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



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

 equation 



referred to the rectangular axes OX', OY' . 



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

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

 P' 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 : 



X '2 + ^2 _ #2 ( I ) 



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



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



x' = x x + y 1 tan a 

 y - y x sec a, 



and the new locus has for its equation 



(^1 + 2/1 tan a ) 2 + (2/i sec °0 2 = R 2 

 or 



x i l + Vi + 2 ^i 2 tan " a + 2a; i2/i tan a = i? 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 i — y sm ^ "^ x cos ^ 



y x = y cos 6 — x sin 6. 

 Substituting and separating terms equation ( 2 ) becomes 



+ 1 



— tan a sin 26 -f y 2 



+ 2 tan 2 a sin 2 6 



+ 1 



-f- tan a sin 26 



+ 2 tan 2 a cos 2 





+ 2xy 



tan a cos 26 — z?2 



- tan 2 a sin 26 ~ U ' < 8 ) 



