For the structure of the Republican Army and its triarii one has to go to the earliest source, Polybius: Livy we believe relied upon Polybius and gives additional information.
Polybius 6.21.7 onwards states that there were 4 types of Roman infantry which formed up in separate lines, the rearmost line consisting of the Triarii. Polybiius goes on to explain that there were 30 maniples of Triarii in a legion, each maniple consisting of two centuries of 60 men each. It appears that each maniple of Triarii would have had 24 velites attached to it.
Livy, Book 8, Chapter 8, goes on to explain the structure and use of the Triarii as well as the interactions betwen the lines:
"The Romans formerly used targets; afterwards, when they began to receive pay, they made shields instead of targets; and what before constituted phalanxes similar to the Macedonian, afterwards became a line drawn up in distinct companies. At length they were divided into several centuries. A century contained sixty soldiers, two centurions, and one standard-bearer. The spearmen (hastati) formed the first line in fifteen companies, with small intervals between them: a company had twenty light-armed soldiers, the rest wearing shields; those were called light who carried only a spear and short iron javelins. This, which constituted the van in the field of battle, contained the youth in early bloom advancing towards the age of service. Next followed men of more robust age, in the same number of companies, who were called Principes, all wearing shields, and distinguished by the completest armour. This band of thirty companies they called antepilani, because there were fifteen others placed behind them with the standards; of which each company consisted of three divisions, and the first division of each they called a pilus. Each company consisted of three ensigns, and contained one hundred and eighty-six men. The first ensign was at the head of the Triarii, veteran soldiers of .tried bravery, the second, at the head of the Rorarii, men whose ability was less by reason of their age and course of service; .the third, at the head of the Accensi, a body in whom very little confidence was reposed. For this reason also they were thrown back to the rear. When the army was marshalled according to this arrangement, the spearmen first commenced the fight. If the spearmen were unable to repulse the enemy, they re treated leisurely, and were received by the Principes into the intervals of the ranks. The fight then devolved on the Principes; the spearmen followed. The Triarii continued kneeling behind the ensigns, their left leg extended forward, holding their shields resting on their shoulders, and their spears fixed in the ground, with the points erect, so that their line bristled as if enclosed by a rampart. If the Principes also did not make sufficient impression in the fight, they retreated slowly from the front to the Triarii. Hence, when a difficulty is felt, "Matters have come to the Triarii," became a usual proverb. The Triarii rising up, after receiving the Principes and spearmen into the intervals between their ranks, immediately closing their files, shut up as it were the openings; and in one compact body fell upon the enemy, no other hope being now left: that was the most formidable circumstance to the enemy, when having pursued them as vanquished, they beheld a new line suddenly starting up, increased also in strength."
Note the explanation of the kneeling pose and its reason and also the proverb "
res ad triarios rediit" meaning
the situation has come down to the triarii, the third line of defense (i.e., the situation is critical) (Livy)
Livy goes on to explain how the Triarii fought in the battle of Mount Vesuvius against the similarly equipped and structured Latins in 340BCE, Book 8 Ch 10-
" while the struggle continued, and in some parts of the field the Latins were prevailing by reason of their numbers, the consul Manlius learned of his colleague's end, and having paid to so memorable a death —as justice and piety demanded —its well —merited [2] meed of tears as well as praise, he was for a little while in doubt whether the moment were yet come for the
triarii to rise; but afterwards deeming it better to keep them fresh for the final push, he commanded the
accensi to advance from the rear before the standards. [3] no sooner had they gone up, than the Latins, supposing their enemies had done [p. 41]the same, sent in their own
triarii. These having
1 fought fiercely for some time, and worn themselves out and broken or blunted their spears, yet were driving back the foe, and supposed that they had already won the field and penetrated the last line, when the consul cried out to the Roman
triarii: [4] “rise up now, and with fresh strength confront the weary enemy, remembering your country and your parents, your wives and your children, remembering the consul who lies dead that you may conquer.” [5] when the
triarii had got to their feet, fresh and sound in their glittering armour, a new and unforeseen array, they received the
antepilani into [6??] the gaps between their files, and, raising a shout, threw the enemy's front ranks into disorder, and thrusting their spears into their faces, disposed of the fine flower of their manhood and went through the other maniples almost scatheless, as though their opponents had been unarmed, penetrating their masses with such slaughter as scarce to leave a fourth part of their enemies alive."
It must have been desperate for the Triarii on a very personal level- as the oldest soldiers in the army, if they did not prevail, escape would probably prove difficult.
Whilst Livy mentions officers within the maniples and centuries, there is no indication at all here that the Triarii acted as NCOs in relation to the rest of the army, nor indeed that they impeded the withdrawal of their fellows, rather the opposite is true- having opened ranks to permit their withdrawal, these were closed up again to protect them.