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any point of the curve is every | tangent to the hyperbola cuts the where equal to the transverse axis. axis in some point T, between A

The subtangent PT =

aa

From this we may deduce an and C. But as the abscissa ineasy method of describing an hy-creases, the line CT diminishes so perbola, whose axis are 2a and 26. that it is infinitely little or nothing Take an interval Ff= 2 (ua+bb), when the abscissa is infinitely and make use of a rule /MO, longer, great. Hence we see that through in proportion as you desire to have the centre C we may draw two a greater proportion of the hyper-right lines cX, Ca which will be bola: fix one extremity of it at one the limits of the tangents to the of the foci, to the point f, for ex-hyperbola, these lines, whose poample, so that it may revolve freely sition we shall soon determine, are about that point. Take then a called the asymptotes of the hy. thread FMO equal in length to perbola. MO-2a. Fasten one of the ends of this thread to the point 0 of the rule, and the other to the focus F. This done, draw the rule away from the axis as far as the thread FMO will allow, and then bring it again gradually towards the axis, taking care to keep the thread always extended by means of a point or style M which glides along the rule fMO. The curve described during this motion by the point M, will be an hyperbolical branch AM, since the difference of the ra dius-vectors will be every where equal to the transverse axis.

This same property will enable us to draw a tangent MT, fig. 2. to any point m of the hyperbola. For if we conceive the arc Mm to to be infinitely little, and draw the lines fM, fm, FM, Fm, we may prove much in the same manner as in the ellipse, that the angles fmM, MmF are equal, and consequently that if we bisect the angle MF by the line MT, this line will be the required tangent.

22-aa

%
bb

aa

and the tangent MT=

· (zz — aa) + zz — 2a2+"

(aa + bb

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a

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ar

=√
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If we draw the norm MN we
shall have the subnormal P'N
bb

aa

2

(zz-aa)

2%

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uu

and the normal

da

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62

at

+

MN=

-a2

The line AT a CT = a

aa
and if we draw AS parallel to
MP, we shall have PT: PMAT: A
S, or zz-ua

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:y=a- :

aa uy
% ≈ + a

(+). Now if we suppose

will

%-a infinite, the quantity ≈ + a not differ from unity. Consequently we shall then have AS= b. Hence it follows that if we draw AD and Ad perpendicular to CA, and each equal to the semi-conju gate axis b, the lines CD, Cd, drawn so as to pass through the points D, d, and the centre C, will be the asymptotes to the hyperbola MA M, and if prolonged in a contrary direction, they will be those of the opposite hyperbola.

If the hyperbola is equilateral, the angle DCd, made by the asymp totes, is a right angle, for then D Ad= CA.

A

The hyperbola referred to its asymptotes has many properties; here follow the principle of them: If though any point N (fig. 3.) of the asymptote we draw the straight line Nn parallel to the line Dd, we shall have CA: DA= CP: bz NP or ab=z: therefore

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bb DA. Since NP2= and that MP2 =

b2 zv
22

line TDt will be a tangent to the point D, and we shall have Fm × mf Dt X DT and fh × hF= Dt = DT Fm X mf, therefore fh (hm + m F) Fm (mh+hf); therefore th Fm, and consequently TD = Dt. But if we draw DE parallel to Ct, or an ordinate to the asymptote C NT, the similar triangles TDE, TIC, will give TE EC. Consequently to draw to the hyperbol a tangent to a point D, corresponding to the ordinate DE, we must take ET E C and draw through the points T and D, the tangent TDt.

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Since we always have fh Fm, in whatever manner we draw the straight line Ff, the two parts Fm, -bb, we have |sh intercepted between the curve always NB>MP. Consequently and its asymptotes will always be the hyperbolio branch can never come in contact with its asympner of describing an hyperbola be Hence we derive an easy man tote, however it perpetually ap proaches towards it, for as the ab. scissa increases, the difference be

tween

b2x2
a2

and

b2 12
a?

-bb becomes

equal.

tween two given asymptotes CT, Ct, and passing through a given

point m.

Draw through this point the right lines Ff, MN, &c. and take fh= less sensible; so that if we suppose Fn, nN Mm, &c. and the points z infinite, this difference vanishes.n, h, &c. will be in the hyperbola. Draw MQ and AL parallel to the asymptote Cd. It is easy to see that the triangles DLA and LCA are isosceles. Let then AL = DL =m, CQ=x, QM=y. If we draw MK, parallel and equal to CQ, because of the similar triangles DLA, NQM, M/Kn, we shall have the proportions MN: DA=QM: LA

Mn: DAMK: DL Therefore Mn × MN: DA QM X MK: AL. But Mn × MND A, hence xy=mn, an equation to the hyperbola referred to its asymptotes, in which mm = (aa +bb) is called the power of the hy. perbola.

According to what we have seen a tangent TMt, terminated at the asymptotes is bisected at the point of contact M. If we draw MCM/, this line is called a diameter; the tangent TMt is its conju gate diameter. Its ordinates are the right lines mQm/ parallel to the conjugate diameter TMt or DCd, and the parameter of any diame ter is a line, a third proportional to that diameter and its conjugate. The line MCM=2CM is also called the first or principal diameter, and the line TMt=2TM = DCd=2DC is the second conjugate diameter.

This premised, it is easy to see that a diameter bisects all its ordinates. For NQ Qn = TM: Mt, and Nm = mn. Call then CM ==m, CD=TM=n, CQ = x, Qm = y; and we shall have m: n:: z : NQ = nz But Nm X mn TM2. Hence n2 z2 m2

If two parallels Ff, Gg, (fig. 4.) terminated at the asymptotes cut an hyperbola in the points m, h, p, K, we shall have GpX pg = Fm mf. For if we draw MmN, PpQ, perpendicular to the axis, we shall have Fm: Mm Gp: Pp and mf: mN: pg pQ therefore Fmxmf: Mmx mN:: Gp pg : n2 = Pp × pQ. But Pp XPQ = = bb

=

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Mmx mN. Consequently Fmxm2), an equation similar to that mf=Cpx pg. And therefore also of the co-ordinates to the trans

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to the co-ordinates of the second diameter CD; it is easy to perceive the analogy which this equation has to that of the co-ordinates to the conjugate axis.

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ly CP2 — CE2= a2.

Firth, a2 62 = CP2 +. PM2 DE CE CM2-CD2. Therefore the difference of the squares of any two conjugate diameters is equal to the difference of the squares of the two axes. Hence also in the equilateral hyperbola the conjugate diameters are equal. Hyperbolas receive various de

arities in their construction, proportions, &c.; as acute, ambigenal, equilateral, &c.

Let now aCA be the transverse axis of the hyperbola, and suppose that BA represents the semi-conjugate axis; if we draw DE, TG, Mnominations from certain peculi PK, perpendicular to this axis, and ML, tK parallel to it, the triangles MTL, MIK, CDE, will be equal and similar. If we call CP-u, PM=z,| CE tK ML = r, MK DE TLs, and as before CM=m, TM =u, CA=a, AB = b, we shall have TG+s, CG=u+r, and x+s: u+r::ba, and consequently az + as = bu + br; besides TL: ML:: MP: PS, or sr u? a?

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:2: and PS=

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aasz b2 ur

Acute HYPERBOLA, is that whose asymptotes form an acute angle with each other.

Ambigenal HYPERBOLA, is that which has one of its infinite legs inscribed, or falling within an angle formed by its asymptotes, and the other circumscribed or falling without that angle. This is one of Newton's triple hyperbolas of the second order. See Newton's "Enusub-meration of the Curves of the Third Order."

aazz

bz u

stituting this value in the equation Anguinal HYPERBOLA, a name az + as = bu+br we have (bu-given by Newton to four of his as) (bu az)=0. Now bu az curves of the second order; viz. cannot be equal to zero; therefore species 33, 34, 35, 36, expressed by bu -as= 0. Hence bu as, and the equation xy2+ey=-ax3 + b consequently az br. Therefore a2+ ca+d; being hyperbolas of we have CP: DE:: ab:: CE a serpentine figure. MP. Consequently,

First, the triangles CED, CMP, are equal in surface.

Second, if we draw DM, we shall have DMC, or CDTM to the

Conic HYPERBOLA, is that which arises from the section of a cone mon or Apollonian hyperbola. by a plane; called also the com

Conjugate HYPERBOLAS, are those formed or lying together, and

trapezium DEMP=(s+x) ("") having the same axis, but in a con

su + uz — sr — rz

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su- rz

2ub

2 aazz

ab; therefore the pa

rallelogram TT constructed on the conjugate diameters is equal to the rectangle of the axes.

b Third, DE - CP.

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Hence D

trary order; viz. the transverse of each equal the conjugate of the other. See the definitions in the preceding article.

Equilateral or Rectangular HrPERBOLA, is that whose two axes are equal to each other, or whose Hence the property or equation of asymptotes make a right angle. the equilateral hyperbola is y2= a x+x, where a is the axis, a the absciss, and y its ordinate; which is similar to the equation of the circle, viz. y=ax-x2; differing

E2= CP2 = b2 + PM3, and DE only in the sign of the second term,

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pressed or defined by general| equations, similar to that of the conic or common hyperbola, but having general exponents, instead of the particular numeral ones, but Bo that the sum of those on one side of the equation is equal to the sun of those on the other side.

Obtuse HYPERBOLA, is that whose asymptotes form an obtuse angle. Nodated HYPERBOLA, a term applied by Newton to a curve of this kind, which by turning round crosses itself.

Rectangular HYPERBOLA, the same as equilateral hyperbola. HYPERBOLIC Arc, is an arc of the hyperbola, the length of which may be found thus:

Let t and c represent any semi. diameter and its conjugate, and y the ordinate which limits the arc, to be measured from the vertex, te + c2 q, and hyp. log.

making

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where A, B, C, &c. represent the preceding terms.

To which may be added the following approximations: viz.

4. Hyp. area =

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+

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5.

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5t + 24ts c2 + 48t+c++ 64t2 c3)y8

3. Arc=y

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{21 √(2x+

x2)+4√2tx } nearly.

HYPERBOLIC Frustum or Zone, is the space included between the curve and two lines parallel to each other, and perpendicular to the axis; the area of which is equal to the difference of two seg ments cut off by those lines. Or,

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HYPERBOLIC Logarithms, or Naperian Logarithms, express the areas or spaces contained between the asymptote and curve of the hy perbola; but as this property is not peculiar to this system, they are now more commonly called Nape. rean logarithms, from the name of the illustrious inventor of this method of computation.

The logarithm of any number is the index of the power to which another number would have to be raised, in order to become equal to the first, that is, if rxa then is the logarithm of a, and r will be the radix of the system; and it is obvious that r may be assumed at pleasure. If we take r = 10, it will be the common, or Brigg's logarithmic system; and if r = 2.7182818, &c. it gives the hyperbolic scale:

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1+x

And this rule is general for every quantity, of which the numerator is the fluxion of the denominator, or any multiple of that fluxion; these being all reducible to the same form.

HYPERBOLICUM Acutum, the solid generated by the infinite space contained between an hy. perbolic leg and its asymptote, by revolving about the latter, which is equal to a finite quantity.

HYPERBOLOID, or HYPERBOLIC Conoid, is the solid formed by the revolution of an hyperbola about its axis.

axis of the generating hyperbola, Let a and c represent the semithe distance of its base from the centre: also let A =

ar

be the a2 + c2 semi-transverse of another hyperbola, whose semi-conjugate is c,

, log. a=(a-1) — } (a− 1) + } (a1)3(a-1)+. &c. or the same with that of the former. h. log. (1+ a)= a — §} a2 + } a3 —la, the area of the frustum of this Then find, by the proper formu

at + a5 --, &c.

This is the simplest form that the logarithmic series admits of, and therefore such as would be natu

rally adopted by any person, be

latter hyperbola, whose ends are distant from the centre by v and a, multiply this area by 3-1416 for the surface; that is,

1. Surface=p× { v Y — c

hyp. log. of

ay — AC.

AY + ev

Aytac

fore he was aware of the advantages and disadvantages of differ ent systems; the former of which, in the tables in present use, much where p = 3.1416, Y and y the or more than compensates for the ad-dinates of the latter hyperbola. ditional labour employed in their computation. We have seen above (writing a instead of a) that hyp. log. (1+x) - {x2 + } x3 − {x2 + 1 x3 —,

=x=

&c.

2. Solidity

where a =
the base,
- P=3.1416.

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par.2 ×

t + z u
t + a

altitude, r = radius of the transverse axis, and

r2 + d2

6

Xap, where a is the diameter, in the middle between the base and vertex.

Frustums of Hyperboloids. Let D and a denote the semidiameters of the two ends, a the altitude, t and c the transverse and conjugate axis, p = 3.1416; then

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