1517.] 



LAGRANGIAN EQUATIONS. 



13 



the particle (a, b, c) at time t are ~ , - , - , and the component 



ctt 



-, ,. ,, j. 



accelerations in the same directions are - , ~j &amp;gt; 



T 

 Let p 



and/o be the pressure and density in the neighbourhood of this particle 

 at time t\ X, Y, Zihe components of the external impressed forces 

 per unit mass acting there. Considering the motion of the mass 

 of fluid which at time t occupies the differential element of volume 

 dxdydz, we find by the same reasoning as in Art. 6, 



d*x 1 dp -\ 



77J = -A -- -j- , 



dt 



dt 



p dx 



1 dp 



p dy 



_ 7 _\dp 

 ~ * J 



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 



, dx dy dz 

 equations by -5- , -- , -3- , respectively, and add ; a second time 



- 



dx dy dz -, 

 db tb&amp;gt; db&amp;gt; and 



3- 



-, . ^ - ^ - ^ dx dy dz 

 ; and agam a thlrd tlme by dc dc dc 



and add. We thus get the three equations 



dx (tfy y\d_l 

 da + (dt* J da 



^z _ \dz +1^-0 

 Sf )d~a + pda~ 



dx 

 ~ 



\dy tfz 

 J dc 



db 

 dz 



p db 



z \dp_ 

 c + dc~ } J 



...(22). 



These are the Lagrangian forms of the dynamical equations. 



17. As before, two additional equations are required. We 

 have, first, a relation between p and p of the form (3), (4), or (5), 

 as the case may be. 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 paral 

 lelepiped having the corner nearest the origin at the point (a, b, c), 

 and its edges di, db. dc parallel to the axes. At the time t the 



