THE EQUATIONS OF MOTION. 



[CHAP. i. 



same element forms an oblique parallelepiped. The corner cor 

 responding to (a, b, c) has for its co-ordinates x, y, z\ and the 

 co-ordinates relative to this point of the other extremities of the 



three edges meeting in it are respectively -,- da , -^ da, -j-da : 



J da da da 



dx 77 dy 77 dz 77 dx 7 dy -, dz , 



-rj- db, -if db, ~jj db ; -y- do, -f- dc, -j- ac. The volume of the par- 



db^ do 

 allelepiped is therefore* 



dc dc 



-jj dadbdc, 



or, as it is often written, 



dadbdc. 



d (a, b, c) 

 Hence, since the mass of the element is unchanged, we have 



pJ^|L| =po (23)&amp;gt; 



where p is the initial density at (a, 6, c). 



In the case of an incompressible fluid p = p ot so that (23) 

 becomes 



d. (or*. 11 ^ 



(24). 



d (a, 6, c) 



Weber s Transformation. 



18. If the forces X, Y, Zh&ve a potential, i.e. if they can be 

 expressed as the partial differential coefficients with respect to 

 x, y, z of a single function which we denote by F (so that V is 

 the potential energy, due to those forces, of unit mass placed in the 

 position (#, ?/, z)), the equations (22) may be written 



tfx dx d^ydy d*zdz__ __ dV__ 1 dp 

 dt 2 da dt? da dt 2 da da p da 

 &c., &c. 



* Salmon, Geometry of Three Dimensions. 



