DR. BOOTH ON THE GEOMETRICAJi PROPERTIES OF ELLIPTIC INTEGRALS. 379 
two algebraical expressions for the same arc of the one logarithmic hyperbola. See 
Art. XLVIIl. In the preceding case, that of the spherical ellipse, the analogous 
formula expresses the sum of the arcs of two inverse spherical ellipses, whose ampli- 
tudes are the same. 
LV. We shall use the term inverse spherical ellipses to denote curves whose 
representative elliptic integrals have reciprocal parameters. The terms reciprocal 
and supplemental have long since been appropriated to curves otherwise related. 
Let a and (3, and (3^ denote the principal semiarcs of two such curves. Since 
the modulus i is the same in both integrals, the orthogonal projections of these 
curves, on the base of the hemisphere, are similar ellipses. (15.) gives 
e^=Psec^(3, ef=P sec'jS^, and we assume eV=:?. 
Hence secjS sec(3/'= 1 (315.) 
Again, as tan*a(l — e^) = tan^/3= sec^fS— 1, and tanV^(l— e^) = tan^(3^=sec^(3^— 1 ; 
multiplying these expressions together, and introducing the relation in (315.), 
, , , , Psec^i3sec^f3, — t^(sec^/3+sec^^,)+i^ 
tan a tan a,r= ^ ■ -o ' ? — = 1. 
‘ ] + 2 “'— (sec^/3+ sec-'p^) 
(316.) 
Hence the principal arcs of the inverse spherical ellipses are connected by the 
symmetrical relations 
tana tana, /= 1, and sec/3 sec(3//= 1 (317-) 
When the inverse curves coincide, a = a„ /3=|S„ and the last equations may be 
reduced to tan’^a— tan^j3=: 1. Now we have shown in (59.) that when the principal 
arcs of a spherical hyperconic section are so related, the curve is the spherical 
parabola, or when the curve becomes its own inverse, it is the spherical parabola. 
TIT , , • / \ 1 -o sin^a — sin^/3 
We have shown m (15.) tliat i = =1 
sin^/3 
sin/3 
but (3.) gives cos;j=^ — , 2/1 
sin^a’ \ / o sina’ 
being the angle between the cyclic arcs of the spherical ellipse. Hence /=sin;?, but 
i is constant. Therefore all inverse spherical ellipses have the same cyclic arcs. 
That portion of the surface of a sphere which lies between the cyclic circles may be 
called the cyclic area. 
The spherical parabola divides the cyclic area into two regions. In the one, between 
the pole and the spherical parabola, lie all the inverse curves, whose parameters range 
from P to i. In the other, between the spherical parabola and the cyclic circles, lie 
all the conjugate inverse curves, whose parameters range from i to 1. 
Let ach, adb be the cyclic circles, the inter- Fig 17. 
section of the sphere by an elliptic cylinder, 
whose transverse axe is equal to the diameter 
of the sphere, and whose minor axe is 2j. Let 
a series of concyclic spherical ellipses be de- 
scribed within this cyclic area, whose semi- 
transverse arcs are 01, 02, 04, 05, and let 03 
be the spherical parabola of the series. For 
every curve, 01, or 02, within the spherical pa- 
rabola, there may be found another without it, 
