifying that the inclosed integral is deduced from an attracted par- 

 ticle in the surface of the fluid. 



The integral / —r- in the last equations is the sum of all the molecules of the fluid 



mass divided by their respective distances from the attracted point. In equation (1.) 

 the pressure p varies through all gradations of magnitude, from zero to the maximum 

 value at the centre of gravity. The exact import of this equation is therefore attended 

 with no difficulty. 



The integral for a point or particle in the surface of the fluid, distinguished by 

 brackets in equation (2.), is a particular or singular value of the general integral. 

 When the attracted point is within the surface, the value of the integral depends not 

 only upon the coordinates of that point, but also upon the limits of the integrations, 

 which are determined by the equation of the surface of the fluid ; but when the at- 

 tracted point is in the surface, the expression of the integral is more simple, because 

 it involves only the coordinates of the surface. The particular integral is obtained 

 from the general one by substituting the coordinates of the surface ; but the integral 

 for a point within the surface cannot be derived by any change of coordinates, from 

 the modified and singular form which the expression assumes at the surface. From 

 this it follows ihat the level surfaces deduced from the equation of the upper surface 

 of the fluid, are different from the interior surfaces determined by making p constant 

 in equation (1.). 



Relatively to the linear dimensions of the mass of fluid c? m is a quantity of three 



dimensions, and therefore the integral^ -^ being extended to all the molecules of 



the mass, is only of two dimensions. Let x, y, z, R represent the coordinates of an 

 attracted point in the surface of the fluid, and the radius drawn from the same point 

 to the centre of gravity ; further, assume 







and the quantities «, b, c will depend only upon the angles that determine the di- 



