16 THE EQUATIONS OF MOTION. [CHAP. I 



Let us integrate these equations with respect to t between the 

 limits and t. We remark that 



p d?x dx , _ Vdx dxV _ [ f dx 

 L~dVda &quot;dtda. J.dt 



f dx d*x 



dadt 



dxdx * 



, 

 * 



where u is the initial value of the ^-component of velocity of the 

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



we find 



dx dx dy dy dz dz dy \ 



_1_ / ; I -it /V . 



dt da dt da dt da da 



dx dx dy dy dz dz d% 



dx dx dy dy dz dz dy 



I &amp;lt;_ *J ^i. MSf A*i - _ ^V 



dt dc dt dc dt dc dc 

 These three equations, together with 



.(2) 



and the equation of continuity, are the partial differential equa 

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

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

 tions of Art. 9. 



The initial conditions to be satisfied are 



x =. a, y l&amp;gt;, z = c, % = 0. 



16. It is to be remarked that the quantities a, b, c need not 

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

 may 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 the 

 dynamical equations of Art. 13 is not altered ; to find the form 

 which the equation of continuity assumes, let x , y Qi Z Q now denote 

 the initial co-ordinates of the particle to which a, b, c refer. 

 The initial volume of the parallelepiped, whose centre is at 



* H. Weber, &quot; Ueber eine Transformation der hydrodynamischen Gleichungen &quot;, 

 Crelle, t. Ixviii. (1868). 



