S93 
the Elliptic Representation of a Circle. 
C and K ; and CP is the shortest radius, and, when M is 
beyond F, KQ is the shortest; and, the further any centre W 
is beyond C, the more does its circumference fall within the 
circle POGN at OIN, and the more without it elsewhere. 
And the like beyond K. Therefore, as the mean point moves 
from the first M to P, the circles OM become smaller, and 
cut LR in points further beyond A ; which are the directing 
points of the minor axes ; and the reverse takes place, while 
the mean point moves from P to U, and so on to F. As it 
approaches to F, the radius of OM increases without limit. 
When it is at F, the arc OM becomes a right line ; which 
passes through A : whence the minor axis is * perpendicular 
to the intersecting line. While the mean point moves from 
F to O, then beyond Q, the radius of OM decreases to KO, 
then increases without limit ; and the intersection of the cir- 
cumference OM with LR moves from A to H, then approaches 
to A without limit. 
The cases of AO equal to AF and greater, bear a resem- 
blance to the second and third cases of Prop. X ; and are not 
perhaps sufficiently frequent, in practice, to justify an exami- 
nation here. And, whatever be the length of AO, if any mean 
point fall within LOR, it is to be remembered -f* that the 
directing point of the major axis is then nearer to A than is 
that of the minor. 
Fig. 13. Now let the centres E, E, of the objective circles, 
^be in a right line. Let LR and AO be as before; also the 
diameters J ST perpendicular to the directing line; D, D, 
being their directing points, B that of the line through the 
* As also by Cor. 4 to prop. V. 
f Prop. IX, last paragraph but one; and X, 3d case. J As in prop. IX, X. 
