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considerations: in (99) let x y = oo , then all the terms must be neglected except the first; and thus the direction of the curve at infinity is given by means of the first term of (99). Now this breaks up into two factors a\ = 0 and u„_j = 0, of which the former represents a straight line, and the latter a curve of the (n —l)th degree: to each of these therefore is the curve infinitesimally near at infinity; and thus ax = 0, and u„_i = 0, are the equations to asymptotic lines. Hereby therefore we have not only rectilinear but also curvilinear asymptotes. Thus, for instance, in Art.235, Ex.2, the equation if + x3ax* = 0 may be expressed in the form

(y+*-f){sM*-f)M*-in + ^-5 = °;

and therefore the asymptotic lines are

y + x-^ = 0, and y2 + (x-|) y - (x - |) = 0;

of which the first is a straight line and the second is a hyperbola.

Similarly, in Ex. 4, the equation may be expressed in the form

(x + y) (xt-xy + y^ + Satx + SPy + c3 = 0;

and x + y = 0 is the equation to a rectilinear asymptote.

Suppose also ai and a2 to be linear functions of the coordinates; and suppose c„_2 to be a function of x and y of «—2 dimensions; and M„ + Mn_i + «„_2 to be capable of expression in the form aia2u„_2; so that the equation to the curve becomes

aia2U„_2 + M„_3+ ... +tti + Wo = 0; (100)

then if we consider the points on the curve where x = y = so , only the first term of (100) is to be taken account of, and we have

aids u„_2 = 0,

which is satisfied by a\ = 0, a2 = 0, cM_2 = 0; of which the first two equations are those of straight lines, and the last is that of a curve of the (n —2)th degree: the curve therefore at infinity breaks up into two straight lines which are asymptotes to it, and into another curve of the (« — 2)th degree which is also asymptotic. Thus, if the equation to a hyperbola is

we may put it into the form

(M)(M)-' = °;

and the lines T = 0, - 4- ^ = 0 are two rectilinear asymp

a b a o

totes.

Again, if the first four terms of the general equation are such that

«« + «n-l + Mn-2 + Mn-S = a^as Vn-3, (101)

then the curve has three rectilinear asymptotes, and a curvilinear one whose equation is u„_s = 0: it is unnecessary to enter farther into the investigation of special cases. It is however to be observed that all the terms in the equation of lower dimensions may be altered when the equation is expressed in the form (101). Such is the case in the first example given above.

It is also to be observed that, generally, a curve of the wth degree has n rectilinear asymptotes. For if x = y = x , all the terms of lower dimensions must be neglected, and we have

un 0,

which is a homogeneous equation of n dimensions in terms of x and y; and which -is therefore capable of resolution into n factors of the form y—miX, y—m2x,... y—m„x; each of which gives the direction in which the curve is going at infinity; and will become the equations to rectilinear asymptotes, or of lines parallel to them, if mu w2,... are real numbers; if the m's are, some or all, imaginary, the asymptotes refer to branches of the curves in other planes.

Section 3.—The direction of curvature, and points of inflexion.

240.2 The value of which we have discussed in the

preceding Sections, enables us to determine the inclination to either of the coordinate axes of a curve at any point, but tells us nothing as to the direction of curvature, that is, as to whether the curve is concave or convex towards a certain line or in a given direction; we use these words in their common meaning; and we proceed to discover criteria which will determine such direction. To simplify the formulae we shall take x to be an equicrescent variable.

, . . . d2u dx rf.tanr

Since under this supposition -~ = — = ——, see

equation (24), Art. 218; and since T = (seer)8, which is

d3y ...

always positive; it follows that if is positive, r and x are

d%y .

increasing and decreasing simultaneously; and if ^-j is negative, as x increases, r decreases, and vice versd.

Now from the geometry it is plain, that if r and x simultaneously increase and decrease, the form of the curve must be such as that of fig. 53, that is, the curve must be convex doicnwards; and if as x increases T decreases, or vice versd, the form must be such as that of fig. 54, that is, the curve must be concave downwards. Hence we have the following criteria of the direction of curvature.

d2v . cPy

If *s positive, the curve is convex downwards; and if

is negative, the curve is concave downwards.

Suppose that at a certain point changes its sign, which

in algebraical curves it can do only by passing through 0 or 00, the direction of curvature changes; if the change of sign is from + to —, then the curvature, having been convex downwards, becomes concave; if the change is from — to +, the reverse is the case. A point where such a change of curvature takes place is called a point of inflexion. To determine

d*v

its position we must equate to 0, and to 00 ; and if at the

corresponding critical value there is a change of sign, then such a point is a point of inflexion; fig. 55, (y) and (8) of fig. 48, fig. 14, and fig. 15 illustrate such points of inflexion. This is also evident from the following considerations.

241.] Let y = f(x) be the equation to a curve; and suppose that we consider it not only at the point {x, y), but also at another point (x + h, y + k), so that

y + k=f(x + h); (102)

then, expanding f(x 4- h) to three terms by Taylor's Theorem,

y + k =/(*)+/(*) J+T$f"(* + 0 *>• (103>

If a tangent is drawn to the curve at the point (x, y), its equation is ,

v = y+ -£((-*); (104)

and therefore its ordinate, when £ is x + h, is

,1 = y+^h; (105)

dv

in which, replacing y by /(#), and ~ by /'(#)> and subtracting

(105) from (103), we obtain the difiference between the ordinates to the curve and to the tangent corresponding to the point x-Vh, and we have

y + k-v= *Lf"(x + 0h); (106)

and taking h to be infinitesimal, that is, considering only the point in the curve which is next to (x + dx, y + dy), whereby

d'ly

f"(r-\ 0h) becomes f"(x), that is, we have

And therefore if is positive, the ordinate to the curve is

greater than the ordinate to the tangent, and the curve lies above the tangent, whatever is the sign of h; that is, the curve

d%y

is convex downwards, as in fig. 53; but if is negative, contrary results follow, and the curve is concave downwards, as in fig. 54.

If therefore at any point the curve passes through the tangent so as to be above it on one side of the point of contact and section, and below it on the other, then y + k rj must change sign as h changes sign, which can only be the case when

d^v d v

= 0 and is a finite quantity; or in general, when the

term in the expansion of f(x + h), which gives sign to the whole, is one involving an odd power of h, in which case the curve will pass from below the tangent to above it, or vice versd, in one or other of the ways indicated in fig. 55. It is plain Price, Vol. i. 3 B

that, at a point of inflexion, and therefore the angle r,

attains to a maximum or minimum, and is in fact at the point stationary; that is, r is the same for three consecutive points on the curve: the corresponding tangent is called a stationary tangent.

The curvature of a curve towards the right or the left may be

d*x

determined in a similar manner by the value and sign of -y—:.

242.] Examples illustrative of the foregoing theory.

Ex. 1. To determine the direction of curvature of the curve whose equation is

_ (a?-l)(<r-3) y ~ x-2"

d*y _ 2_

dx2 (.r-2)3'

d2y

that is, -r^y is positive or negative according as x is less or

greater than 2; and therefore the curve is convex downwards for all values of x less than 2, and concave downwards for all values of x greater than 2; and when x = 2, there is a point of inflexion.

Ex. 2. To determine the direction of curvature of the Witch of Agnesi.

/2a-x\t d2y 2a2(Sa-2x)
y = 2a[ -); T4 = — i>

therefore the upper branch of the curve in the plane of reference is convex downwards for all values of x between 0 and

3a . . 3a d2y . ,. ,

x = —; and when x is greater than —, -~ is negative, and

the curve is concave downwards; at the point therefore,

x = , there is a point of inflexion. This investigation ex

plains the form of the curve as drawn in fig. 35.

Ex. 3. To prove that the equitangential curve, see Art. 200, is always convex downwards.

*i _ _ y

dx (a2_y2)3'

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