It is convenient to introduce also parabolic coordinates A.^, \„, nearly as in Section 87, 

 so that 



a; = — (A^-A?), S'=A,A„, A,^0, A„^0; 



Aj ^yfr-x, X^ ^\jr+x, r^^Jx-^+TS^ [143b, c, d] 



The surfaces, Aj = constant or A, = constant, are confocal paraboloids opening, respectively, 

 toward positive and negative x. Their traces on a plane through the axis are illustrated in 

 Figure 137. On the a;-axis, A^ = and x - A /2 for x = Q, whereas A. = and x - -\W2 for 

 x^O. 



By introducing also for the moment y = wcos w and z = STsin co where « is the angle 

 about the axis, it is found from Equation [136a] that 



Ss^ 8s \ 



1 2 0T1/0 ^ 1 1 



= ^ CX2, a2\1/2 



Ss, 



(A2 + a2)1/2 = =-2- [143e, f] 



SAj SA2 ^ ^ 8<o AjA2 *^ 



The surface of the given solid, on which x<a, is the paraboloid A^ =\fa', as is easily 

 verified from Equation [143c]. On this surface the stream function ip must be constant. Fur- 

 thermore, in the surrounding space, as A,-*"", a;-»oo, and in the limit it is necessary that 

 xfj-* t/fcr^/2 = V >\.^^X^/2\ see Equation [I19b] for a uniform stream. The differential Equa- 

 tion [136m] for ip becomes here 



This equation and the two boundary conditions are satisfied if 



^i- = -t/A2(A2-a) = -f/[ar2-a(r-a;)]. • [143h] 



Then i/i = on the solid and also where a;>0 on the a;-axis. 



376 



