25.] FUNDAMENTAL ASSUMPTIONS. 3 



Let p, p l9 p , p s denote the intensities of the stresses* across the 

 faces ABC, PBC, PCA, PAB, respectively, of the tetrahedron 

 PABC. If A be the area of the first-mentioned face, the areas 

 of the others in order are ZA, ??iA, ?iA. Hence if we form the 

 equation of motion of the tetrahedron parallel to PA we have 



p l JA=pl. A, 



where we have omitted the terms which express the rate of 

 change of momentum, and the component of the external im 

 pressed forces, because they are ultimately proportional to the 

 mass of the tetrahedron, and therefore of the third order of small 

 quantities, whilst the terms retained in the equation of motion 

 are of the second. We have then, ultimately, p =p v and similarly 

 p=Pt=p a , which proves the theorem. 



4. 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 know 

 ledge 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 each individual 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 f. 



The Eulerian Forms of the Equations. 



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

 axes, of the velocity at the point (#, y, z] at the time t. These 

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

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



* Reckoned positive when pressures, negative when tensions. Ordinary fluids 

 are, however, incapable of supporting more than an exceedingly slight degree of 

 tension, so that p is nearly always positive. 



t Principes g&ieraux du mouvemeut des fluides. Hist, de VAcad. de Berlin, 

 1755. 



De principiis motus fluidorum. Novi Comm. Acad. Pctrop. t. 14, p. 1, 1759. 



Lagrauge starts in the Mecanique Analytique with the second form of the 

 equations, but transforms them at once to the Eulerian form. 



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