10 THE EQUATIONS OF MOTION. [( HAP. 1. 



upper limit of integration in (25). We find % = V -- , where tz 



is the impulsive pressure and V the potential of the external 

 impulsive forces at the point (a, ft, c). Since x = a, y b, z = c, 

 we have, by (26), 



cU dT Id 



dt~ U &quot; ~ da pfa\- 



which agree with the equations of Art. 15. 



20. In the method of Art. 16 the quantities a, b, c need not 

 be restricted to mean the initial co-ordinates of a particle ; they 

 may be considered to be any three quantities which serve to 

 identify a particle, and which vary continuously from one particle to 

 another. If we thus generalize the meanings of a, b, c, the form of 

 equations (22) is not altered ; to find the form which (23) assumes, 

 let # , ?/ , Z Q now denote the initial co-ordinates of the particle to 

 which a, 6, c refer. The initial volume of the parallelepiped, three 

 of whose edges are drawn from the particle (a, b, c) to the particles 

 (a -I- da, 6, c), (a, b -f db, c), (a, b, c + dc), respectively, is 



d (a, 6, c) 

 so that instead of (23) we have 



d (a?, y, z) _ d (.r , 

 ~~ 



and for incompressible fluids 



d(a,b,c) 



21. If we compare the two forms of the fundamental equations 

 to which we have been led, we notice that the Eulerian equations 

 of motion are linear and of the first order, whilst the Lagrangian 

 equations are of the second order, and also contain products of 

 differential coefficients. In Weber s transformation the latter are 

 replaced by a system of equations of the first order, and of the 

 second degree. The Eulerian equation of continuity is also much 

 simpler than the Lagrangian, especially in the case of liquids. In 

 these respects, therefore, the Eulerian forms of the equations 

 possess great advantages over the Lograngian. Again, the form in 



