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points (Tembrassement; and if they are in the plane of reference on one side of the point, and on the other side pass out of it, then the curve at the point is such as one or other of those drawn in fig. 60, where the dotted lines indicate the course of the curve out of the plane of the paper, and the points are called cusps, and by French writers points de rebroussement. That in the fig. 60, marked (a), is called a ceratoid cusp, or a cusp of the first species, in which the two branches touch the common tangent on opposite sides of it; that marked (/3) is called a ramphoid cusp, or cusp of the second species, in which the two branches touch the common tangent on the same side. Cusps in other positions are shewn in fig. 61.

These several subordinate varieties of double points must be distinguished by examining the form and nature of the equation du d u

to the curve, and of -—^ and , when x = x0 + h, y = y0 + k,

h and k being taken very small, and (xq, yo) being the point of the curve in question; as, for example, in fig. 60, if when d*y

x = x0 + h, is positive for one and negative for the other

dy

branch of the curve, and when x = x0 h, is affected with

d2y

the curve is of the form drawn in fig. (a); but if -~ is

(IX

positive for both branches of the curve, and the curve is out of the plane of the paper when x is less than x0> then it is such as that delineated in fig. (j9).

249.] Of these double points we shall give instances, and apply to particular cases the principles of the foregoing method, without adapting them directly to the general forms.

Ex. 1. Examine the nature of the point at the origin of the lemniscata whose equation is, see Art. 197 and fig. 37,

(x2 + y*)* = a*(x*-y*).

Differentiating we have

4(;r2-f y2) (xdx + ydy) = 2a2 (xdx—ydy);

dy dlx—1x{xi + y1') 0 i. „ , „

. •. -r- = o x—--s—- = K > when x = 0, and y = 0,

a2dx—... dx ~ n2dy + ... ~ dy' ^1 - 1 <ty _ , 1 .

that is, two branches of the curve pass through the origiu, cutting the axis of x at angles respectively of 45° and 1353, and which are in the plane of reference on both sides of the point; and therefore all the characteristics of a true double point are satisfied.

Ex. 2. Determine the nature of the point at the origin of the curve, y2 = x2 (1 — x2).

dy x 2x3 0 „,
• • Tx = ~y— = at the °ngm'
__ (1 —6x2) dx _ dx
~ dy ~ dy'"

• • dx* ~' dx ±'

and two branches of the curve pass through the origin, which are inclined to the axis of x at angles respectively of 46° and 135°; and there is at the origin a true double point; see fig. 62.

Ex. 3. Determine the nature of the point at the origin of the Folium of Descartes, the equation to which is x3 Saxy + y3 = 0.

dy ay-x2 0

dx= y^ = 0' &ttteTM&a>

ady—2xdx dy

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that is, two branches of the curve pass through the origin and touch the axis of x on different sides of it, and are both in the plane of reference.

If the equation to the curve were y2(x2 a2) = x*, the two branches which pass through the origin would touch the axis of x, but both would be out of the plane of reference; see fig. 77.

Ex. 5. Discuss the nature of the point at the origin of the curve whose equation is y2 = ax3.

Saxdx

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••• (i)'=3a*; g=±(3o*)» = 0,if* = 0;

and is affected with +, when x is positive, but with + -J , if x is negative: hence at the origin there is a cusp, both branches of which touch the axis of x, and the curve is out of the plane of the paper when x is negative. The curve is, on account of the form of its equation, called the Semicubical Parabola.

dhj 3a*

Also since -r4 = A :j

ax1 4#4

which is positive or negative according as the branch of the curve is above or below the axis of x, the cusp is of the first species; see fig. 64.

Ex. 6. Discuss the nature of the point at the origin of the curve y3 = ax2 Xs.

dy 2ax—Sx2 n , .i_

£ = 3y» = 6' at the °ngln'

_ (2a—6x) dx _
6y dy'

Hence the curve has two branches at the origin touching the axis of y, but as is affected with + \/—\ when y is negative, it shews that at the origin there is a cusp of the first species, and such as is drawn in fig. 65.

Ex. 7. The equation to a curve is yi = x (x + af; determine the nature of the point where x = — a, and y 0.

dy 3x2 + 4,ax + a2 0 , , n

j- = n = ^, when x a, and y = 0.

(6,r + 4a) dx = 2~dy'

.•.(|)*= i-i^i:

that is, there is a conjugate point, both branches of the curve being out of the plane of the paper, and piercing it at the point; and their directions making, with the axis of x, angles whose tangents are + Va.

250.] The equations of many curves admit of being reduced to an explicit form, which is well adapted to exhibit the peculiarities of cusps of both species.

Suppose that the equation to a curve can be put into the form

y =/(*) ± <*>(*), (130) of which f(x) is possible for all values of x through which we consider it, and <f> (x) is possible for some and impossible for others; and, to fix our thoughts, suppose that </> (x) is imaginary or real, according as x is less or greater than a. The curve whose equation is y = f(x) is aptly called the diametral curve of (130), the ordinates of (130) being equal to f(x) increased and diminished by the same quantity, viz. <J> (x). Then, if <t>(x) is such that <f>(a) = 0 and $'(a) = 0, the curve (130) has, when x = a, a common point, and is coincident in direction, with y = f(x); but as two branches unite at the point, and are distinct when x = a + h, and affected with + v—-, when x— a—h, we have a cusp of the first or second species, according as the curvature is turned in opposite or in the same directions. The following examples will illustrate the method.

Ex. 1. (y a—x)1 = (x—c)6.

y = a + x + (x—c)*,

| = i ±

Hence the diametral line is that whose equation is y = a -f x, and as y is affected with + v —, for all values of x less than c, it follows that the curve is in or out of the plane of refer

dy

ence, according as x is greater or less than c; and as -j- = 1

Q/3C

when x = c, it'is plain that at that point both branches touch a common tangent, one being above and the other below it; d2v

and as is + when x is greater than c, the cusp is of the first species; see fig. 66.

Ex.2. iy-x2)2 = ax6;

y = x2 ± a$x%,

"v. - 2x+ xi

d'2y _ g 15 a* ^
dx2 ~ 4

From the above equations it appears that the diametral curve is the parabola whose equation is y = x2; that the curve is in the plane of reference on the positive side of the axis of y, and out of it on the negative side; that there is a cusp of the second

d2y .

species at the origin, since is positive for both branches; see fig. 67.

Ex. 3. y = a + bx + cx2 ± x$.

-/ = b + 2cx + - xi,
dx — 2

dx2 4xi

The diametral curve is plainly a parabola, and the curve is in or out of the plane of reference, according as x is positive or negative; there is a cusp of the first species at the point where the diameter cuts the axis of y, and the tangent at the point makes, with the axis of x, an angle whose tangent is b; see fig. 68.

251.] For a double point on a curve we have the two following conditions,

at the point. Let us suppose the equation to the curve to be Price, Vol. 1. 3 n

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