186, 187] EQUATION OF CONTINUITY. 499 



for the dilatation found in equation 75) 169, the divergence, which 

 we shall still call <?, being now the time rate of dilatation. Accord- 

 ingly the equation of continuity is purely kinematical in character 

 and expresses the conservation of mass of every part of the .fluid. 

 If the fluid is incompressible, Q is constant, and consequently 



that is, the velocity of an incompressible fluid is a solenoidal vector. 

 This is the property that give the name to such vectors, and we 

 see, as in 117, that the flux across every cross -section of a tube 

 of flow is constant. 



Besides the three dynamical and one kinematical equation there 

 will be a physical equation involving the nature of the fluid, giving 

 the relation connecting the density with the pressure,- 



making five equations to determine the five functions u,v, 

 of the four variables x^y^z^t. 



We have here made use of two distinct methods. In one we 

 fix our attention on a definite point in space and consider what 

 takes place there as diiferent particles of fluid pass through it. This 

 is called the statistical method, for by the statistics of all points we 

 get a complete statement of the motion. This method is commonly 

 associated with the name of Euler and the equations 6) are called 

 the Eulerian equations of motion. The second method consists in 

 fixing our attention upon a given particle and following it in its 

 travels. In this we use the notation of ordinary derivatives. This 

 is called the historical or Lagrangian method. Obviously if we know 

 the history of all particles we also have a complete representation 

 of the motion. Both methods are due to Euler. We shall not here 

 make use of the Lagrangian equations and shall therefore not write 

 them down. The student will find them in the usual treatises on 

 Hydrodynamics of which Lamb's and Basset's Hydrodynamics, Kirch- 

 hoffs Dynamik and Wien's Hydrodynamik may be especially com- 

 mended. 



187. Hamilton's Principle. We shall now deduce the equa- 

 tions by means of Hamilton's Principle. 



The kinetic energy of the fluid contained in an element dr 



being -~gdv times the square of its velocity, we have for the kinetic 

 energy of the fluid contained in a given fixed volume, 



32* 



