134 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 





which shews that tzru . dx vru . diz 



is an exact differential. Denoting this by d^jr we obtain the 

 relations (1)*. 



So far the motion has not been assumed to be irrotational ; 

 the condition that it should be so is 



dv du 

 dx dtp 

 which leads to 



The differential equation of (f&amp;gt; is obtained by writing 



dd&amp;gt; dd&amp;gt; 



u = -p- , v = 



dx d^ 



in (2), viz. it is *+* + ?. 1=0 (4). 



It appears that the functions &amp;lt;/&amp;gt; and i/r are not now (as they were 

 in Art. 62) interchangeable. They are, indeed, of different dimen 

 sions. 



The kinetic energy of the liquid contained in any region 

 bounded by surfaces of revolution about the axis is given by 



**--&amp;lt; 



--&amp;lt;//* 



(5), 



Bs denoting an element of the meridian section of the bounding 

 surfaces, and the integration extending round the various parts of 

 this section, in the proper directions. Compare Art. 61. 



* The stream-function for the case of symmetry about an axis was introduced 

 in this manner by Stokes, &quot;On the Steady Motion of Incompressible Fluids,&quot; 

 Canib. Trans., t. vii. (1842) ; Math, and Phys. Papers, t. i., p. 14. Its analytical 

 theory has been treated very fully by Sampson, &quot;On Stokes Current-Function,&quot; 

 Phil. Trans. A., 1891. 



