Help needed in understanding the procedure to calculate the envelope of a two parameter family of surfaces.

I was reading the book Linear Partial Differential Equations for Scientists and Engineers (Fourth Edition) written by Tyn Myint-U and Lokenath Debnath.

In Section 2.3 titled as "Construction of a First-Order Equation" I am having trouble in understanding the following:

We obtain the general solution of $F (x,y,z,p,q)=0\tag{2.3.3}$ from its complete integral $f(x,y,z,a,b)\tag{2.3.1}$ as follows.

First, we prescribe the second parameter $b$ as an arbitrary function of the first parameter $a$ in the complete solution $(2.3.1)$ of $(2.3.3),$ that is, $b = b(a).$ We then consider the envelope of the one-parameter family of solutions so defined. This envelope is represented by the two simultaneous equations $f (x,y,z,a,b(a)) = 0\tag{2.3.5},\\$ $f_a (x,y,z,a,b(a)) + f_b (x,y,z,b(a))b(a)=0,\tag{2.3.6}$ where the second equation $(2.3.6)$ is obtained from the first equation $(2.3.5)$ by partial differentiation with respect to $a.$ In principle, equation $(2.3.5)$ can be solved for $a = a(x,y,z)$ as a function of $x, y,$ and $z.$ We substitute this result back in $(2.3.5)$ to obtain $f (x,y,z,a(x,y,z), b(a(x,y,z)))=0,\tag{2.3.7}$ where $b$ is an arbitrary function. Indeed, the two equations $(2.3.5)$ and $(2.3.6)$ together define the general solution of $(2.3.3).$ When a definite $b(a)$ is prescribed, we obtain a particular solution from the general solution.

Since the general solution depends on an arbitrary function, there are infinitely many solutions. In practice, only one solution satisfying prescribed conditions is required for a physical problem. Such a solution may be called a particular solution.

In addition to the general and particular solutions of $(2.3.3),$ if the envelope of the two-parameter system $(2.3.1)$ of surfaces exists, it also represents a solution of the given equation $(2.3.3);$ the envelope is called the singular solution of equation $(2.3.3).$ The singular solution can easily be constructed from the complete solution $(2.3.1)$ representing a two-parameter family of surfaces. The envelope of this family is given by the system of three equations $f (x,y,z,a,b)=0, f_a (x,y,z,a,b)=0, f_b (x,y,z,a,b)=0. \tag{2.3.8}$ In general, it is possible to eliminate $a$ and $b$ from $(2.3.8)$ to obtain the equation of the envelope which gives the singular solution.

I studied about envelopes a long time ago when I took a course on ODEs. As far as I remember:

"The envelope of a family of curves is defined to be a curve which is tangent to all those curves in the family. For a family of surfaces, the envelope is defined to be a surface that touches each surface of the family at points, such that the tangent plane of the surfaces in the family at those particular points of contact are same as that of the envelope at those points of contact."

I hope by understanding of the envelope is correct. (Is it?)

But there are a few issues that I faced. I mention them below:

Issue 1:

The procedure of calculating the enevelope of two parameter family of surfaces $f(x,y,z,a,b)$ was stated as:

First, we prescribe the second parameter $b$ as an arbitrary function of the first parameter $a$ in the complete solution $(2.3.1)$ of $(2.3.3),$ that is, $b = b(a).$ We then consider the envelope of the one-parameter family of solutions so defined. This envelope is represented by the two simultaneous equations $f (x,y,z,a,b(a)) = 0\tag{2.3.5},\\$ $f_a (x,y,z,a,b(a)) + f_b (x,y,z,b(a))b(a)=0,\tag{2.3.6}$ where the second equation $(2.3.6)$ is obtained from the first equation $(2.3.5)$ by partial differentiation with respect to $a.$

But again after a few lines the procedure to calculate the envelope of $f(x,y,z,a,b)$ is given as:

The envelope of this family is given by the system of three equations $f (x,y,z,a,b)=0, f_a (x,y,z,a,b)=0, f_b (x,y,z,a,b)=0. \tag{2.3.8}$

and this procedure is completely different from the one given previously.

So, are both the procedure valid in general when trying to find the envelope of an arbitrary family of surfaces having 2 parameters?

Issue 2:

I had a question if the envelope represented by the two equations $2.3.5,2.3.6$ can be represented with the help of only one equation? This is because, it's said that this envelope constitutes a singular solution of the given PDE. But to obtain the singular solution I need to have the envelope in the form $u=g(x,y)$ so that I can verify if $f(x,y,u,p,q)=0$ holds true?

But this question seems to have been answered in the paragraphs that followed, i.e in the portion,

The singular solution can easily be constructed from the complete solution $(2.3.1)$ representing a two-parameter family of surfaces. The envelope of this family is given by the system of three equations $f (x,y,z,a,b)=0, f_a (x,y,z,a,b)=0, f_b (x,y,z,a,b)=0. \tag{2.3.8}$ In general, it is possible to eliminate $a$ and $b$ from $(2.3.8)$ to obtain the equation of the envelope which gives the singular solution.

So, as it stands that the resulting equation obtained after eliminating the parameter in the "set of equations that constitute the envelope" gives us the equation of the envelope as well, but this time, we can only represent the envelope with the help of only one equation instaead of two.

But I don't understand the justification behind this. I mean why does eliminating the parameter $a$ and $b$ provides us with an equation that represents the enevelope?

I understand that all the points of the enevelope will satisfy the resulting equation obtained by eliminating those parameters. But that does not mean that all the points satisfying that equation will belong to the enevelope.

Does this mean that eliminating the single parameter $a$ in the envelope obtained with the help of the first process mentioned in the quoted text namely, equation $2.3.5,2.3.6$ gives us the equation of the enevelope as well?

Any help regarding these two issues will be greatly appreciated.