342 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
Now this product may be resolved into the sum of two terms. Let 
dcr_ P ^ , 
— l)sin""‘<p] a/ 1 — sin^ip — l)sm®(p] Vl — i^ sia^^p' 
Or reducing these expressions to a common denominator, 
dtr (P + Q) — (P + Q)e^sm®(p+ a/ — 1(P — Q)2_;sin^ip 
d<p [I — i(i -fy v' — 1 ) sin^cp] [1 — i{i—j A^ — 1) sin®^] V'l — z^ sin®^ ' 
Hence P+Q=^', P— Q=0; *.• P=j, Q=J (e.) 
Integrating (c.), we get 
^ — V — 1) sin®<p] a/ 1 — — i(i— y a/ — 1) sin^f] a/ 1— i^sin^p (108.) 
Now if we multiply together the imaginary parameters 
(*'+y\/^) and (P-yV— 1). 
their product is or the parameters are reciprocal. 
Since the parameters are each affected with a negative sign, and one is equal to 
P+ a certain quantity, while the other is equal to P— a certain quantity, the 
former parameter is of the circular form, while the other is of the logarithmic form. 
It is very remarkable, that although the spherical parabola is a spherical conic, 
the imaginary parameters satisfy the criterion of conjugation which belongs to 
the logarithmic form, and not that which belongs to the circular form. Let 
m=i{i—j^—\), n=i{i-\-j^ These values of m and n satisfy the equation 
of logarithmic conjugation, m-\-n—mn=P, and not m — n-\-mn—P, the equation of 
circular conjugation. 
On Spherical Conic Sections with Reciprocal Parameters. 
0^ 'Ip' 
XXXI. Let^+|,= l be the equation of an ellipse, the base of an elliptic cylinder. 
Let two spheres be described, having their centres at the centre of this elliptic base, 
and intersecting the cylinder in two spherical conic sections. These sections will 
have reciprocal parameters, if the radii of the spheres, are connected by the 
equation 
( P - a^) {P - a^) = aH\ (109.) 
P being, as before, equal to 
-^2 
When k and k' are equal, we get k‘^=a^{\-\-i). This value of k agrees with that 
found for k in (96.), or, in other words, when the two spheres coincide, the section 
of the elliptic cylinder by the sphere is a spherical parabola. Hence also the spherical 
parabola abvays lies between two spherical conic sections with reciprocal parameters. 
Let e^ and e^ be the parameters of those sections of the cylinder made by the 
spheres. Then, as shown in (12.), 
2 sin®« — sin^/3 — IPP 
^ sin^a cos''^/3 (P{tP—l)^') IP — cP cPP ' 
