134 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



s- B+w " = .................. > 



which shews that TZV . dx tan . dtz 



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 



1 --- 7 - - ^J 



dx der 

 which leads to 



__ 

 da? + d^ * dv ~ 



The differential equation of </> is obtained by writing 



dd> dd> 



u = - -f , v = -J- 



dx der 



in (2), viz. it is ?*+*,+ - f* = .................. (4). 



dx 2 d^ OT dvr 



It appears that the functions </> and ty 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 



(5), 



&s 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, "On the Steady Motion of Incompressible Fluids," 

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

 theory has been treated very fully by Sampson, " On Stokes' Current-Function," 

 Phil. Trans. A., 1891. 



