(a) Keep ^ > 0. Then from Equation [82c, d] it is easily seen that, as the a;-axis is crossed 

 between the points x = - c, sin 17 must change sign discontinuously without change of cos 77; 



T] itself may change discontinuously to -7/, or to 2nn -jj where n is an integer, positive or 

 negative. Elsewhere rj may vary continuously: in this case tj will differ on the two halves of 

 each hyperbola and is many valued; in going once around both of the points {- c,0) in the 

 same direction, 77, like a polar angle, changes by - 277. 



A possible choice to make -q single valued is the range -ttKti^tt. Then rj changes 

 sign discontinuously in crossing the a;-axis wherever |a;l < c. Values of f and 77 according 

 to this convention are indicated in Figure 130a. 



(b) As an alternative, ^may be given everywhere the same sign as y. Then ^will have 

 opposite signs on the two halves of each ellipse and will change sign discontinuously at 

 the a;-axis where \x\ > c, whereas 77 may be made to vary continuously and will then have a 

 fixed value on each hyperbola. A possible range is ^ 7/ < w. This latter convention is 

 illustrated in Figure 130b, and in more detail in Figure 129. 



In any case, if d-q = 0, dx = c sinh f cos 77 <^^ and dy ■= c cosh ^ sin 77^^; if d^ - 0, 

 dx = - c cosh f sin r/dj] and dy - c sinh ^ cos 7/ dr]. 



Figure 130a 



Figure 130b 



Figure 130 — Symbolism for flow parallel to major axis past an elliptic cylinder. 



Hence the slope angles of the ^ and 77 coordinate directions are 



0/:=tan~ 



. , = tan ^ (coth /^tan 77); d =tan ^ 1 



-tan (tanh ^cot77). [82o,p] 



19-1 



