144, 145.] STEADY VORTEX MOTION. 173 



It follows that 



dP dP dP 



u- r - + v^- + w-j- = 0, 

 dx dy dz 



t dP dP dP 



f ~T~ + V ~j h -j- &amp;gt; 



* dx dy dz 



so that each of the surfaces P = const, contains both stream -lines 

 and vortex-lines. If further dn denote an element of the normal 

 at any point of such a surface, we have 



-y- = qa) sin 6 (53), 



where q is the current-velocity, o&amp;gt; the angular velocity, and 6 the 

 angle between the stream-line and the vortex-line at that point. 



Hence the conditions that a given state of motion of a fluid 

 may be a possible state of steady motion are as follows. It must 

 be possible to draw in the fluid an infinite system of surfaces 

 each of which is covered by a network of stream-lines and vortex- 

 lines ; and the product of qco sin 6 dn must be constant over each 

 such surface, dn denoting the length of the normal drawn to a 

 consecutive surface of the system. 



These conditions may also be deduced from the considerations 

 that the stream-lines are, in steady motion, the actual paths of 

 the particles, that the product of the angular velocity into the 

 cross-section is the same at all points of a vortex, and that this 

 product is, for the same vortex, constant with regard to the 

 time. 



The theorem that the quantity P, defined by (52), is constant 

 over each surface of the above kind is an extension of that of 

 Art. 28, where it was shewn that P is constant along a stream 

 line. 



The above conditions are satisfied identically in all cases of 

 irrotational motion. 



In the motion of a liquid in two dimensions, the product 

 qdn is constant along a stream-line; the conditions then reduce 

 to this, that o&amp;gt; (or f, if the axes of co-ordinates be the same as 



