Explanation of the Prayers and Ceremonies of Holy Mass, Dom Prosper Guéranger - Memento of the Living

Memento, Domine, famulorum famularumque tuarum N. et N. ... and the Priest, joining his hands, recalls in secret, those whom he wishes to recommend to God.  Thus has the Priest first of all prayed for the whole Church in general, for the Pope, the Bishop, and all Orthodox Catholics, that is to say, all who are of the Faith of Holy Church.  But this great Sacrifice, the fruits of which are infinite, operates in a more particular manner on all those, for whom special prayer is made; therefore the Priest is allowed here to mention those whom he wishes to recommend to God more especially.  We learn from Tradition that in all ages, the Priest has been free thus to pray more expressly for those in whom he was interested, because the fruits of the Holy Sacrifice can be applied to them in particular, without prejudice to the principal intention.

Again stretching out his hands, the Priest continues his prayer, saying: Et omnium circumstantium, quorum tibi fides cognita est et nota devotio. ... The Priest prays for all those who are present around him, because their faith has urged them to leave alone everything else, and to come gathering about the Altar, and for this reason, they deserve a special share in the Holy Sacrifice.  See here, how good it is to assist at Mass as often as possible.  But if we do so, it must be with faith and devotion, for the Priest particularly says: quorum tibi fides cognita est et nota devotio.  It is quite clear that the Priest could never speak thus to God in behalf of such Christians as conduct themselves no differently in Church than they would anywhere else, who are in no way preoccupied with what is going on at the Altar, and who seem to have nothing else to do, but to distract themselves as far as they can, more or less respectably.  So then of those who are present it is only such as assist with faith and devotion that can participate in the fruits of Holy Mass.  As to those who are absent, they too can participate of the Sacrifice, by uniting themselves spiritually thereunto, and by desiring to assist thereat, with faith and devotion, were it in their power to come.  If such be their dispositions, they do really share in the fruits of the great Sacrifice, how far soever distant they may be.  Observe from all that has been said, how the Priest can have no mere personal idea, when approaching the Altar to offer Sacrifice.  He then holds the whole Church in his hands, and he prays with outstretched arms, like Christ Himself, offering Sacrifice for all men.

The Priest here adds further instance to his prayer, singling out before God, those divers persons for whom he is offering the Holy Sacrifice: pro quibus tibi offerimus, vel qui tibi offerunt hoc Sacrificium laudis.  The Church here uses this term Sacrifice of praise (though more properly applied to the Psalmody), because Holy Mass is likewise for the praise and honour of God; besides, this is a Scripture phrase, often to be met with, elsewhere.

For whom is the Sacrifice being offered  The Priest, still speaking of those whom he has mentioned, continues his thought, adding: pro se, suisque omnibus, pro redemptione animarum suarum, pro spe salutis et incolumitatis suae.  Thus does the Holy Sacrifice embrace all, extend to all.  The soul holds the first place in this enumeration; and we have here come across that petition, so frequently found in Foundation-Charta of the Middle Ages, namely, pro redemptione animarum suarum, &c.  The Church next occupies herself with the bodily needs of her children; she begs of God to keep the body safe and sound amidst all the perils by which it is surrounded.  Finally, the Priest concludes by offering to the living God, the desires and wants of all the Faithful, in these words: tibique reddunt vota sua aeterno Deo vivo et vero.

The Priest cannot here pray either for Jews or for infidels, no more than he can for heretics, who by the very fact of heresy alone, are excommunicates, and consequently out of the pale of the holy Catholic Church.  Neither can he pray for such as, without being heretics, are excommunicated for other causes; it would be a profanation to utter the names of any such in the midst of the Holy Sacrifice.  They may be prayed for in private, but not in official prayers.  They are excluded from the Sacrifice, as they are out of the Church; consequently, it is impossible to mention them during the Sacred Celebration.