42 Trans. Acad. Sci. of St. Louis. 



Referring to Fig. 1, let the circle P'BCA be given by its 

 equation 



x 2 + y' 2 = R 2 



referred to the rectangular axes OX' , OY'. 



Take any point P' in the circumference whose co-ordinates 

 are x' , y' and turn the ordinate y' through an angle a bringing 



r to p. 



The co-ordinates of P referred to OX' , OY' are x lt y r 

 Treat every point of the circumference in the same way and 

 we shall have (Fig. 2) the circle transformed into an ellipse. 



Proof : 



jb'2 + y' 2 = R 2 (1) 



is the equation to the circle referred to OX', OY'. 



The co-ordinates of P' in terms of those of P are 



x' = Xj + 2/j tan a 

 y' = y l sec a, 



and the new locus has for its equation 



(x x + .Vi tan a) 2 + (y x sec a) 2 = R 2 

 or 



x 2 + y t 2 + 2y x 2 tan 2 a -f- 2^, tan a — R 2 . (2) 



which is the equation of an ellipse. 



Now let us refer the conic to the axes OX, OY, which 

 make an angle # with OX', OY' respectively, and let the new 

 co-ordinates of P referred to OX, OY be x, y. 



For changing from the axes OX' , OY' to the axes OX, OY 



x x = y sin 6 + x cos 6 

 y 1 = y cos 6 — x sin 0. 



Substituting and separating terms equation ( 2 ) becomes 



x 2 



+ 1 



— tan a sin 20 + y 2 



+ 2 tan 2 a sin 2 6 



+ 1 



+ tan a sin 26 



+ 2 tan 2 a cos 2 



+ 2ar?/ 



tan a cos 20 



-tan 2 a sin 20 = ^' ^ 



