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BELL SYSTEM TECHNICAL JOURNAL 



perimeter remains constant, while the major diameter changes, we have 

 expressed our results in terms of an average diameter, defined as 



_ perimeter 



Figure 5 shows the elHpse and various relations of interest. 



Y 



P=PERIMETER 



e=ECCENTRICITY = 



_ Co 



a 



E=ELLIPTICITY=^^^' . 



A= AREA = TTab 



D = "average" DIAMETER = £: 

 TT 



b=aYi-e2 = a(i-E) 

 A=Tra2'Yi-e2=Tra2 o-e) 



Fig. 5 — The ellipse 



Elliptic Coordinates and Functions 



The elliptic coordinate system is shown on Fig. 6. Following Stratton,'^ 

 we have used ^ in place of the table's z, since we wish to use z as the coor- 

 dinate along the longitudinal axis. Stratton also uses tj = cos if as the angu- 

 lar coordinate; this is frequently convenient. 



Analogous to cos (6 and sin (d in the circular case, there are even and 

 odd* angular functions, denoted by 



^Sf{c, cos <f>) and °Sf{c, cos ^) 



which reduce to cos Id and sin Id respectively when c 

 are even and odd* radial functions, denoted by 



'Jf^c, k) and "Jfic, 



* For ^ = 0, only even functions exist. 



0. Similarly, there 



