550 M. OSTROGRADSKY ON A SINGULAR CASE OF 



and the variation of the volume will consequently become 



rrfXrfYrfZ_rfXrfYrfZ rfXrfYrfZ_rfXrfYrfZ 

 \_dx dy dz dx dz dy dy dz dx dy dx dz 



rfXrfYrfZ_^^rfZ_-| 



dz dx dy dz dy dx J ^ 



by substituting for X, Y, Z their values a; + Ja:, y + Jy, 2 + J z, and 

 rejecting (on the same ^irinciple as they are rejected in the differential 

 calculus) all the infinitely small quantities, except those of the lowest 

 order, we have as the variation of the volume 



(dSx,dSydSz\ , , 

 -dx+~d^ +-d^)d^dydz: 



or, if we consider the possible displacement alone, the volume can only 

 increase or continue unchanged. The foregoing variation must then be 

 either zero or positive for all the possible displacements, and must be so 

 whatever be the volume under consideration, that is to say, whatever 

 be the limits of the integral 



^(m-"^*"-^)'''^"'- 



which cannot be the case unless we have 



d^ X dl y dl z 



d X dy d z 



for all possible displacements. We might have employed the polar or 

 any other coordinates whatever. We might liiiewise, if it were neces- 

 sary, express the invariability of a portion of a mass, &c. 



The geometers who have treated of the equilibrium of fluids in Euler's 

 manner have considered the equilibrium of the differential parallelopi- 

 peds also, but the equilibrium might be determined for any portion of 

 the volume, whether finite or infinitely small. Let us imagine, in the 

 interior of the liquid, any volume at pleasure. The condition of equi- 

 librium of this volume must be established in virtue of the moving forces 

 applied to it and of the pressures on its surface. If we employ dm to 

 represent an element of its mass answering to the coordinates x, y, z, 

 and X, Y, Z to represent the accelerative forces parallel to the coordi- 

 nate axes, the moving forces will be X c? m, Y d m, Tidm, and even 

 other elements will be acted upon by similar forces. 



This being supposed, let p be the pressure at the surface of the vo- 

 lume in question. \i d s represents an element of this surface, and 

 X, /*, V the angles formed with the coordinate axes by the normal to 

 d s produced beyond the volume, p d s will be the pressure sustained 

 by the element d s, and — p cos. \ds, — p cos fi.ds, — pcosvds 

 the components of that pressure. Now, each element of the volume 



