14 THE EQUATIONS OF MOTION. [CHAP. I 



The Lagrangian Equations. 



13. Let a, b, c be the initial co-ordinates of any particle of 

 fluid, x, y, z its co-ordinates at time t. We here consider x, y, z as 

 functions of the independent variables a, b, c, t', their values in 

 terms of these quantities give the whole history of every particle 

 of the fluid. The velocities parallel to the axes of co-ordinates of 

 the particle (a, b, c) at time t are dx/dt, dy/dt, dz/dt, and the 

 component accelerations in the same directions are d-x/dt 2 , d 2 y/dt~, 

 d^zjatf. Let p be the pressure and p the density in the neigh- 

 bourhood of this particle at time t', X, Y, Z the components 

 of the extraneous forces per unit mass acting there. Consider- 

 ing the motion of the mass of fluid which at time t occupies 

 the differential element of volume SxSySz, we find by the same 

 reasoning as in Art. 6, 



d?x _ I dp 

 d? = ~~~pdx' 



^y = Y --^> 



dt 2 p dy 



d*z _ 1 dp 

 dt 2 " ~pdz m 



These equations contain differential coefficients with respect to 

 x, y, z, whereas our independent variables are a, b, c, t. To 

 eliminate these differential coefficients, we multiply the above 

 equations by dx/da, dy/da, dz/da, respectively, and add ; a second 

 time by dx/db, dy/db, dzjdb, and add ; and again a third time by 

 dx/dc, dy/dc, dz/dc, and add. We thus get the three equations 



\dy (d*z \ dz 1 dp _ 

 ~ ^^~ Z ' + ~ 



d? db dtf db dt* db p db ~ 



dx /d*y \dy (d?z \ dz I dp 



- + ~ = 



These are the ' Lagrangian ' forms of the dynamical equations. 



14. To find the form which the equation of continuity 

 assumes in terms of our present variables, we consider the 

 element of fluid which originally occupied a rectangular parallel- 



