Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

a

111

PM=y, AP=s. We shall then have PY: PM :: AQ : AB, or (aa-yy):y :: Fav(aa-yy) : 6. Consequently xy = 6+yn (aa-yy); this is the equation to the co-ordinates of the upper or superior conchoid.

The same calculus gives xy=b-y (aa-yy) for the inferior conchoid. The equation is still the same for the nodated conchoid.

This curve is therefore algebraic, and disembarrassing its equation from the radical quantity we shall find that it is a line of the fourth order or a curve of the third, having for its equation

yt+2by? + (64--4*+ **) ya = 2a* by=a* b* It may be described by the continual intersection of a rule BCM, moveable about the point B, with a circle

G described with the radius CM = a, made to move along GH, so that the centre C may always be on that line. For this purpose it

B is only necessary that the rule should constantly pass through the centre of the circle.

63. We may even construct an indefinite number of different conchoids. For if instead of a circle we

D cause any curve CM to move along GH, its intersection with a rule BM moveable about the point B, and subjected to

No А pass through a fixed point Q in the axis

1 P of the curve CM, will describe a conchoid of which the equation will be easily found. For if we draw MP and AB perpendicular on the directrix, and suppose AP=x, PM=y, CP=z, CQ=a, AB=b, we shall have PQ : PM::AQ : AB, or s– -a:y::Ita-2; 6; hence z=a+

ху

b+y Then substituting this value in the equation of the curve CM, we shall have that of the conchoid MD.

For example if the curve CM is a circle, whose centre is Q, we have

yy 2az-zz, which gives xy=(6+y) (aa-yy) for the equation of the ordinary conchoid.

64. But if the moveable curve is a parabola whose equation y'= pz, then y? + by apy--apb=pry becomes the equation of the parabolic conchoid which Descartes employed to resolve a general equation of the sixth degree.

M

II. THE CISSOID OF DIOCLES.

И

M

TGC

12

In

65. Let ANBn be a circle whose diameter is AB, and to which QBq is a tangent at the point B. If after having drawn from the point A the right lines AQ to different points of the tangent, we take QM = AN, the curve Mam which passes through the the points M, m, thus determined is called a cissoid.

A

P

B We easily perceive that it is composed on two similar and equal parts AM, Am forming at A, a cusp, or point of reflexion, and which after having cut the circumference at the points C, c, equally distant from A and B, recede indefinitely from each other without ever reaching the tangent QBq, which therefore is their

asymptote. 66. To find the equation of the cissoid, draw OM parallel to AP, and MP, NG perpendicular to d. Call AP=x, PM=y, and AB or the diameter of the generating circle=a. Since AN MQ we have AG-PB, and AG: GN

:: AP : PM that is, -a: (xx-ar):: :: y =

vla—x): z3 consequently y' =- the equation required.

67. This equation shews I'. That the cissoid is an algebraic curve of the second order ;

II'. That to each abscissa AP correspond two equal ordinates PM, Pm, one positive, the other negative ; and therefore that the curve has too perfectly equal and similar branches;

III'. That when x = 0, y is also =0; consequently the curve passes through the origin of the abscissæ ;

IV°. That when x = La, then y = + a; that is, the two branches of the cissoid cut the circumference in two points C, c, equally distant froin A and B;

Vo. That if x=a, y is infinite, and therefore the tangent BQ is the asymptote of this curve, as we had already concluded from its description.

The conchoid and cissoid were employed by their inventors Ni. comedes and Diocles to find the duplication of the cube, a problem celebrated among ancient geometers, but which is no longer considered as either difficult or interesting by modern analisis.

[ocr errors]
[ocr errors]

a

IIL THE LOGARITHMIC CURVE.

1

68. If after having taken any point A in an indefinite right line GH, we raise ordinates PM which have for logarithms their

M. abscissæ AP, the curve BMm, which passes through the extremities of these ordinates is called the logarithmic curve.

E Call AP *, PM =y, m =the modulus, e=the number 2.7182818, &c. whose hyper- 0 bolic logarithm is unity, and we shall have x=m log y=x loge, HQ

на

AT E PPG or ym=e, whence y=-*, the equation of the logarithmic curve.

yo= en

This shews 19, that this curve is of the number of transcendental curves ;

2ly. That when x=0, y or AB=1; 1

3ly. That if x=AE=AB=1, y or EF=e", and therefore that making EF-a we shall always have y=a"; hence if the abscisse form the arithmetical progression 1, 2, 3, 4, &c. the ordinates will form the geometrical progression a', a, a, a, &c. Consequently the logarithmic extends indefinitely above AP. But if on AQ we take negative abscissæ x=- 1, 2

2,

1 1 1 -3, &c. the ordinates will successively become

a'a'a &c. ; that is, the curve has an infinite branch BO which approaches nearer and nearer to the directrix, or axis GH, without ever touching it.

69. The most remarkable property of the logarithmic, is that its subtangent is always of the same magnitude. This is proved with the greatest facility by the fluxional or differential calculus; but not to anticipate matters, we shall here give a demonstration on nearly similar principles.

Draw the ordinate mp infinitely near MP, and prolong the side Mm to have the tangent MT. Then, if we draw Mr parallel to the axis, and call Pp = e, mr=i, we shall have xte=Ā. log (y+i)

ja = y (

&c.); there y ya By Ai

1 fore since =A log y, we must have e = (1 +

29 By But the quantity i being infinitely little, its powers , , &c. may be rejected; we have therefore or PT=A. Hence the sub

-
a

=A. log 9 (1+5)=

= A log yta

+

-&c.)

[ocr errors]

tangent is always equal to the modulus ; and since in general r=A log y, it is clear that in different logarithmic curves, the abscissæ of the same ordinates are as the subtangents, or which amounts to the same thing, the logarithms of the same numbers in different systems are to each other as the moduli.

THE CYCLOID.

70. If a circle AG be made to roll along a right line Aa, till the

' point which first touched this right line at A, again touches it at ā, this point will describe a curve called a cycloid, or trochoid. The G E B

Fig. I.

[ocr errors][merged small][merged small]
[ocr errors]

A

F labours of Pascal, Huyghens, Bernoulli, and other eminent mathematicians have rendered this curved very celebrated.

The curve will be a common cycloid when the generating circle has no other movement but that of its revolution.

B But if it has also a movement of translation

М.

P in the same direction the point A will describe curtate cycloid.

A If this movement be in a contrary

B direction the curve is called a prolate or lengthened cycloid.

M

P It is clear that in the common cycloid, fig. 1. the base Aa is equal to the circumference of the generating circle ;

A that it is shorter in the curtate cycloid, and that it is longer in the prolate cycloid.

The diameter BC of the generating circle is called the axis of the cycloid when it is perpendicular to the middle of the base. The point B is the vertex, and therefore BC is its greatest height.

71. This premised, draw MP perpendicular on BC, (fig. 1) and draw the equal chords MF and OC, we shall have FC – MO, because of the parallel lines; therefore since FC=AC—AF=BOC_ FKM=BOC-OLC=BIO, it is evident that the part MO of the ordinate MP is always equal to the corresponding arc BIO of the generating circle. Besides the other part OP is the sine of that same arc; therefore calling MP=y, BIO=u, we shall have for the equation of the common cycloid y = 4 + sin u, and to render this

6 equation general, make MO = BIO, which agrees to the com

a

a

y = +sin u.

mən, curtate or prolate cycloid, according as 6 is equal to, less, or greater than a : so that the general equation will be

Hence the cycloid is a transcendental or mecha6 nical curve.

72. To draw a tangent MT to the point M, conceive the arc Mm to be infinitely small, draw

I the ordinate mp, and the little

dá line Mr parallel to the tangent OT at the point of the cire

g... cumference of the generating circle. We shall have MO = b

6 BIO and mo ==

Blo, this

6
gives me = Oo. Also, be- А
cause of the similar triangles, we have

MOx Mr
mr : Mr::MO: OT=7 = MO=BIO,

Oo

a

a

[ocr errors]

we must take therefore on the tangent of the generating circle the part OT=BIO, and draw through the points M and †, the line MT which will be a tangent to the cycloid, whether common, curtate, or prolate. However in the first, the construction may be simplified : for_since MO= BIO = OT, we have the angle TOP, or 2BOP = 2TMO; that is, a right line MT parallel to the chord OB is necessarily a tangent to the point M of the common cycloid.

73. Now let there be drawn the indefinite line BQQ perpendicular on the axis BC, and Qq, Q'm, parallel to the same axis ; be cause of the similar triangles, we shall have mq : Mq, or Q'Q : Pp ::OP: BP; therefore Q'QxBP=Pp X OP, or MmQQ = Ppo0; and consequently the circular space BIOP = BQM, and the semicircle BOČB – BDAB. But the rectangle AB in the common cycloid is quadruple this semicircle, therefore the cycloidal area is triple the generating circle.

74. If instead of taking a point of the circumferenee of the cir. cle to describe the cycloid, we had taken a point either within or without the circle, then the curve described would have been another species of cycloid ; and if in place of causing the circle to revolve along a right line, we Ład made it revolve

upon

the circumference of another circle, the curve described by one of its points would have been of the species of curves called cpicycloids. V. THE QUADRATRIX OF DINOSTRATES.

75. Suppose that a straight line AG, tangent at A moves uniformly parallel to itself along the diameter Aa, and that at the same

« ΠροηγούμενηΣυνέχεια »