42 Trans, Acad. ScL of St. Louis. 



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

 equation 



aj'2 + y"^ = R^ 



referred to the rectangular axes 0X\ OY' . 



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

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

 P' to P. 



The co-ordinates of P referred to 0X\ OY' are a;^ y^. 



Treat every point of the circumference in the same way and 

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



Proof: 



is the equation to the circle referred to 0X\ OY' . 



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



cc' = cCj + y^ tan a 

 y' - y^ sec a, 



and the new locus has for its equation 



(^1 + 2/i tan a)2 + (y^ sec a)^ = B^ 

 or 



x[^ + y'x + ^y^ tan2 a + 2a;jyi tan a = P^. ( 2 ) 



which is the equation of an ellipse. 



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

 make an angled with 0X\ OY' respectively, and let the new 

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



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



ajj = 2/ sin ^ -t- aj cos Q 

 y^'=y cos B — x sin 6, 



Substituting and separating terms equation ( 2 ) becomes 



+ 1 



— tan a sin 20 + y'^ 



+ 2 tan2 a sin2 ^ 



+ 1 



+ tan a sin 26 



4- 2 tan^ a cos* ^ 



4- 2321/ 



tan a cos 2^ __ 02 o 



- tan2 a sin 2^ ^^' ^^^ 



