the Elliptic Representation of a Circh, 5gri 
of AO with the objective circles before given, and in L on the 
other side. Then,* if one of those circles, as ?r', have its mean 
point in the circumference OR, the directing points oF the two 
axes are L and R. And, for any circle between ii and AO, 
the directing point of the major axis is between R and A, and 
nearer to A than that of the minor is ; which is beyond L. 
And, for all circles beyond ii, the directing points of the minor 
axes are between L and A. Draw LO, RO. Then LOA, 
ROA, or triangles similar to them, transferred to the picture, 
as 'f GOA in the first case and HOA or HFA in the second, 
give lines to which the axes will be respectively parallel : and, 
of the minor axes, each, as its circle is further, is nearer to a 
right angle with the intersecting line ; and without limit. 
This is a law observed. 
Any requisite approximation to accuracy may be made ; by 
the mode used in the second case, when N was found. 
A figure, constructed as fig. ii, may, if required, be 
reduced by a scale.]; 
Prop. XI, and last. Prob. V. Fig. 12, 13. To find the same, 
when the centres are in a right line oblique to the intersecting 
line of the plane. 
Fig. 12. If the circles had been so placed that (instead of 
the centres) the mean points, M, M, &c, were in such line 
having any directing point X ; a law observed would have 
been as follows. Let LR be the directing line ; and let AO 
be as before, but less than AF, and in F let AO produced meet 
XM. With the centre A describe^a circle, having the diameters 
ON, LR ; L being on the same side of A with X : and let XM 
* Construction of prop. VI. 
J See the end of prop. IX. 
f As LOA in prop. IX. 
