CHAPTER XIII.

ON CONTACT OF CURVES AND ON ENVELOPES.

SECTION 1.-On the theory of contact of plane curves.

304.] IN Art. 214 a tangent is defined to be that line which passes through two consecutive points on a curve; and therefore it follows that there are two points common to the tangent and to the curve; and in Art. 290, it is shewn that three points are common to the curve and to the circle of curvature. This property of curves having consecutive points in common, or, as it is called, having contact, it is our object to generalize, and with reference to it we define as follows:

Curves which have two consecutive points in common are said to have contact of the first order: those which have three consecutive points in common are said to have contact of the second order; and similarly two curves have contact of the nth order, if they have (n + 1) consecutive points in common.

Thus ordinarily, a tangent line has contact of the first order with a curve; and the circle of curvature has contact of the second order.

Curves which possess these relative properties are also called osculating curves, and curves are said to osculate to each other.

Nothing is said as to curves having only one point in common, because such a condition implies no more than that they intersect, and does not enable us to determine the relative direction of the curves.

305.] Hence then it appears that if for two curves whose equations are,

y = f(x), n = $(8),

(1)

we have the series of common points indicated in the following

(x,y), (x+dx, y+dy) { the two curves have contact of the (§, n), (§ + d§, n+dn)' first order;

(x,y), (x+dx, y+dy), (x + 2 dx + d2 x, y + 2 dy+d2y) ) (§, n), (§ + d§, n + dn), (§ +2 d§ + d2 §, n + 2 dŋ + d2n) S

the two curves have contact of the second order;

and similarly as to contact of the nth order, for which it is necessary that the successive differentials of the variables up to the nth should be equal in both curves.

These conditions become greatly simplified if we consider x and to be equicrescent, and each to increase by the same augment, in which case the several differentials of x and §, after the first, vanish; and we have, if

and if, besides, all the several successive differential coefficients, up to the nth inclusively, are equal in both curves, there is contact of the nth order.

Hence, if two curves have contact of the first order, they have a point in common, and the same tangent at the point; and therefore the tangent has contact of the first order. And if two curves have contact of the second order, they have not only a common tangent at the common point, but the curvature is the same, and is turned in the same direction; that is, they have the same circle of curvature.

306.] Hereby then the criterion of the order of contact assumes a new form; it depends on the number of the successively-derived functions of the equations to the curves which are equal; hence we are led to the following mode of viewing the subject, from which many important properties may be deduced.

Let

y = f(x), n = $(§),

(2)

be the equations to the two curves, which have a common point; that is, let x = , y = n; and let y' and ' be the ordinates corresponding to the abscissa x+h; then

* is used as the general symbol of a proper fraction, and does not necessarily represent the same quantity in (3) and (4), or in the subsequent equations: this is manifest from the argument of Chapter IV.

Therefore if the contact is of the first order, that is, if n = 2, f(x) = (x), f'(x) = '(x); and

that is, if h is infinitesimal, the difference between the ordinates corresponding to x+h is an infinitesimal of the second order.

And no other line can pass between these two curves, unless it has with each of them a contact of at least the first order; for suppose it to be possible that the ordinate y, corresponding to the abscissa x+h of the curve y = F(x) should be such that F(x) is not equal to f (x), then

which difference is obviously greater than that given in (5), if h is infinitesimal; and therefore the curve y = F(x) does not come between the curves y = f(x) and y

=

φ (α).

If the contact is of the second order, then, besides the former conditions, we have

ƒ" (x) = "(x);

and, if n = 3, then subtracting (4) from (3), we have

that is, the difference between the ordinates corresponding to the abscissa xh is an infinitesimal of the third order.

And if there is another curve, y = F(x), such that r'(x) is not equal to f'(x), although r'(x) = f'(x); then, if y1 is the ordinate of this third curve corresponding to the abscissa +h,

h2

{f"(x+0h)-F" (x+0h)};

(8)

y' — y1 = 1.2 which difference, being an infinitesimal of the second order, is obviously greater than that given by equation (7); and therefore this third curve does not come between the first two curves. Similarly, if the contact between two curves is of the nth order, the difference of the ordinates corresponding to xh, when h is infinitesimal, is an infinitesimal of the (n + 1)th order. Hence we have the following theorems:

"Two curves which have contact of the nth order are infi

nitely nearer to one another than two curves which have contact of an order lower than the nth."

"A curve which has contact of the nth order cannot come between two curves which have contact of an order higher than the nth."

"Two curves which have contact of the nth order with a third curve, have contact of at least that order with each other."

307.] An inspection of the equations (5) and (7) above, and of other equations formed in a similar manner, and giving the difference between the ordinates y' and n', corresponding to the several orders of contact, leads to the following theorem:

"If the contact of two curves is of an odd order, they touch and do not intersect; and if the contact is of an even order, they touch and intersect."

For suppose the contact to be of the nth order,

-

Then, if n is even, y' — n' changes its sign as h changes sign; and therefore f(x− h) — 4(x− h) and f(x+h)−4(x+h) have different signs, and therefore the curves intersect at the point of contact. But if the contact is of an odd order, n is odd and n+1 is even, and y'- n' does not change sign with h; that is, the curve which was nearer to the axis of a before contact is nearer to it afterwards, and the curves do not intersect.

x

308.] Suppose two curves, of one of which the equation is given, and contains certain fixed constants so that its form and position are completely determined; and the equation to the other involves arbitrary constants, in the determination of which we may make the curve fulfil certain conditions; we will shew that these latter may be satisfied by making the curve have with the former curve a contact, the order of which depends on the number of undetermined constants.

Let

F(x, y) = 0,

be the equation to the former curve; and

f(E, n, C1, C2, ... Cn)

......

= 0,

(10)

(11)

that to the latter, in which C1, C2, Cn are n arbitrary constants, and to be determined. By the theory of algebraical elimination it is plain that there must be n independent equa

tions to determine the n unknown quantities; let n equations be formed by making the curve pass through n given points, that is, by substituting successively the coordinates to the given points for the current coordinates to the curve in its equation; and let us suppose these n points to be on the curve (10), and, which is allowable, to be infinitesimally consecutive points; then making x and έ equicrescent, and variables with the same augments, in the two curves, by the latter part of Art. 305, all the successively-derived functions up to the (n-1)th of the equations of the two curves must be equal; up to the (n-1)th, I say, for thereby will n consecutive points be common, and sufficient conditions will have been introduced for the determination of all the constants.

And if the latter curve has with the former a contact of an order lower than the (n-1)th, the conditions will not be sufficient to determine all the n constants; and therefore the form and position of the latter curve may vary; and the number of curves satisfying the conditions is infinite.

309.] Suppose the equation to the latter curve to contain two arbitrary constants, and to be of the form,

f(, n, C1, C2) = 0;

then, differentiating,

(12)

(13)

by means of which equations, when x and y are substituted for and 7, in combination with r (x, y) = 0, we may determine c1 and c2, and find a curve which will have contact of the first order with the latter curve.

Ex. 1. To determine the values of the constants a and b, so that the straight line whose equation is

may have contact of the first order with the curve

Since (14) is to pass through a point on (15), we have

(14)

(15)

(16)

a

Also differentiating (14),

PRICE, VOL. I.

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