THE EQUATIONS OF FLUID-MOTION. 143 



In spite of the simplicity of this fornix it does not appear 

 to yield a convenient form of condition affecting T>ta; nor 

 for Y'Va. The following property, however, can be 

 deduced. 



In a perfect fluid under conservative forces we must 

 have (So-vjcr of the form v?- But 



2 



(ScrV)a-= V — +VcrVvo-; 



.*. Vo-Vvo"= vQ^ say, 

 or 



whence 



and -y— and ^- become o when acted on by D<, whence 

 d(p, d<p, ^ 



Q is a function of f^ and (p^ only. The existence of this 



surface Q, on which both stream-lines and vortex-lines lie, 



is dependent on the existence of vortex motion ; but if the 



surface exists we may take it in place of the surface ^j to 



indicate the stream-lines ; and then we get 



2,=o 



and 



or 



■^2 • , 



and the rotation would be given by 



Vvor=S.Vv9xVQ+ ' VvQVf.. . . (3) 



