See Figure 129, on which possible values of ^ and 77 are indicated in terms of n-/30 as a unit. 



30 X 



Figure 129 - Illustrating elliptic coordinates. See Section 82. 



The ellipse for ^ = reduces to the segment of the a^-axis between -c, on which 

 a; = c cos r\. The remainder of the a;-axis can be regarded as a hyperbola on which sin 7; = 

 while cos 77 = 1 for a; > c or cos 77 = -1 for a; < -c, and on which a; = i c cosh ^. 



The variables ^, 77 can obviously be used as coordinates on the 2-plane; they are 

 called elliptic coordinates. They have the disadvantage of being doubly many-valued. Not 

 only is 77 many-valued like an angle, with a period of 2??, but the values -f, -77 define the 

 same point (a;, y) as do ^, 77. If both £, and 77 are required to vary continuously with x and y, 

 ^ must change sign in crossing the a;-axis between a; = - c, since there |cos 77I < 1 and 

 sin 77 ?^ 0, whereas in crossing at \x\ > c, \^\ > and sin 77 must change sign with y. Hence 

 it is easily seen that ^does not change sign but 77 changes by - 2tj in going once around 

 both of the points {- c, 0); whereas, if only one of these points is encircled, upon returning 

 to the starting-point, 77 has returned to its initial value but 1^ has changed sign. In appli- 

 cations it is usually convenient to suppress at least the ambiquity as to ^. The two most 

 useful alternative conventions are the following. 



193 



