H 



Ellery W . Davis 



same coaxial and similar blue ellipse. All the red vectors be- 

 longing to the line have the like relation to a pair of ellipses 

 coaxial and similar to these. The motion set up is elliptic and 



Fig. 15. 



compounded of simple harmonic motions parallel to conjugate 

 diameters of the ellipses ; that is, parallel to a red vector and the 

 corresponding black vector from the center of the line (see Fig. 

 15). For all values of a and b 



y — b = re' 1 * \x — a ) 



has the same elliptic vector at infinity. The conjugate line 



y — b = re- 1 * (x — a ) 



has its vectors tangent to the same ellipses. The pair together 

 give the point ellipse 



y — b -\- (.r — a) cos<£) 2 -\- r-(x — a) 2 sin 2 </> = o; 

 for the two can be written 



(y — k) — r (- r — a ) cos <£==»' (-t' — c) sin</>, 

 (y — b) — r (x — a) cos <£ = — ir (x — a) sin <£■. 

 We now perceive that to multiply the coordinates of a red vec- 



14 



