620 



PROCEEDINGS OP THE AMERICAN ACADEMY. 



tial functions. Eliminating \p and <f> successively between the equations 

 above gives : 



+'- 



ir 



cosh 2 </> sinh 2 <f> 

 x 2 y 2 



cos \p sin 2 if/ 



Figure 3. 



<£ defines a family of confocal ellipses, one of which reduces to the 

 straight line joining (1,0) with ( — 1, 0). ip defines a family of hyperbolas 

 with the same foci as the family of ellipses. One of the hyperbolas is 

 the X axis with the segment joining (1, 0) to (— 1, 0) omitted. <£ and \p 

 define then a system of orthogonal curvilinear co-ordinates. Now if 

 we examine the origiual equations I, we shall see that x and y do not 

 uniquely define (/> and if/. In the first quadrant, for example, and if/ 

 may be either both positive or both negative. This ambiguity enables 



