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136 CONOIDAL SECTIONS. TRACT 30.

the greatest of the parallel sections ; but in the hyperboloid,it is the smallest; and in the paraboloid, all the sectionsparallel to the axe are equal to one another.—For, the axeis the greatest parallel chord line in the ellipse, but the leastin the opposite hyperbolas, and all the diameters are equalin a parabola.

C M

Corol. 6. If the extremities of the diameters kl, mn, bejoined by the line kn, and ao be drawn parallel to kn, andmeeting geh in o, e being the middle of ac, or ae thesemi-axe, and gh parallel to mn. Then eo will be equal toef, the other semi-axe of the section.—For, by similar tri-angles, ki : in :: ae : eo.

Or, upon gh as a diameter, describe a circle meetingeq, perpendicular to gh, in q; and it is evident that eqwill be equal to the semi-diameter ef.

Corol. 7. Draw ap parallel to the axe bd of the solid, andmeeting the perpendicular gh in p. Then it will be evidentthat, in the spheroid, the semi-axe ef = eo will be greaterthan ep ; but in the hyperboloid, the semi-axe ef = eo, ofthe elliptic section, will be less than ep; and in the parabo-loid, ef = eo is always equal to ep.

scholium.

The analogy of the sections of an hyperboloid to thoseof the cone, are very remarkable, all the three conic sec-tions being formed by cutting an hyperboloid in the sameposition as the cone is cut.

Elf =-b-