S 90 Dr. R. Key's Propositions concerning 
in the parallel through F ) be taken between F and the M of 
a, or beyond that M ; and the circumference cuts GA between 
G and A, approaching to A as the radius is increased. And 
both the increase of the radius and the approach of the cir- 
cumference to AO are without limit, while circles are added. 
The i and iv are so placed, in the figure, as to have the 
same directing point for their minor axes : C being the centre 
of OMM. 
Second ease. Let AO be AF. Produce AG to H, making 
AH equal to AF. Then, the nearer M is to F, the less is the 
radius of the circle OM or FM ; never less than AF. At the 
same time, its centre is the nearer to A, and the directing point 
of the minor axis is tlie nearer to H : which approaches may- 
be made without limit. Place, as before, in the plane of the 
picture, a triangle ; now HFA, or one similar. The minor 
axes will be parallel to lines from the base of such triangle to 
its vertex : each, as its circle is further, being nearer to a right 
angle with the intersecting line ; and without limit. This is 
a law observed. 
If requisite, find accurately the directing points of the minor 
axes for a few circles nearest to AF : or for one ; as N for 
iv. In HFA, transferred to the picture, mark N. Then, for 
all the circles nearer than iv, the minor axes will be parallel 
to lines from points between H and N ; and, for ail beyond 
iv, between N and A. 
In this second ca^se, if M be at F, the representation is * a 
circle. 
Third case. Let AO be greater than AF. With the centre 
A describe a circle LOR ; cutting AD in R on the same side 
I 
♦ Cor. 3 to prop. V. 
