Figure 228 — Choice of signs for 



prolate-spheroidal coordinates. 



See Section 137. 



Axis of Symmetry 



1/2 



1/2 



Toward infinity, 1^ -♦ oo and, approximately, 6J' = /^iC (1 - fi ) and r = (a; +'ar ) =kC,., 

 so that C = ''A) ^ = ^/kC = os-./t = cos 6 in terms of the polar angle 6. 



The elements of distance in the coordinate directions, calculated from Equation [136a], 



5s/-= k 



1/2 



1/2 



<C'-1 



5^, bs. = k 



H. 



1 - 



[137k, 1] 



_ o 1/2 9 1/2 



§s =~5aj = A (^2 _ ]^) (1 - M ) Sw. 



[137m] 



The coordinate direction for u> is perpendicular to the plane through the a;-axis; that for ^ is 

 perpendicularly outward across the ellipsoids, that for \i is tangential to them and from fi < 

 or a; < around toward /i > or a; > 0. These two directions make angles Q y, B with the 

 positive a?-axis such that ^ >- < tt-, -nj'i ^ ^ ^ "■/2? ^nd, from Equations [136b, c,di, 



cos Oy=- sin 



e-i 



1/2 



[137n] 



y^ J. 



Q - V- 

 The components of velocity in the coordinate directions are, from Equation [136e], 



H- 



8s y dC 



8fi d<f> 

 Ss du. 



8<jj d(, 

 8s,. dc 



[137o,p,q] 



347 



