380 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
05, or 04, such that their principal arcs are connected by the equations 
tanatana^^=l, secjS sec/3^z= 1 . 
The algebraic expressions for the arcs of these curves, having the same amplitude, 
give elliptic integrals with reciprocal parameters. 
The concyclic spherical ellipses will be ortho- 
gonally projected on the base of the hemisphere 
into as many concentric and similar plane ellipses, 
whose semiaxes are 01 , 02 , 04, 05. The cyclic 
area will be projected into the plane ellipse, and 
the spherical parabola into the area of the plane 
k 
ellipse, whose transverse semiaxe is — 7 ==^. Let 
^ \/l+^ 
E be the area of the plane ellipse, the projection 
of the cyclic area, and IT the area of the plane 
ellipse, the projection of the spherical parabola. 
. E — XT 
Then E='^, and 11= whence or 
the ellipse, the projection of the spherical para- 
bola, divides the area of the ellipse, the projection of the cyclic area, into two por- 
tions, such that the outer is to the inner as i'. 1 . 
The reader must have observed the importance of this curve, the spherical paia- 
bola, in the discussion of the geometrical theory of elliptic integrals. 
We may determine the principal arcs of two inverse spherical ellipses by a simple 
geometrical construction. Let AZB be a vertical section of the hemisphere, on which 
the curves are to be described. Let F be the focus of the elliptic base of the maxi- 
mum cylinder, whose principal transverse axis is accordingly equal to the diameter 
of the sphere. Join OZ, FZ, and draw ZC at right angles to ZF, meeting the line 
AO in C. Produce ZO until OD=AC, and on OD as diameter describe a circle. 
We are required, given one principal arc Za, to determine the corresponding prin- 
cipal arc Za' of the inverse hyperconic. Draw the tangent ZG. Through a draw 
