35 



where the double integrals are taken over the boundary of the body and dn 

 denotes an element of the outwardly-directed normal to the surface S. Also 

 derivable from Green's formulas is the well-known expression for a potential 

 function in terms of its values and the values of its normal derivatives on 

 the boundary 2 



*<«>-feIT[-r 1 g + *anF> s ^ 



where r is the distance from the element dS on the body to a point Q exterior 

 to the body. 



When a distribution of 4> or d^/dn over the surface of the body is 

 given then [85] may be considered as an integral equation of the first kind 

 for finding d^/dn or <f> respectively, on the surface. If the integral equation 

 can be solved, [86] would then give the value of <f> at any point Q of the 

 region exterior to the body. 



AN INTEGRAL EQUATION FOR AXISYMMETRIC PLOW 



Equation [85] will now be applied to obtain an Integral equation for 

 axisymmetric flow about a body of revolution. Let y be the ordinate of a 

 meridian section of the body and ds an element of arc length along the boundary 

 in a meridian plane. Then we may put 



dS = 2rry ds [87] 



It will be supposed that the body is moving with unit velocity in the negative 

 x-direction, which is taken to coincide with the axis of symmetry. The con- 

 dition that the body should be a solid boundary for the flow is that the com- 

 ponent of the fluid velocity at the body normal to body is the same as the 

 component of the velocity of the body normal to itself. This gives the bound- 

 ary condition 



d£ 

 dn 



■sin y 



where y is the angle of the tangent to the body with the x-axis. Substitution 

 of Equations [87] and [88] into [85] now gives 



y< * dn ds = " yw sin ydi 



J(\ •'0 



where 2P is the perimeter of a meridian section and the arc length s is meas- 

 ured from the foremost point of the body. 



