NO. 4 ■ WIND TUNNEL EXPERIMENTS IN AERODYNAMICS 79 



We have to find a solution of this equation which is the same at all 

 points of circles about the axis of rotation (owing to the assumed 

 rotational symmetry), which reduces to ux at all points at great 

 distances from the ellipsoid (the axis of x being taken along the axis 

 of revolution and u being the general velocity of the stream), and 



which satisfies the condition that the normal derivative -^ vanishes 



an 



at each point of the ellipsoid (as the flow must be tangential to the 

 ellipsoid). 



We first introduce cylindrical coordinates with the axis of revolu- 

 tion as axis. Then 



3; = rcos^, s = r sin 6, 



and 



d'-cj> ^d~cl> ^ I aj> ^ I d-<i> ^^ 



dx- ' dr- }■ dr r- dO' 

 As we are interested only in solutions which do not depend on 6, 

 Laplace's equation reduces to 



p: + ft+'l*=o- (■) 



o-i'- oy' ■'" or 

 We next replace x and r, the rectangular coordinates in a plane 

 through the axis, by a system of coordinates (C, fi) derived from the 

 system of ellipses and hyperbolas in the .rr-plane confocal with the 

 ellipse in which that plane cuts the ellipsoid.' If c be the distance 

 from focus to center, we write 



x = ciJi^, r = cVi-/x^Vr+i- (2) 



The elimination of /x and ^ respectively gives 



A small value of t, gives a narrow ellipse ; a large value, a large ellipse. 

 The set of ellipses may therefore be represented by values of t, from 

 o to 00. A small value of ix gives a sharp hyperbola nearly coincident 

 with x = o, a value of ju. equal to i gives the line r=o, the axis of 

 revolution. By assigning different signs to /u. and to one of the 

 radicals in (2) we may represent all points {x, r) of the plane; but 

 the symmetry of the figure is such that we may work only in the 



^This system of coordinates is that used by Lamb, Hydrodynamics, 2d ed. 

 (1895), p. 150, for treating the motion of an elHpsoid of revolution in a fluid 

 at rest at infinity. We could use Lamb's analysis and make a correction to 

 bring the ellipsoid to rest in a moving fluid. It seems as easy to solve our 

 problem independently with all the simplifications it admits. 



