Ad Victoriam

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

Detailed walkthrough
After destroying the Railroad and getting the beryllium agitator, speak to Proctor Ingram at the Boston Airport. She will now allow you to 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 that 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 makes 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

 * When prompted to install the beryllium agitator inside Liberty Prime - the hatch is closed. Fix is unknown. It's also unconfirmed if loading the previous save helped; also how far back you should load is unknown, because it's not clear what makes the hatch be closed.
 * 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 choosing to wait for Liberty Prime (I.E sleeping or sitting and waiting) he can get stuck near Monsignor Plaza rendering him unable to continue on; attempting to fix this like the waiting shed will make the problem worse as he will then reload on the roof next to him. If you haven't completed Banished from the Institute, fast travel to the Institute and finish the quest; you will then be transported to the C.I.T. ruins where you will fight the synths. Liberty Prime will then teleport to that location.
 * Collateral damage caused by Liberty Prime's nuke may make the settlers and traders in Bunker Hill turn hostile to him. If that happens, they will fight endlessly unless the player wipes out the settlement, since many of the characters living there are protected from death and thus cannot be killed by Liberty Prime's attack. Reaching Bunker Hill prior to Liberty Prime's arrival and killing any and all enemies close to the settlement will prevent this.
 * It is possible for Liberty Prime to just stand in place and not move making the quest uncompletable.

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