What does "meaning" mean in Whitehead and Russell's PM? In Principia Mathematica's Introduction, there is a definition for "incomplete" symbol:

By an "incomplete" symbol we mean a symbol which is not supposed to
  have any meaning in isolation, but is only defined in certain
  contexts.  -- Chapter III, Principia Mathematica, 1st edition, page 69.

I am not sure what "meaning" means in this definition. Judging by the passage that follows, I guess if a symbol has a "meaning," that symbol stands for an object, like "Socrates" stands for a certain man. As a matter of fact, in Russell's An Inquiry of Meaning and Truth, the meaning of an object-word is the thing it stands for. Please let me know if "meaning" in PM has the same definition as in An Inquiry of Meaning and Truth. Thanks.
 A: Russell is developing his theory of descriptions here. Roughly, he takes an "incomplete symbol" to be one that does not refer -- one that does not have a denotation in the way that proper names do. So, for example, the expression "the author of Waverley" -- according to Russell's theory -- does not denote; you cannot ask what its referent is. (EDIT 2: Russell does use the notation $\psi(\iota x\phi (x))$, but this doesn't mean that $\iota x\phi(x)$ denotes an object that can serve as the argument of $\psi(y)$, because $\psi(\iota x\phi(x))$ is really shorthand for $\exists x(\phi(x)\land\forall y(\phi(y)\rightarrow y=x)\land \psi(x))$.) 
So noun phrases like "the author of Waverley," or "the more famous author of Principia Mathematica," cannot be thought of as proper names; and since Russell thinks of "meaning" in terms of reference, the expression cannot be said to have a meaning "in isolation." (One reason Russell makes this argument is to help explain how to formalize language about objects that don't exist, e.g. "the present King of France is bald" or "the greatest prime number.") Only in a sentence like "Scott is the author of Waverley" does the expression acquire a referent: that is what he means by the meaning only being "defined in certain contexts." In short, some expressions that look like object-words or proper names -- i.e., expressions that look like they refer (to an object) -- actually turn out not to be, if you're careful about the logical analysis.
Whether or not you are a proponent of the Russellian theory of definite descriptions, the mathematical examples Russell gives at the beginning of chapter 3 should not be taken too seriously. He says, for example, that the symbol $\frac{d}{dx}$ should be thought of as an incomplete symbol -- that it should not be thought of as having a denotation or referent by itself, when it isn't "completed" by a function symbol: e.g $\frac{d}{dx}(x^2)$.
To a contemporary mathematician, though, this might sound silly. A contemporary mathematician would likely say that $\frac{d}{dx}$ denotes a certain operator on a function space (e.g. the space of smooth functions from $\mathbb{R}$ to $\mathbb{R}$). Of course, there may be some ambiguity about which function space is meant, but that doesn't pose much of a problem: it is usually clear from context which space of functions one is working over.
A: Following your own answer, recall that [page 66] :

By an "incomplete" symbol we mean a symbol which is not supposed to have any meaning in isolation, but is only defined in certain contexts.

The issue is similar to that of terms [i.e.names] without denotation.
Basically, the "philosophy of meaning" of PM is based on propositions which "speak of" facts: the "ideal" language of PM's logic must express only true or false propositions.
Due to the paradoxes, R&W - in order to avoid the risk of propositions which are not true nor false - introduce the concept of "meaningless" expression. In their system, a menaingless proposition is detected by the fact that it violates syntactic constraints (see: types theory).
The paradigmatic case of "incomplete" symbol is that of definite descriptions, like :

the King of France is bald.

What is the meaning of the above proposition if there is no King of France ?
The solution Russell discovered was based on the well know paraphrase :

$(\exists x) (king_F(x) \land bald(x)) \land (\forall y)[(king_F(y) \land bald(y)) \rightarrow x = y]$.

Thus, in conclusion, if there is no King of France, the left conjunct is false, so that the complete sentence is false. We have found a way to give "meaning" to an expression also when it "seemingly" includes a non-denoting term.
Russell's solution is : circumvent the necessity of a term to safeguard "compositionalty" of meaning (i.e.the meaning or truth-value of the complete expression is based on the meaning of its components).
All this "machinery" is used in PM for classes [page 71] :

The symbols for classes, like those for descriptions, are, in our system, incomplete symbols: their uses are defined, but they themselves are not assumed to mean anything at all. That is to say, the uses of such symbols are so defined that, when the definiens is substituted for the definiendum, there no longer remains any symbol which could be supposed to represent a class. Thus classes, so far as we introduce them, are merely symbolic or linguistic conveniences, not genuine objects as their members are if they are
  individuals.

In other way, also class terms (i.e."modern" set terms : $\alpha = \{ x : \phi x \}$) like $\alpha = \hat{x} \phi x$, are "eliminated" in the same way, because non-denoting. 
