74 DEFINITE INTEGRALS. [INT. III. 



be a volume integral, we may define its variation by the equation 



(V + SV)da;dydz, 



where 8V is any arbitrary function of x, y, z multiplied by an in- 

 finitesimal constant e. We may also vary an integral in another 

 manner. Suppose we consider the volume in question to be occu- 

 pied by material substance, and that to each material point 

 belongs a value of the function V. Now let every material 

 point be displaced in any manner by an infinitesimal amount 

 defined by the projections Bx, By, Bz. The material point which 

 arrives at x, y, z brings with it a different value of V, and the 

 value of the integral through the same portion of space, since the 

 latter is filled with different material points, is different. It is to 

 be noticed that this is the exact converse of the process exemplified 

 in 29, 31 for there the functions X, F, Z were associated with 

 fixed points in space, while the integral was over a field which was 

 varied, whereas here the function V goes with the varied point, 

 while the field of integration is fixed. As an example, let us 

 consider the integral 



m = 1 1 1 pdxdydz 



representing the mass of a body r whose density at any point is 

 p, the density being defined as the limit of the ratio of the mass of 

 a portion of the body to its volume, both being decreased in- 

 definitely. Let us consider the mass in an infinitesimal rect- 

 angular parallelepiped, whose sides are dx, dy, dz, and whose 

 mass is dm pdxdydz. When all points are displaced by the 

 amounts Bx } By, Bz, particles in the face normal to the Jf-axis and 

 nearest the origin move to the right a distance Bx, and the volume 

 of new matter that enters the parallelepiped through that face 

 is dydzBx, whose mass is pdydzBx, p and Bx having the values 

 belonging to the face in question. At the opposite parallel face, 

 farthest from the origin, pBx has the value 



and the amount of matter that moves out of the parallelepiped to 

 the right is 



j j ( * d(p&r) , ) 

 ay dz < p ox -I ^ ax j- . 



