If you're reading for the first time, you assume his "departure" is just his booking off for Gondor, about which he's constantly harped hitherto. But alas! it's the other definition.
Tricksy.
Dan
So now as the clues add up, the mystery becomes clear...
'How then do you read this riddle?' asked Gimli.Aragorn did not answer at once, but went back to the camping-place and looked at the baggage. 'Two packs are missing.' he said, 'and one is certainly Sam's: it was rather large and heavy. This then is the answer: Frodo has gone by boat, and his servant has gone with him. Frodo must have returned while we were all away. I met Sam going up the hill and told him to follow me; but plainly he did not do so. He guessed his master s mind and came back here before Frodo had gone. He did not find it easy to leave Sam behind!'
'But why should he leave us behind, and without a word?' said Gimli. 'That was a strange deed!'
'And a brave deed,' said Aragorn. 'Sam was right, I think. Frodo did not wish to lead any friend to death with him in Mordor. But he knew that he must go himself. Something happened after he left us that overcame his fear and doubt.'
'Maybe hunting Orcs came on him and he fled,' said Legolas.
'He fled, certainly,' said Aragorn, 'but not, I think, from Orcs.' What he thought was the cause of Frodo's sudden resolve and flight Aragorn did not say. The last words of Boromir he long kept secret.
'Well, so much at least is now clear,' said Legolas: 'Frodo is no longer on this side of the River: only he can have taken the boat. And Sam is with him; only he would have taken his pack.'
'Our choice then,' said Gimli, 'is either to take the remaining boat and follow Frodo, or else to follow the Orcs on foot. There is little hope either way. We have already lost precious hours.'
'Let me think!' said Aragorn. 'And now may I make a right choice and change the evil fate of this unhappy day!' He stood silent for a moment. 'I will follow the Orcs,' he said at last. 'I would have guided Frodo to Mordor and gone with him to the end; but if I seek him now in the wilderness, I must abandon the captives to torment and death. My heart speaks clearly at last: the fate of the Bearer is in my hands no longer. The Company has played its part. Yet we that remain cannot forsake our companions while we have strength left. Come! We will go now. Leave all that can be spared behind! We will press on by day and dark!'
So Aragorn comes to a decision... but he withholds some of the reasoning. Frodo certainly left alone (or tried to) to spare his friends from his hopeless journey, but also to spare himself from the evil temptation The Ring might bring. (does Aragorn fear the same fault that Boromir had?)
Who among us would make the other choice and follow Frodo? - After all, Frodo's mission is more important, is it not, than saving Merry and Pippin. Would you have held to the mission, or to plight of the Merry and Pippin?