508 XL HYDRODYNAMICS. 



' = x + dx + dt [u + <^-dx + ^dy + 

 - (x + udf) 



cy z 



and therefore the change per unit time in the projections are 



du 7 



K.I\ d /? \ dv -, . dv ^ , cv 



51 ) 



Thus we have 



B 



K ON ^w , dv dw 



52) 



and substituting from equations 49) and 51) 



B 



-ON df fAB C[3U' . cU' . 8U' 



53 ) -dr=Jb^ dx + i^ d y + -w d * 



A 



du . 3v . 3w 

 ^ + V d-x + W Zx 



du . 3v , dw 



^ + v 8j + w 



du . dv . 



A 



which vanishes for a closed curve. 



Therefore if the forces are conservative, the circulation around 

 any closed path moving with the fluid is independent of the time. 

 Thus if the circulation around any closed path is zero at one time, 

 it is always zero, or in other words if a velocity potential once 

 exists, it always exists. This theorem is due to Lagrange. 



