Ad Victoriam

Ad Victoriam is a Brotherhood of Steel main quest and achievement in Fallout 4.

Detailed walkthrough
After your successful return from the Mass Fusion building with the Beryllium agitator, speak to Proctor Ingram at the Boston Airport. She will allow you to now power up Liberty Prime. Go to the top of the gantry and plug the agitator into Liberty Prime. After that, return to Ingram and press the button. Liberty Prime will activate, leave the gantry and start walking towards the C.I.T. ruins, with the Brotherhood of Steel personnel saluting it as it walks away.

You must now follow Liberty Prime as it walks through the ruins of Boston to reach the Institute. As it reaches the East Boston Preparatory School, synths will be teleported to attack it. From that point on, you will encounter various enemies: raiders, super mutants (including one or two behemoths), and synths. Due to a broken bridge, Liberty Prime will have to cross through the water in order to continue. Liberty Prime will reach the Bunker Hill monument, after what it will continue its journey through the streets of Boston.

Liberty Prime will annihilate everything thrown at it, using its powerful eye laser as well as nukes, even effortlessly crushing the behemoths that will attack the robot near Ticonderoga. Do note however that, unlike in Take it Back!, Liberty Prime will mostly use its Mark 28 bombs rather than its eye laser; thus, it is advised to stand back from Prime's enemies and let it destroy them, as you may get caught in the blast radius of one of Prime's bombs, which will deal a significant amount of damage if it does not kill you. Since Prime is entirely immune to his own weaponry and well aware of the fact, he has no qualms about using his nukes at point-blank range. During your trip to the Institute, you will be followed by a small group of Brotherhood of Steel knights, comprised of Knight Lucia and a few unnamed Brotherhood of Steel members, as well as a vertibird.

Eventually, Prime reaches the CIT ruins, which are guarded by a large group of synths. At this point, Elder Maxson, armed with his unique weapon, as well as more Brotherhood of Steel personnel, will join in the fight. Liberty Prime will analyze the area and detect the Institute, and proceed to use his weaponry to create an access. While Prime uses his eye laser and nukes to create an access, he will be unable to attack the synths, meaning you will have to deal with them, although Maxson's gatling laser often make short work of them. After some time, Prime will throw nukes at the front of the ruins, creating a hole that will allow the Brotherhood of Steel to gain direct access to the Institute. Jump down the hole to enter the Institute, completing this quest and starting the next one, The Nuclear Option.

Behind the scenes

 * Ad Victoriam is an accusative construct in Latin; ad being a preposition that is paired with an accusative, ad having the meaning meaning 'toward', and victoriam being the accusative form of the noun Victoria, ae, a first declension feminine noun meaning 'victory', translating to: 'toward victory'. Brotherhood of Steel members in Fallout 4 often use this quote, during combat or to celebrate victory.

Bugs

 * While following Liberty Prime he may stop walking and not continue, make sure to regularly quicksave in case this happens. This can be fixed by shooting him (which won't cause him to become aggressive) or by fast travelling to C.I.T. ruins, or by fast travelling to the institute and completing Banished from the Institute.
 * After crossing the river and eliminating all synths, Liberty Prime may get stuck and keep throwing mini-nukes on the waiting shed. You have to fast-travel to the nearby locations like Bunker Hill or Weatherby Savings & Loan to "reload" his position.
 * When Liberty Prime reaches Bunker Hill, he may turn hostile to settlers and caravans living there. It will make this quest stuck since Liberty Prime and most of named characters in Bunker Hill are essential and thus the battle continues endlessly.