An Ambiguous Definition of Envelopes with reference to Differential Equations and Singular Solutions.

Question

Recently, I was, going through a definition of envelope. I know that the definitions can be written and the exposition of it, might vary, but the following definition, which I am hereby mentioning, has the definition of an envelope given with the reference to a differential equations of 1st order(, and probably 2nd degree,) and I am unable to make up /decipher the meaning. Here it is:

Note:Throughout this post, the variable $p$ designates, $\frac{dy}{dx}.$ The function, $f(x,y,p)=0$ respresents a differential equation and $\phi(x,y,c)$ represents the corresponding general solution of $f$ with $c$ as the arbitrary constant.

The envelope: If in $\phi(x,y,c)= 0,$ c be given all possible values, there is obtained a set of curves, infinite in number, of the same kind. Suppose that the $c'$s are arranged in order of magnitude, the successive $c$'s thus differing by infinitesimal amounts, and that all these curves are drawn. Curves corresponding, to two consecutive values of $c$ are called consecutive curves, and their intersection is called an "ultimate point of intersection". The limiting position of these points of intersection includes the envelope of the system of curves. It is shown in works on the differential calculus, that the envelope is part of the locus of the equation obtained by eliminating $c$ between $$\phi(x,y,c)=0,$$

and

$$\frac{\partial \phi}{\partial c}=0,$$

that is, the envelope is part of the locus of the $c$ discriminant relation. This might have been anticipated, because in the limit the $c$'s for two consecutive curves become equal, and the $c$ discriminant relation represents the locus of points for which $\phi(x,y,c)= 0,$ will have equal values of $c.$ It is also shown in the differential calculus, that at any point on the envelope, the latter is touched by some curve of the system; that is, that the envelope and some one of the curves have the same value of $p$ at the point.

The definitions used in this article for reference are :

p-discriminant-When the equation is quadratic, the discriminant can be written immediately; but when it is such that the condition for equal roots is not easily perceived, the discriminant is found in the following way. The given equation being $F=0,$ form another equation by (partially) differentiating $F$ with respect to the variable, and eliminate the variable between the two equations. For example,

$\phi(x,y,c)= 0,$

may be looked on as an equation in $c$, its coefficients then being funetions of $x$ and $y.$ The simplest rational function of $x$ and $y,$ whose vanishing expresses that the equation $\phi(x,y,c)=0$ has equal roots for $c,$ is called the $c$ discriminant of $\phi,$ and is obtained by eliminating $c$ between the equations,

$$\phi(x,y,c)= 0,$$

and

$$\frac{\partial \phi}{\partial c}=0.$$

Thus the $c$ discriminant relation represents the locus, for each point of which $\phi(x,y,c)= 0,$ has equal values of $c$. Similarly, the $p$ discriminant of $f(x, y, p) = 0,$ the differential equation corresponding to $\phi (x, y, c)= 0, $ is obtained by eliminating $p$ between the equations,

$$f(x,y,p)= 0,$$

and

$$\frac{\partial f}{\partial p}=0.$$

Thus the $p$ discriminant relation represents the locus, for each point of which $f(x, y, p)= 0$ has equal values of $p.$ In order that there may be a $c$ and a $p$ discriminant, the above equations must be of the second degree at least in $c$ and $p.$ Here, we assumed, that these equations are of the same degree in $c$ and $p,$ and hence, if there is a $p$ discriminant, there must be a $c$ discriminant.

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First of all, I dont understand the fact where it says,

"Curves corresponding, to two consecutive values of $c$ are called consecutive curves, and their intersection is called an "ultimate point of intersection."

I feel this is erroneous , for $c$ is an arbitrary real constant and there is nothing as, "consecutive real numbers," since there are infinite real numbers between any two real numbers.

Finally, where they are supposedly asserting the definition of the envelope as :

The limiting position of these points of intersection includes the envelope of the system of curves.

I don't have an idea, what are they trying to imply by the phrase, "limiting position of these points of intersection". I don't understand the meaning at all.

This whole article is turning to be much confusing. Any explanations addressing these issues will be greatly appreciated. An intuitive geometrical picture concerning this particular notion in the article will be much helpful.

The above definitions/contents appear in the book ,"An Introductory course in Differential Equations " by D.A Murray, on the chapter titled "Singular Solutions."

 

Answer 1

The writing style is typical "nineteenth-century" (or physicists') style. You look at the intersection point $P(c,\Delta c)$ of the curve $F(x,y,c)=0$ with the nearby curve $F(x,y,c+\Delta c)=0$. The point on the envelope is the limit $Q(c)=\lim\limits_{\Delta c\to 0} P(c,\Delta c)$. Here is the heuristic for why you solve for the envelope by solving $F(x,y,c) = \frac{\partial F}{\partial c}(x,y,c) = 0$.

We use the linear approximation as always: $$F(x,y,c+\Delta c) \approx F(x,y,c) + \frac{\partial F}{\partial c}(x,y,c)\Delta c,$$ so the point $P(c,\Delta c)=(x_0,y_0,c)$ lies on both curves provided $F(x_0,y_0,c) = 0$ and (approximately) $F(x_0,y_0,c) + \frac{\partial F}{\partial c}(x_0,y_0,c)\Delta c=0$. Since the approximation becomes better and better as $\Delta c\to 0$, this leads us to the equation $$F(x_0,y_0,c) = \frac{\partial F}{\partial c}(x_0,y_0,c) = 0,$$ for $Q(x_0,y_0,c)$, since $\Delta c\ne 0$.

You can find a rigorous discussion of this with the Implicit Function Theorem in numerous posts on this site. I have addressed the issue several times, myself.