1-4] EULERIAN EQUATIONS. 3 



3. The equations of motion of a fluid have been obtained in 

 two different forms, corresponding to the two ways in which the 

 problem of determining the motion of a fluid mass, acted on by 

 given forces and subject to given conditions, may be viewed. 

 We may either regard as the object of our investigations a 

 knowledge of the velocity, the pressure, and the density, at all 

 points of space occupied by the fluid, for all instants; or we 

 may seek to determine the history of any particle. The equations 

 obtained on these two plans are conveniently designated, as by 

 German mathematicians, the 'Eulerian' and the 'Lagrangian' 

 forms of the hydrokinetic equations, although both forms are in 

 reality due to Euler*. 



The Eulerian Equations. 



4. Let u, v, w be the components, parallel to the co-ordinate 

 axes, of the velocity at the point (x, y, z) at the time t. These 

 quantities are then functions of the independent variables x, y, z, t. 

 For any particular value of t they express the motion at that 

 instant at all points of space occupied by the fluid; whilst for 

 particular values of x, y, z they give the history of what goes on 

 at a particular place. 



We shall suppose, for the most part, not only that u, v, w are 

 finite and continuous functions of x, y, z, but that their space- 

 derivatives of the first order (du/dx, dv/dx, dw/dx, &c.) are 

 everywhere finite t ; we shall understand by the term 'continuous 

 motion/ a motion subject to these restrictions. Cases of excep- 

 tion, if they present themselves, will require separate examination. 

 In continuous motion, as thus defined, the relative velocity of 



* " Principes ge"neraux du mouvement des fluides." Hist, de VAcad. de Berlin, 

 1755. 



" De principiis motus fluidorum." Novi Comm. Acad. Petrop. t. xiv. p. 1 (1759). 



Lagrange gave three investigations of the equations of motion ; first, incidentally, 

 in connection with the principle of Least Action, in the Miscellanea Taurinensia, 

 t. ii., (1760), Oeuvres, Paris, 1867-92, t. i.; secondly in his " Memoire sur la ThSorie 

 du Mouvement des Fluides ", Notiv. mem. de VAcad. de Berlin, 1781, Oeuvres, t. iv.; 

 and thirdly in the Mecanique Analytique. In this last exposition he starts with the 

 second form of the equations (Art. 13, below), but translates them at once into the 

 ' Eulerian ' notation. 



t It is important to bear in mind, with a view to some later developments 

 under the head of Vortex Motion, that these derivatives need not be assumed to be 

 continuous. 



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