14 DR. J. BOTTOMLEY ON COMPOUND 



oc at the origin and at the point a; = c ; it cuts the circle at 

 the point r=^c, (9=45°; ^^ ^^i^ point the inclination of its 

 tangent is tan~'^^. Below the axis of x there is a branch 

 PBC similar to the one above the axis^ and to the left of 

 the axis of y there is a branch PDEF similar to the one 

 on the right. 



The major axis of the ellipse is generally greater than 

 the radius of the circle. But of the curve just described 

 a portion lies within the circle, and for such points the 

 radius is less than c ; the connection of this portion of the 

 curve with the axes of the ellipses may be established as 

 follows. Let r, be the length of the minor axis of the 

 projectrix; then from (8) we have 



r,= 



Eliminating m between this equation and (7) we obtain 



the equation 



c 



\/i+{cosec^e-iy 



But if 0^ be the inclination of the minor axis to the axis 

 of X, measured in the positive direction_, we shall have 



1 "^ 



),=-+, 



hence the polar equation to the minor axis is 



r.= 



^|+(sec^^,- 



This equation is of the same form as (9), but the minor 

 axis is generally less than c ; hence it follows that those 

 portions of the curve which lie outside the circle are 

 traced out by the extremities of the major axis, and those 

 portions lying within the circle are traced out by the 



