GO IRROTATTONAL MOTION. [CHAP. III. 



&amp;lt;to 



lu -f my + nw = ~ , 



C/71 



where dn is an element of the inwardly-directed normal to the 

 surface of S. Substituting in (17) and performing the differentia 

 tions indicated, we find 



(21) 



By simply interchanging &amp;lt;j&amp;gt; and -fy we obtain (provided ^ be single- 

 valued), 



(22). 



Equations (21) and (22) together constitute Green s theorem. 



Corollary 3. In (21) let &amp;lt;/&amp;gt; be the velocity-potential of a liquid, 

 and let &amp;gt;|r = 1 ; we find, since y 2 &amp;lt; = 0, 



which is in fact what (20) becomes for the case of irrotational 

 motion. Compare Art. 44. 



Corollary 4. In (21) let ^ = ^&amp;gt;, and let $ be the velocity- 

 potential of a liquid. We obtain 



If we multiply this equation by ^p it becomes susceptible of a 

 simple dynamical interpretation. On the right-hand side 53P d e _ 



CLYL 



notes the normal velocity of the fluid inwards, whilst p&amp;lt; is, by 

 Art. 2G, the impulsive pressure necessary to generate the actual 



