50 IRROTATIONAL MOTION. [CHAP. Ill 



By interchanging &amp;lt;j) and &amp;lt; we obtain 



f f_i&amp;gt; &amp;lt;ty j a f f /W d $ d&amp;lt; t&amp;gt; d&amp;lt;t&amp;gt; d&amp;lt;/&amp;gt; d &amp;lt;t&amp;gt; \ 777 

 \\&amp;lt;i&amp;gt; -j dS=- I I j 2 - j + -J^-T^+T^ - J- \dxdydz 

 JJ^ dn JJJ\dx dx dy dy dz dz ) 



-Jf!&amp;lt;t&amp;gt; V*&amp;lt;f&amp;gt;dxdydz ..................... (6). 



Equations (5) and (6) together constitute Green s theorem*. 



44. If $, &amp;lt;p be the velocity-potentials of two distinct modes 

 of irrotational motion of a liquid, so that 



v^ = o, vy = o ........................ (i), 



we obtain tf &quot;* ^ .................. (2) 



If we recall the physical interpretation of the velocity-potential, 

 given in Art. 19, then, regarding the motion as generated in each 

 case impulsively from rest, we recognize this equation as a 

 particular case of the dynamical theorem that 



where p r , q r and p r t q r are generalized components of impulse and 

 velocity, in any two possible motions of a systemf. 



Again, in Art. 43 (6) let &amp;lt;/&amp;gt; = &amp;lt;, and let &amp;lt;f&amp;gt; be the velocity- 

 potential of a liquid. We obtain 



[[[ \ /cty\ a (d&amp;lt;l&amp;gt;\* , fd&amp;lt;l&amp;gt;\*\j , , ff .&amp;lt;tyjo 



III M j I +? + j I \dxdydz = 6 ~d8 ...... (3). 



JJJ \\dxj \dy) \dz&amp;gt;} J ] ^ dn 



To interpret this we multiply both sides by ^ p. Then 

 on the right-hand side d^&amp;gt;/dn denotes the normal velocity of 

 the fluid inwards, whilst p&amp;lt;f&amp;gt; is, by Art. 19, the impulsive pres 

 sure necessary to generate the motion. It is a proposition in 

 Dynamics J that the work done by an impulse is measured by the 

 product of the impulse into half the sum of the initial and final 

 velocities, resolved in the direction of the impulse, of the point to 

 which it is applied. Hence the right-hand side of (3), when 

 modified as described, expresses the work done by the system of 

 impulsive pressures which, applied to the surface S, would 

 generate the actual motion; whilst the left-hand side gives 

 the kinetic energy of this motion. The formula asserts that 



* G. Green, Essay on Electricity and Magnetism, Nottingham, 1828, Art. 3. 

 Mathematical Papers (ed. Ferrers), Cambridge, 1871, p. 23. 



t Thomson and Tait, Natural Philosophy, Art. 313, equation (11). 

 I Thomson and Tait, Natural Philosophy, Art. 308. 



