32 Prof. Challis on the Principles of Hydrodynamics. 



I proceed now to the consideration of the equations of con- 

 tinuity. 



Axiom I. Let it be granted that the motions of a fluid must 

 be consistent with the physical circumstance that the mass of 

 the fluid is constant. 



This limitation, to which the motion is subject, is expressible 

 by an equation which may be thus investigated. 



P>-aposition V. To express by an equation the condition that 

 the motion is consistent with the principle of constancy of mass. 



The whole mass is the sum of all the elements plL>xI)yJ)z, the 

 variations Dx, Dy, J)z being independent of each other and of 

 the variation of time. Hence the condition to be satisfied is, 

 that 



S{pDxT)yT>z)= a constant, 

 or 



8.S{pI>x'Dy'Dz) = 0, 



the symbol S having reference to change of time and position. 

 On account of the independence of the symbols S and S, the 

 last equation is equivalent to 



8{8.pI>xJ)yJ)z) = 0, 



which signifies that the sum of the variations of all the elements 

 by change of time and position is equal to zero. Now 



S . pI>xJ)yJ)z-p{J)yJ)zmx + UxBzDBy + DxDyDSz) 



+ BxJ)yJ)z(^^ Bt + ^Bx+ ^By+^ Sz). 

 \dt ax dy dz J 



And since Bx, By, Bz are the variations of the coordinates of a 

 given element in the time Bt, we have 



Bx=uBt, By^vBt, dz=wBf. 

 Hence 



mx=^J)xBt, my=^J)yBt, mz=^J)zBt. 

 Consequently, by substituting, 



which equation is satisfied if at every point of the fluid 

 dp d.pu , d.pv , d.piv _^ 



dt'^~d^ + -df+~dr-^- ' ' • (^-^ 



This is the equation it was requii-ed to obtain. 



Another equation of continuity depends on the following axiom. 



Axiom II. Let it be granted that the directions of motion in 

 each element of the fluid mass may at all times be cut at right 

 angles by a continuous surface. 



