17 19.] WEBER S EQUATIONS. 15 



Let us integrate these equations with respect to t between the 

 limits and t. We remark that 



[t^xdzj. |~^^T_ [ tdx d ** 

 J df da &quot; \_dt da\ Q J dt dad 



i* 

 dadt 



dxdx . d 



where U Q is the initial value of the ^-component of velocity of the 

 particle (a, b, c). Hence if we write 



we have 



dx dx dy dy dz dz _d% 

 dt da dt da dt da da 



. _i dx dx dy dy dz dz dy 

 and, similarly, J^ + J jf + j JJ ~* - jf,| W 



cfce cfcc cZy cZv cfe cfe c?v 1 



j E -3- _j W ^ -^ I 



dt dc dt dc dt dc dc J 



These three equations, together with 



and the equation of continuity, are the partial differential equa 

 tions to be satisfied by the five unknown quantities x, y t z, p, % ; 

 p being supposed already eliminated by means of one of the rela 

 tions of Art. 7. 



In the case of a liquid, p occurs in (27) only, so that (26) and 

 (24) may be employed to find x, y, z, and ^, while p may be found 

 afterwards from (27). 



The initial conditions to be satisfied are x = a, y = b, z = c, % = 0. 

 The boundary conditions vary with the particular problem under 

 investigation. 



19. The equations (26) and (27) may be applied to find the 

 equations of impulsive motion of a liquid. Let the impulse act 

 from t = to t T, where T is infinitely small, and let T be the 



* H. Weber, Crelle, t. 68. 



