59-60] STREAM-FUNCTION. 71 



which is the analytical condition that udy vdx should be an 

 exact differential*. 



The foregoing considerations apply whether the motion be 

 rotational or irrotational. The formulse for the component angular 

 velocities, given in Art. 31, become 



f=o, 77=0, rtiVi 



V fi *Y 



so that in irrotational motion we have 



$H|r c 2 -\/r ... 



~d^ + w = (&amp;gt; 



60. In what follows we confine ourselves to the case of 

 irrotational motion, which is, as we have already seen, character 

 ized by the existence, in addition, of a velocity-potential &amp;lt;/&amp;gt;, 

 connected with u, v by the relations 



u = -^ v--^ m 



dx dy&quot; 



and, since we are considering the motion of incompressible fluids 

 only, satisfying the equation of continuity 



dtf + ~df = 



The theory of the function $, and the relation between its 

 properties and the nature of the two-dimensional space through 

 which the irrotational motion holds, may be readily inferred 

 from the corresponding theorems in three dimensions proved in 

 the last chapter. The alterations, whether of enunciation or of 

 proof, which are requisite to adapt these to the case of two 

 dimensions are for the most part purely verbal. 



An exception, which we will briefly examine, occurs however in the case 

 of the theorem of Art. 39 and of those which depend on it. 



If 8s be an element of the boundary of any portion of the plane xy which 

 is occupied wholly by moving liquid, and if 8n be an element of the normal to 

 8s drawn inwards, we have, by Art. 37, 



* The function ^ was first introduced in this way by Lagrange, Nouv. mem. de 

 VAccid. de Berlin, 1781 ; Oeuvres, t. iv., p. 720. The kinematical interpretation 

 is due to Rankine, 



