163-164] CONDITIONS FOR STEADY MOTION. 263 



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

 and vortex-lines. If further 8n denote an element of the normal 

 at any point of such a surface, we have 



n^ ........................ (2), 



where q is the current-velocity, co the rotation, and /3 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 qw sin /3 Sn must be constant over each 

 such surface, Sn 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 function ^ , defined by (1), is constant 

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

 Art. 22, where it was shewn that % is constant along a stream 

 line. 



The above conditions are satisfied identically in all cases of 

 irrotational motion, provided of course the boundary-conditions be 

 such as are consistent with the steady motion. 



In the motion of a liquid in two dimensions (xy) the product 

 cftn is constant along a stream-line ; the conditions in question 

 then reduce to this, that the angular velocity f must be constant 

 along each stream-line, or, by Art. 59, 



t. 



where f(^r) is an arbitrary function o 



* See a paper &quot; On the Conditions for Steady Motion of a Fluid,&quot; Proc. Lond. 

 Math. Soc., t. ix., p. 91 (1878). 



t Cf. Lagrange, Nouv. M6m. de VAcad. de Berlin, 1781, Oeuvres, t. iv., p. 720 ; 

 and Stokes, 1. c. p. 264. 



