the Elliptic Representation of a Circle, 387 
another be ix, x, xi, &c. : all having their centres E, E, &c. 
From the directing line AN, find the centroids V, and the 
mean points M, in as many circles as may be requisite. On 
. the same side of AN as the circles are, take AO, in the cen- 
tral line, equal to the distance of A from the eye ; and make 
A the centre of a circle ON. In either rank, as /, zV, &c, if 
all the mean points be without ON, find * the directing point 
L of the minor axis of i the nearest circle. By the construc- 
tion, the directing points of the minor axes for all the circles 
a, z/V, &c, will be between L and A. Draw LO. In the plane 
of the picture place the triangle LOA, or any similar triangle, 
in the same situation with regard to the indefinite intersecting 
line, or some parallel line, as LOA has here with regard to 
tile directing line. Then 'f will all the minor axes, for zz, zzz, 
&c, be respectively parallel to lines drawn from points in the 
base of such triangle to its vertex corresponding with O ; each 
of such points will be nearer to the point corresponding with 
A, as the circle is further from the directing line ; and the 
approach to A (and to its corresponding point) will be with- 
out limit, while circles are added. This, then, is a law 
observed by the minor axes : to which the major will be per- 
pendicular, respectively. 
If the circles be unequal, the law holds good ; unless the 
inequality cause the mean point of a further circle to be not 
further than that of a nearer circle. 
If the angle LOA be inconveniently large, find also the di- 
recting point F for the circle zz. Then all the directing points, 
for zzz, &c, will be between F and A. Or, in some cases, find 
the directing point G for the furthest of the rank : and then the 
♦ Prop. VI. t 
