135 137.] KINETIC ENERGY. 161 



dv du\ du div 



ff (dv du\ fdu di 

 u \y(T^Ty)- z (dz--dx 



+ w ^ -T - w * 1 



O.JC J 



Transforming the remaining terms in the same way, adding, and 

 making use of the equation of continuity 



du dv dw _ 

 dx + ~(h/ + ~fa~ 

 we obtain 



ff/Y 2,2 2 &amp;lt;% U dv dw 



P I II ( u v w xu ~J~ ~*~ y v J~ zw ~/T 

 or, finally, on again transforming the last three terms, 



i.e. T. 



The value (30) of T must of course be unaltered by any dis 

 placement of the axes of co-ordinates. This consideration gives 



fff(wrj vQ dxdydz = } 

 ff!(uZ-w%)dxdydz = () j ................... (31), 



//JW ~ *&quot;?) dxdydz = J 

 relations which of course admit of independent verification. Thus 



- **** .///{. ( - )-.-! 



and similarly for the others. 



137. The rate at which the energy of any mass of liquid is 

 increasing at any instant is 



(32), 

 I, 11 



