Achilles himself was not destined to a long life. Having by chance seen Polyxena, daughter of King Priam, perhaps on occasion of the truce which was allowed the Trojans for the burial of Hector, he was captivated with her charms; and to win her in marriage, it is said (but not by Homer) that he agreed to influence the Greeks to make peace with Troy. While the hero was in the temple of Apollo negotiating the marriage, Paris discharged at him a poisoned arrow, which, guided by Apollo, fatally wounded him in the heel. This was his only vulnerable spot; for Thetis, having dipped him when an infant in the river Styx, had rendered every part of him invulnerable except that by which she held him. 

The body of Achilles so treacherously slain was rescued by Ajax and Ulysses. Thetis directed the Greeks to bestow her son's armor on that hero who of all survivors should be judged most deserving of it. Ajax and Ulysses were the only claimants. A select number of the other chiefs were appointed to award the prize. By the will of Minerva it was awarded to Ulysses, -- wisdom being thus rated above valor. Ajax, enraged, set forth from his tent to wreak vengeance upon the Atridae and Ulysses. But the goddess robbed him of reason and turned his hand against the flocks and herds of the Argives, which he slaughtered or led captive to his tent, counting them the rivals who had wronged him. then the cruel goddess restored to him his wits. And he, fixing his sword in the ground, prepared to take his own life:

"Coma and look on me,
O Death, O Death, -- and yet in yonder world 
I shall dwell with thee, speak enough with thee:
And thee I call, thou light of golden day,
Thou Sun, who drivest on thy glorious car,
Thee, for this last time, -- never more again!
O Light, O sacred land that was my home;
O Salamis where stands my father's hearth.
Thou glorious Athens, with thy kindred race;
Ye streams and rivers here, and Troia's plains,
To you that fed my life I fid farewell;
This last, last word does Ajax speak to you;
All else, I speak in Hades to the dead."

(Sophocles)

Then, falling upon his sword, he died. So, in the words of his magnanimous foe, Ulysses, passed to the god that ruleth in gloom.

"The best and bravest of the Argive host,
Of all that came to Troia, saving one,
Achilles' self"

(Sophocles)

On the spot where his blood sank into the earth a hyacinth sprang up, bearing on its leaves the first two letters of his name, Ai, the Greek interjection of woe.

It was now discovered that Troy could not be taken but by the aid of the arrows of Hercules. They were in possession of Philoctetes, the friend who had been with Hercules at the last and had lighted his funeral pyre. Philoctetes had joined the Grecian expedition against Troy; but he accidentally wounded his foot with one of the poisoned arrows, and the smell from the wound proved so offensive that his companions carried him to the isle of Lemnos and left him there. Diomede and Ulysses, or Ulysses and Neoptolemus (son of Achilles), were now sent to induce him to rejoin the army. They succeeded. Philoctetes was cured of his wound by Machaon, and Paris was the first victim of the fatal arrows.