13-15] 



LAGRANGIAN EQUATIONS. 



15 



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

 Sa, 8b, &c 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 ~ 

 da 



f&a, 

 da 



dz_ 

 da 



dz 



db 



dx 

 dc 



db 



T 

 dc 



The volume of the parallelepiped is therefore 



Sa&bSc, 



or, as it is often written, 



a, b, 



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



d (x, y, z) _ 



P J ( A ~\ ~ P Q ' 



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



In the case of an incompressible fluid p=po, 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 H, 

 the dynamical equations of Art. 13 may be written 



d?xdx d-^dy ___ 



~dtf da dt 2 da d^da~~^da~da' 



