90 
Construction. — From A, 
the extremity of the longer 
diameter (or from the ex- 
tremity of either if they be 
equal), draw AE at right 
angles to OD and make AF 
equal to OD: on the line OF 
as diameter describe a cir- 
cle; join A with the centre 
G and produce AG to meet 
the circumference in K. OH 
and OK will be the direc- 
tions of the major and minor 
axes respectively, and 
OL“AK= semi-major axis. 
OM= AH = semi-minor axis. 
Proo/’.— The ellipse of which OL and OM are the semi- 
axes will evidently be traced by the point A if the line AK 
be made to move so that H and K move along the axes. It 
only remains therefore to show that this ellipse will pass 
through the point D and have AB and CD for conjugate 
diameters. 
It is well known that when a circle rolls inside another 
of twice its diameter, every point on the circumference of 
the rolling circle moves along a diameter of the other. If 
then the circle OKH be rolled inside the circle FNP the 
points H and K would move along OH and OK so that the 
point of which A is the initial position, moving with the 
circle, would describe the ellipse of which OL and OM are 
the semi-axes. Now as the circle OHK rolls inside FNP 
the points E and F move along the diameters EC and FP ; 
