DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 345 
XXXIV. The projections of supplemental spherical ellipses on the plane of xy are 
confocal plane ellipses. 
For sin;? = sine', sin>?'= sins. Hence 
.bf P-P 
P-hf af ~~P-P' 
This gives as the resulting value or P—P=af—bf. 
Two supplemental cones are cut by a plane at right angles to - their common 
internal axe. The sections are concentric similar ellipses, having the major and the 
minor axes of the one, coinciding with the minor and major axes of the other. 
For 
tan^a— tan ®?3 
tan^« 
■ P, and e; 
tanka' — taP^' cot ®(3 — coPa tap a. — taPjS 
taPa! 
cor/3 
tan^a 
or e' = e. 
Section IV. — On the Logarithmic Ellipse. 
XXXV. The logarithmic ellipse is the curve of symmetrical intersection of a para- 
boloid of revolution with an elliptic cylinder. This section of the cylinder by the 
paraboloid is analogous to the section of the cone by the concentric sphere in IX., 
for this cylinder may be viewed as a cone, having its vertex at the centre of the para- 
boloid, i. e. at an infinite distance. 
Let the axes of the paraboloid and cylinder 
coincide with the axis of Z ; the vertex of the 
paraboloid being supposed to touch the plane 
of xy at the origin O. 
Let k be the semiparameter of the para- 
boloid Onb, and let a and b be the semiaxes 
of the base of the elliptic cylinder ACB ; then 
the equations of these surfaces, and con- 
sequently of the curve in which they inter- 
sect, are 
j:^-1-/=2A-z, and^+^=l. . (II7.) 
Let d 2 be an element of the required curve, 
then 
dd 
=v/(i)WKi)’- 0 
18 .) 
X, y and z being dependent variables on a fourth independent variable 9 . 
Assume cos6/, 3/=:6 sin 0 , then cos^ 0 +Z>* sin^0=:2A:;z. . . . 
Differentiating and substituting, 
de. 
• (P_bZ\i 
=a^ cos"04-- ^2 ' sin^0 cos"0. 
To reduce this expression to a form suited for integration, it may be written, 
/cl2\ ^ 
^ s\P 9 — {a^~ Py siPO 
2 Y 2 
( 119 .) 
( 120 .) 
( 121 .) 
