How does Polonius proceed to prove his theory about the cause of Hamlet's madness?act 2 scene 2

Polonius, in Shakespeare's, Hamlet, first tries to convince Claudius and Gertrude that he knows the cause of Hamlet's odd behavior, then he and Claudius try to prove or disprove the theory by sending Ophelia to engage Hamlet in a conversation while they listen in. 

The section begins at line 86 of Act 2.2, although if you skim through Polonius's long-winded introduction it really begins at line 102:

Mad let us grant him [Hamlet] then.  And now remains

That we find out the cause of this effect,

Or rather say, the cause of this defect,

For this effect defective comes by cause [he's still talking in circles].

thus it remains, and the remainder thus.  Perpend.

I have a daughter--have while she is mine--

Who in her duty and obedience, mark,

Hath given me this.  Now gather and surmise.

(reads a letter) "To the celestial and my soul's idol, the most

beautified Ophelia"--That's an ill phrase, a vile phrase....

Polonius continues attempting to convince the king and queen that unrequited love from Ophelia has led to Hamlet's madness until they accept that it is a possiblity.  Polonius then says he'll attempt to prove it:

At such a time I'll loose my daughter to him.

(to CLAUDIUS) Be you and I behind an arras then,

Mark the encounter.  If he love her not

And be not from his reason fall'n thereon,

Let me be no assistant for a state

But keep a farm and carters.  (Act 2.2.163-168)

Polonius wages his position as adviser to the king that his theory will be proven correct.

Of course, their spying proves nothing, and Polonius's fondness for listening in behind an arras gets him killed later in the play.