There not content free as such, but Russell attempted to show that when stated in particular way they were no longer a metaphysical sentence and consisted of assertions which could be shown to be clearly false. Russell achieved this through denoting.

Take the sentence 'The Present King of France is bald'. Russell said that the sentence is equivalent to :
1) There is a king of France
2) There is not more than one King of france
3) Whatever is the king is bald

1) is an existence claim
2) is a uniqueness claim
3) is the prediction

The Orginal sentence is shown to be true when all the three assertions are true, and the sentence is false if anyone of them is false. This was used by others such Ayer and other Logical Positivist's with the aim to show that metaphysical statements where meaningless.

How does Russell's Theory of Descriptions show metaphysics is meaningless? Russell held that the sentence, "The present king of France is bald" is false, not meaningless, because it implies that there is a present king of France, and there is none.

A.J Ayer a logical positivist use's Russell's theory of descriptions in his work Language,Truth and Logic as part of his arguement against metaphysical statements, I forget which chapter it is included in but I think it may be in the chapter entitled the Elimination of metaphysic's. And A.J Ayer would probably say that the sentence 'The Present king of France is bald' is not meaningless but has no meaning in use. I never asserted that Russell held the view that his theory of descriptions rendered metaphysical sentence's meaningless, but rather groups and individuals used his developments in this area with the aim of Eliminating metaphysic's.

I don't think you are right about Ayer using Russell to show that TPKOFB is meaningless. Ayer did not think it was meaningless. The concept of "meaning in use" is quite a different matter. You are just guessing. The Logical Positivists (like Ayer) employed the verification theory of meaning to "elimininate metaphysics". Russell's theory of descriptions had nothing to do with it.

I'm not guessing it is actually in Language, Truth and Logic which I have a copy off and have read. Russel's theory of descriptions is talked about from page 49-53 in the Nature of Philosophical Analysis and then again in the Appendix from page 183-197, Ayer did not use Russell's theory to show that the sentence was meaningless but it was certainly part of his attempt to bring about the eliminination metaphysics using certain tool's of logical analysis.

YouTube - A.J. Ayer; The Metaphysician's Nightmare

If you say so. How did he use that tool to "bring about the elimination of metaphysics".

Russell held that the sentence, "The present king of France is bald" is false, not meaningless, because it implies that there is a present king of France, and there is none.

If "The present king of France is bald" is false, then its negation
"It is not the case that the present king of France is bald"
must be true. But surely the second sentence is equivalent to
"There is a present king of France and it is not the case that he is bald".
But the last sentence is obviously not true. How do you resolve this contradiction?

He showed that metaphysics cannot give us knowledge about the world which is what it claims to do.

There are two negations of, the present king of France is bald, depending on whether the negation has wide scope, or narrow scope Thus:

It is not the case that the present king of France is bald; and, The present king of France is not bald.

The latter sentence is false, since there is no present king of France. And, of course, the former sentence is true, for the same reason. What is the contradiction?

Take the sentence: "It is not the case that the present king of France is bald". By using the definite article before "present king", it implies that there is such a king of France, and then states that it is not the case that he is bald. But the implied assertion is false; there is no such person. So the sentence as a whole is not true, since one of its assertions is false.

On the other hand, it seems that if "The present king of France is bald" is false, then "It is not the case that the present king of France is bald" must, as its negation, be true. Hence the paradox.

The only way of resolving this difficulty would be to regard "The present king of France is bald" as meaningless, not false. Then "It is not the case that...." could also be regarded as meaningless.

Metaphysics is systematic, what's wrong with it as an exercise? I value it a lot, it just can't be meaningless!

Take the sentence: "It is not the case that the present king of France is bald". By using the definite article before "present king", it implies that there is such a king of France, and then states that it is not the case that he is bald. But the implied assertion is false; there is no such person. So the sentence as a whole is not true, since one of its assertions is false.

On the other hand, it seems that if "The present king of France is bald" is false, then "It is not the case that the present king of France is bald" must, as its negation, be true. Hence the paradox.

The only way of resolving this difficulty would be to regard "The present king of France is bald" as meaningless, not false. Then "It is not the case that...." could also be regarded as meaningless.

Metaphysics is a branch of philosophy that investigates principles of reality transcending those of any particular science. Cosmology and ontology are traditional branches of metaphysics. It is concerned with explaining the fundamental nature of being and the world.[1]

That the human mind will ever give up metaphysical researches is as little to be expected as that we should prefer to give up breathing altogether, to avoid inhaling impure air.
There will therefore always be metaphysics in the world; nay, every one, especially every man of reflection, will have it, and for want of a recognized standard, will shape it for himself after his own pattern.
What has hitherto been called metaphysics, cannot satisfy any critical mind, but to forego it entirely is impossible;

Metaphysics, as a natural disposition of reason, is actual, but if considered by itself alone (as the analytical solution of the third principal question showed), [is]dialectical and illusory.
If we think of taking principles from it, and in using them follow the natural, but on that account not less false, illusion, we can never produce science, but only a vain dialectical art, in which one school may outdo another, but none can ever acquire a just and lasting approbation.