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Section 2.— Tlie theory of envelopes.

313.] In the discussion of the relative properties of evolutes and involutes, in Chapter XII, Arts. 294 and 303, it is proved that the normal to the involute is a tangent to the evolute; or, in other words, that as each normal to the involute passes through two consecutive points of the evolute, the latter curve may be imagined to be made up of an infinite number of infinitesimal straight lines, each of which is a part of a normal to the involute; thus we say that the evolute is formed by the intersection of consecutive normals. To take another case: let us conceive a system of straight lines infinite in number, and varying in position infinitesimally from each other, such that the perpendicular from a given point on each of them is the same; then the curve formed by the intersection of all is a circle. Or again, suppose that an infinite number of equal circles have their centres along a straight line, and infinitesimally near to each other; then they all intersect in, and by their intersections form, two straight lines parallel to the given line. Curves formed in this manner, by the ultimate intersection of straight lines or curves drawn according to some given law, are called envelopes, and are said to envelope the family of straight lines or curves. We proceed to discuss the general theory of them.

It is plain from algebraical as well as geometrical reasoning, that if an equation to a curve is given, involving one or more constants, as well as the current coordinates, the position and dimensions of the curve will be changed by a change in the constants, and yet generally the kind of curve will remain the same; that is, a variation of a constant may involve a specific though not a generic change of curve. A constant that enters into an equation, and varies in the way above explained, is called a variable parameter. Thus in the equation to a parabola, y2 = 4 ma?, as m varies, the form of the parabola will vary, because its latns rectum varies, though its vertex and principal axis are unaltered. In the equation of the ellipse

a and b may be variable parameters; in which case, changes of them will involve a change of an individual ellipse, though the family represented by the equation will remain that of ellipses still.

314.3 Let the equation to the family of curves, of which it is our object to determine the envelope, involve only one parameter, and be „ . 0 ,00.

F (ar, y, a) = 0, (38)

in which a is the variable parameter; so that for every value of a we have some particular curve, but if we make a to vary infinitesimally and continuously, we have a series of curves, the position of each one differing infinitesimally from that of the next. Suppose a to receive a variation da, then the two curves whose equations are (38) and

v(x,y,a-j-da) = 0, (39)

are in position infinitesimally near to another; but owing to the variation of a they will in general intersect in some point, which will be determined by x and y being the same in both

(38) and (39), and which will be a point on the envelope. If therefore we eliminate a between (38) and (39), the resulting equation will involve only x and y, and will be the equation to the envelope.

Before however we proceed to apply the method, we may put

(39) under a more convenient form. By (21), Art. 116, v(x,y, a + da) r (x,y,a) + da r' (x,y, a + dda) = 0;

and therefore by reason of equation (38)

F' (x, y,a + d da) = 0; and therefore in the limit, when da is infinitesimal,

*{x,y,a) = 0. (40)

To determine therefore the envelope of the family of curves whose general equation is F (x, y, a) = 0, and of which the several individuals are formed by making a to vary, we must


eliminate a between F = 0, and -7- = 0.


The geometrical conception of such envelopes evidently requires that the particular curve and the envelope should have the same tangent at their common point. And this truth is also manifest from the following considerations:

Differentiate (38), making x, y and a to vary, then

(£>* + ($* + (£«■-<». m dr

but by reason of (40), ^- = 0; therefore, whether o varies or not, we have the same equation, viz.

whereby to determine and therefore the tangent is the same to the envelope and to each curve at their common point.

315.] Ex. 1. To determine the equation to the curve formed by the intersection of the straight lines whose equation is m .

w = ax H , where a vanes.

Differentiating with respect to a, x and y being constant, we have ,

a2 v x 1

■'■ V = ± {*/(mx) + V{mx)}; y2 = 4-mx; which is the equation to a parabola.

Ex. 2. To determine the envelope of the straight lines, of which the general equation is y = ax + (a2a2 + 62)4, wherein a varies.

Differentiating with respect to a, we have

. a2a b x
0 = x H .-. a=

(a2a2 + A2)i' « (a2-*2)*'

substituting which in the general given equation, and reducing, we have , ,

a2 + b2

On examination of these two examples it will be seen, that the determination of envelopes produced by straight lines is the inverse one to that of finding the equation to a tangent to a curve; for the two equations to the straight lines given in the two preceding examples are those known by the name of the "magical" equations to the tangents severally of the parabola and the ellipse. In this case then we have the equation to the tangent given, and the problem is, to determine the curve; in the other case the equation to the curve is given, and we have

Price, Vol. 1. 3 p

to determine that to the tangent. Hence the method of envelopes has been sometimes called "the inverse method of tangents."

The geometrical property involved in Ex. 1 is, "From a point in the axis of x, at a distance m from the origin, lines are drawn cutting the axis of y; at the points of intersection other lines are drawn perpendicular to these; to find the envelope of these latter lines." And that involved in Ex. 2 is, "To find the envelope of a series of straight lines drawn, so that the product of the two ordinates at distances + a from the origin may be equal to b2."

Ex. 3. To determine the envelope of all parabolas expressed by the equation y = ax - ^ ° x2, wherein a varies.


Differentiating, with respect to a, we have

0 = x— -x2; .'. a = ~;

p x

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The envelope therefore is another parabola, having its focus at the origin.

Ex. 4. It is required to find the envelope of normals drawn to a given curve.

Let the equation to the curve be y =f{x); then the equation to the normal is

0»-y>5£ + f-* = o- (42)

Differentiating, considering »; and f, which are the current coordinates to the normal, to be constant, we have

From which, and from (42),

1 + dy* l + dy

= y+_^f, t = x--J£d£; (44) d2y d2y dx

dx2 dx2

which are the same expressions as (33), Art. 288; and therefore

the evolute is the envelope of the normals. The method pur

sued above is manifestly the same as that of Chapter XII, but expressed in different language.

316.] If the equation representing the family of curves involve many, say w, variable parameters, and these parameters are related by (w —1) other and independent equations, which conditions are equivalent to there being only one variable parameter, instead of eliminating (n — 1) parameters, and then differentiating with respect to the remaining one, and proceeding as in the last Article, the following method is more elegant.

Let the equation to the family of curves be

r (*> y, «i, «2, as, a«) = 0; (45)

and let the (w — 1) equations of condition be f\ «2 a») = 0,

fa (<*!> 02, «n) = 0,

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Now, if x and y are the same in (45) and (47), they refer to the point in the envelope where the two particular curves of the famUy intersect; and therefore if the several variable parameters and their differentials can be eliminated, the resulting equation between x and y will represent the locus of the points of intersection, which will be the envelope required.

By equations (45) (48) we have 2n different relations, from which (2n 1) quantities, viz. a\, a*, a3, o„,

^a"~1, are to be eliminated: which is, of course, a

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possible problem. To solve it, multiply the (»—1) equations

in (48) by (» —1) indeterminate multipliers Xu A2, X3, K-u

and add all and (47) together; by which means there will be n conditions involved in one equation; and therefore we are at

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