08 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV. 



The existence of a function ^ related to u and v in this 

 manner might also have been inferred from the form which the 

 equation of continuity takes in this case, viz. 



? + ?=0- ..(4), 



dx dy 



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. 38, become 



so that in irrotational motion we have 



70. 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 = f, v = f ...................... (6), 



dx dy 



and, since we are considering the motion of incompressible fluids 

 only, satisfying the equation of continuity 



(7). 



The theory of the function &amp;lt;, and the connection of its proper 

 ties with 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, both in the enunciation and in the 

 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. 46 and of those which depend on it. 



