Elements of the conic sections

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TH CONIC SECTIONS

BOOK III.

Osthe perbola.

DEFINITIONS.I. I in a potnt ahen pon a plane, the X Fig. i. tremit nos a uter H is o fixe that therule is test freeno revolverabout the potnt Ras centre and Done endi a string shorte thanthe uter is fixe in the extremit H, and theother endis it in the oin F, hicli is in the fame plane illi the poliata; ut the distancebet een the poliat E, F greater than the X-ees of the tength of the ruter above that of the

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string and si means os asin G, the stringis applied to the si de H of the uter thenwith the string o applied, and kepi uniform lytense, is the uteri movexabo ut the Centre, theso in of the pira illi escribe pon the plane a line, called the hyperbola. But is the ab ove orderi reversed, and theeny of the uterie Xed in the poliata, and the enda of the string in theto in E and thena simila operationi repe a ted another line, opposite to the .sermer, ill be describe d whichis also called the hyperbolis and both togetherare called opposite hyperbotcs. These lines maybe extende beyond any give distanc homili potnis E, F is a stringi taken the tengilios hicli Xceeds that distance.

II. The minis E, F are called the foci. III. An the po in C, whic bisecis thestra ight line bet ween the foci, is called the centre of the hyperbola, or of the opposite hyper

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IV. An strato line passing through thecentre an meeting the hyperbolas, is calle a transcerse diameter and the oinis here a transverse diameter meet the hyperbolan recalled iis vertices Also an Straight Iine whicli passes through the Centre, and bisecis a straight line terminated by the opposite hyperbolas, butno Passing through ah centre, is calle a right

V. That diameter hichiasse through lis foci, is calle the tranSPer Se azis. VI. I from et ther extrerruit A of the transverse axis, a traight line Ambe placed equalto the distance bet een the centre C and ei thersicus F, and Domin a a Centre, illi the distanc AD, a circle e described, meetin astraight line drawn through the centre C, triglit an gles to the transverse axis in theloinis B, b the traight linei is calle the secondatis menc the second axi Lis bisected In the centre C Elem.

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VII. w diameters, ach of hic bisecis ali straight lines parallel to the other, and ter minared both ways by the hyperbola, O OPp sit hyperbolas, arema med conjugale diameters. VIII. When a stratot in no drawnthrough the centre, et terminate both ways by the hyperbola, or opposite hyperbolas, Isbisected by a diameter, Dis sal to te ordia nateis applied to that diameter or it is called Simply, an ordinate to that diameter. Also a diameter paralle to a traight in ordinatet'

applied to another diameter, is sal tole orta nateis opplied to this other diameter. IX. A straight line hic meet the hype hola in only one potnt, and whicli, ein produce both ways, salis ithout the opposite hyperbolas, is sal to ouch the hyperbola in that potnt.

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the foci, the exces of the oneabove the ther is qua to the

Let EGH represent the ruter, and FGH thestring the pin by whicli the hyperbola is describe bein supposed to remain a G. stomEH, GH ake way the common part GH; and the excessi GDabove GF will e qu lto the excessis the tength of the uter bovectat of the string and this conclusion illinoldwhereve the poliat G stallie siluate in the

hyperbol . Aiad since theloinis A a the vertices of the transverse aXis, re in the opposite hyperbolas, the eKces of AE above AF andalso the excessi a above E are acti finem equat to the exces of the tength of thsruler above statis the string that is, oethe X-ces of EG above FG and therefore these two excesses re eo ualla each other: ut et AF

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be adde to ach of the two traight lines AE, AF, and the excessi AE above AF ill be

I from a potnt two traight linesare drawn to the sociis opposite hyperbolas cis the exces of the