An early account taken from the book - When Elephants Weep* - recounts the following by R.G. Cummings, a hunter in South Africa:
"He first shot it in the shoulder, so that it could not run away.
The elephant limped over to a tree and leaned against it.
Deciding to contemplate the elephant before killing it,
Cummings paused to make coffee and then chose to determine experimentally which were an elephant's vulnerable spots.
He walked up to it and fired bullets into various parts of the head.
The elephant did not move except to touch the bullet wounds with the tip of his trunk.
Surprised and shocked to find that I was only tormenting and prolonging the sufferings of the noble beast, which bore his trials with such dignified composure,"
Cummings wrote, he decided to finish him off and shoot him nine times behind the shoulder.
"Large tears now trickled from his eyes, which he slowly shut and opened; his colossal frame quivered convulsively, and, falling on his side, he expired.'"

This animal must have been in great pain and that alone would have caused him to shed tears.