13-15] 



LAGRANGIAN EQUATIONS. 



15 



epiped having its centre at the point (a, b, c), and its edges 

 &, 56, Sc parallel to the axes. At the time t the same element 

 forms an oblique parallelepiped. The centre now has for its 

 co-ordinates x, y, z\ and the projections of the edges on the 

 co-ordinate axes are respectively 



dx 



dy 



dz ~ 



j- oa ; 

 da 



db 



dx 

 dc 



db db 



dc 



dz 



-T 



dc 



The volume of the parallelepiped is therefore 



or, as it is often written, 



d (a, 6, c) 

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



d (a?, y, z) = ^ 



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



In the case of an incompressible fluid p=p , so that (1) 



becomes 



d (x, y, z) 

 d (a, b, c) 



.(2). 



Weber s Transformation. 



15. If as in Art. 11 the forces X, Y, Z have a potential 

 the dynamical equations of Art. 13 may be written 



___, 

 dt 2 da dt~ da dP da da p da 



