202 Wisconsin Academy of Sciences, Arts and Letters. 



II. 



Preliminary Demonstrations Developing required New Expressions Per- 

 taining to the Ellipse. 



First: A, = B/l + E/ cos' 3=A/1-E= sin' Sr 



Second: sin* ar= 



B, = Bi/1 + ,E; sin' 3=At/1-E- cos' ^. 

 E^ sin^ 3- COSTS' 



1— E^ + E'* sin- 3 cos^3- 



In the above expressions A and B represent the semi-major and semi- 

 minor axes of the ellipse, A, any semi-diameter and B, its semi-con j negate. 



A2 132 A^ — B' 



E' equals "^^ — - — and E,' equals — — — Angle a is the difference between 

 A' B'^ 



the angle made by any diameter with its conjugate and a right angle. 

 3^ is further on defined. "■ 



9, To find any diameter and its conjugate, the major and minor axes of 

 the ellipse being given. 



A 



T' 



Diagram 3. 



In Diagram 3* let A B A B be an ellipse having major axis A A and minor 

 axis B B, and let B b B b be a second ellipse having B B major axis and 

 similar in construction to first ellipse; also let B dB d be a circle with ra- 

 dius B C. 



* In center of Diagram 3 C is left out. 



