I reproduce the relevant section here:
3. The Meaning Of The Maxim "Outside The Church No Salvation"
It is at the precise point at which God, by the two-fold power of the apostolic hierarchy, makes contact with men that we must look for the created soul of the Church, and then go on to study the body it animates. For the created soul and the body of the Church are, of themselves, coextensive—in other words, the created soul does not extend beyond its body, nor the body beyond its soul. "The faithful" wrote St. Augustine," must become the Body of Christ if they would live by the Spirit of Christ. Understand, my brothers, what I have said. You are a man, you have a spirit and you have a body. A spirit, I say, called the soul, by which it appears that you are man, because composed of soul and body. You have then an invisible spirit and a visible body. Tell me now, which of these two lives by the other—is it your spirit that lives by your body, or your body by your spirit? Every living man will know how to reply, and if anyone cannot reply I know not whether he lives. And what does every living man reply? That it is the body, of course, that lives by the spirit. Would you then, for your part, live by the Spirit of Christ? Then be in the Body of Christ. Does my body live by your spirit? Mine lives by my spirit, and yours by yours. The Body of Christ cannot live at all, if not by the Spirit of Christ." He adds a little further on: "It is the Spirit that quickens, for it is the spirit that makes the members live. It gives life to those members only whom it finds in the body it quickens. The spirit that is in you, O man, and by which you are a man—does it then quicken any member that has been separated from your flesh? Your spirit is what I call your soul, and your soul quickens only those members that are in your body; if you cut off any one of them it soon ceases to be quickened by your soul, since it has no longer any share in the unity of your body. I say these things that you should learn to love unity and fear separation. The Christian should fear nothing so much as separation from the Body of Christ. Once he is separated he is no longer a member of Christ; and if he is not a member of Christ he is no longer quickened by the Spirit of Christ. For, says the Apostle, he who has not the Spirit of Christ, is not of Christ." The fundamental law of the coextension of the created soul and body of the Church is not contradicted by the fact that sanctifying grace may be found among the unbaptized or the non-Catholic baptized. There are two reasons for this. First of all, grace has to be sacramental and duly orientated before it contributes to the constitution of the perfect soul of the Church; and secondly, although the soul of the Church is only prefigured where the sacramental character, or sacramental grace, or orientated grace are lacking, yet the body of the Church begins to be prefigured there too. Nor is the fundamental law of the co-extension of the created soul and body of the Church contradicted by the fact that many sinners lacking grace continue to be members of the Church; for it can be said that to the extent to which they still adhere to the Church these sinners receive spiritual influences which emanate from the entire soul of the Church, which in this sense is in them by its efficiency and, as it were, dynamically.
The preachers and apologists of the nineteenth century rather lost sight of St. Augustine's great doctrine. How were they to reconcile the axiom "Outside the Church, no salvation" with the doctrine, everywhere received, that those who remain ignorant of the Church in good faith may nevertheless be in a state of grace and in the way to save their souls? Protestantism, prompt to dissociate invisible realities from visible, answered that there exists an "invisible Church" to which the just of all times belong, and a "visible Church" (or many visible Churches) which nobody is bound to enter. A certain number of Catholic writers, without wishing to dislocate the Church in this manner, imagined that her soul, i. e. sanctifying grace as they said, extended far beyond the limits of her body. They added that the just who in good faith remain ignorant of the Church, belong to the soul of the Church, and are therefore not outside her.
In the first place, however, such a mode of distinguishing the soul and body of the Church is without foundation in the authentic documents of the magisterium. It would seem to have been influenced by the Protestant conception of a "spiritual Church", distinct from the "visible Church", and its use appears to be dangerous. On the other hand we can easily see that the soul of the Church is not sanctifying grace pure and simple, as found in those who remain ignorant of the Church in good faith, but sanctifying grace as transmitted by the sacramental power and ruled by the jurisdictional power.
To reconcile the axiom "Outside the Church, no salvation", with the doctrine of the possible salvation of those who remain ignorant of the Church in all good faith, there is no need to manufacture any new theory. All we have to do is to apply to the Church the traditional distinction made in connection with the necessity of Baptism, the door by which the Church is entered. To the question: Can anybody be saved without Baptism? St. Thomas, who here draws on the thought of St. Ambrose, replies that those who lack Baptism re et voto, that is to say who neither are nor want to be baptized, cannot come to salvation, "since they are neither sacramentally nor mentally incorporated into Christ, by whom alone is salvation". But those who lack Baptism re, sed non voto, that is to say "who desire Baptism, but are accidentally overtaken by death before receiving it, can be saved without actual Baptism, in virtue of their desire for Baptism, coming from a faith that works by charity, by which God, whose power is not circumscribed by visible sacraments, sanctifies man interiorly". Conformably with this distinction we shall say that the axiom "No salvation outside the Church" is true of those who do not belong to the Church, which in herself is visible, either visibly (corporaliter) or even invisibly, either by the sacraments (sacramentaliter) or even in spirit (mentaliter); either fully (re) or even by desire (voto); either in accomplished act or even in virtual act. The axiom does not concern the just who, without yet belonging to the Church visibly, in accomplished act (re), do so invisibly, in virtual act, in spirit, by desire (mentaliter, voto), that is to say in virtue of the supernatural righteousness of their lives, even while, through insurmountable ignorance, they know nothing of the sanctity, or even of the existence, of the Church.
4. The Just "Without" Belong To The Church By Desire, Not In Accomplished Act
I do not say that there is no supernatural life at all outside the Church, but simply that there is none that does not look towards her. As preliminary to a deeper study of the soul of the Church, let us examine more closely the position of the just "outside".
They are to be found either in those groups which lack the sacraments of the New Law (paganism, Islam, Judaism, and Protestant sects such as the Quakers), or in those groups which, while separating themselves from the Church, have kept, among other good things, various genuine sacraments. (We may call them dissidents: Graeco-Russians, and traditionalist Protestants.) 
1. The just of the first category enjoy supernatural life—i. e. sanctifying grace issuing in the infused virtues and the gifts of the Holy Spirit. The whole tendency of this life is to grow towards completion, to enrich itself with those modalities which grace possesses in the sacred humanity of Christ, to open out, in a word, into that sacramental grace which, as we have seen, is a primary and fundamental element of the soul of the Church. It thus creates in those who have it a kind of living aspiration to that soul, a real and ontological desire for the Church. Men of this sort are of the Church, say the theologians, not yet re but already voto, mentaliter, by desire. Membership re and membership voto are here opposed, not as real membership to unreal, but as actual, consummated ontological membership to virtual, prefigured ontological membership, as membership in achieved act to membership in virtual act. Membership re, visible, corporeal, terminal, achieved, may be compared with membership voto, invisible, spiritual, prefigured, of desire; as the plant in flower is compared with the plant in bud; or, to take Bellarmine's comparison, as the man with the child still hidden in his mother's womb.
However reduced may be the activities of grace in such souls, they will still be in need of speculative and practical directives. They must know, for example—if they are to believe in them supernaturally—of the existence and providence of God, of the principles of morality and so on. Data of this sort are doubtless woven into the religious and cultural web within which they live, but are bound to be vitiated by endless errors. Each will have to do what he can on his own account, under the inner influence of the Holy Spirit who fails no one—though we may all too easily mistake our own voice for His—to sift the true from the false, the good from the bad. There will be omissions and inaccuracies, more or less serious according to the religious group concerned—Judaism for example or Islam being more helpful than paganism, itself a thing of many degrees. In so far as these religions shut out the truth they are instruments of darkness; but by such truths as they have retained (or perhaps regained), they may, however accidentally and imperfectly, be sources of light for millions of souls inwardly sustained by the Holy Spirit.
2. The just of the second category, the dissident groups, are in a better position. Like the rest, they belong to the Church not completely—not re—but in a way that is initial, virtual, beginning, voto. Note however that membership by desire is realized in an analogous or proportional manner; more feebly in the first category, and to a greater degree of perfection in the second, where certain genuine sacraments of the New Law have been retained, along with numerous traditional data in the speculative and practical orders.
The Graeco-Russian dissidents have kept the power of order with its three degrees: bishops, priests, and deacons. It has been perpetuated among them in virtue of the validly transmitted consecration of those who first made the schism. Thanks to the power of order the redemptive sacrifice is offered and the sacraments preserved: Baptism undoubtedly, and Confirmation, enabling the laity to partake to a certain extent of the sacerdotal power of Christ; the Eucharist too, the end of all the other sacraments, which of itself, whenever it is received with the right dispositions, tends to bestow spiritual life—not, like Baptism, in an inchoate state, but in a consummated state —and to form the Church, the Body of Christ, the "sacrament of piety, the sign of unity, the bond of charity." The just who belong to these Graeco-Russian groups truly possess, besides the triple sacramental character that enables them validly to continue the celebration of the Christian rite, that sacramental grace which though not, in isolation, the soul of the Church, is nevertheless a primary and fundamental constituent of the soul of the Church.
Those dissident groups of the Reformed in which Baptism is still validly administered—whose marriages are therefore held by the Roman Church to be authentically sacramental —can still participate, but in an attenuated way, in the sacramental benefits: the sacerdotal power of Christ is imparted to them only in Baptism, sacramental grace only in Baptism and the sacrament of Matrimony.
As to the supernatural directives needed to give sacramental grace its collective orientation and the final perfection which will make it the soul of the Church, an immanent form uniting, ruling, and vivifying the whole Mystical Body of Christ, they exist outside the Catholic Church as doctrinal patterns—much more important and closely organized in Graeco-Russian Christianity, where the process of separation has not gone so far, than in the Protestant variety, and much more important in Protestantism, where the two Testaments are respected, than in the religions of the non-baptized. That the number of the sacraments should diminish with the value of these doctrinal patterns is easy to understand. When the Protestants of England ceased to believe in the Eucharist their ordinations ceased to be valid, and the power of order lost its divine significance. Today, the Protestant modernists, who do not believe in original sin, no longer attach any great importance to the reception of Baptism. The denial of any divine power of jurisdiction, and the consequent denial of any infallible truth in dogmatic pronouncements, tends of itself to the suppression of the sacramental power.
3. In the unbaptized just, and in those of the Protestant and Graeco-Russian groups, the soul of the Church is, as it were, in formation, yet can nowhere come to fulfilment. For even where sacramental grace attains the fullness of its being and of its modalities, as among the Graeco-Russians, it lacks light, encountering directives which are not always sufficient and not always certain, neither infallibly guaranteed as a whole nor protected from the corrosive influence of modern errors; and it cannot possibly achieve that plenitude which would issue in the created soul of the Church, the immanent ruling form of the Mystical Body of Christ.
It is important to note here that when we say that the Church is in formation outside the Church, we are looking at things in a way which, from an ecclesiological standpoint, is accidental and secondary. We mean that those who broke with the Church took with them certain good things which by their very nature belong to her. In themselves, in virtue of their own internal exigencies, these scattered fragments demand to be reintegrated in the Church, and we know that the universal saving virtue of the God of mercy works mysteriously and incessantly for their reintegration. But clearly this reintegrating movement works in precisely the opposite direction to the original movement by which the dissident Churches cut themselves off from the true Church, and it can gain ground only by sapping the specific principle by which these Churches willed, and still will, to differ from the true Church. Outside the Church the Church is in formation, but this comes about accidentally, by violence done to the course things have taken. Outside the Church, the Church, of itself, is in decomposition. Any fragments of life broken off from her are no sooner detached from their native whole and subjected to the influence of the principle of dissidence, than they begin to disintegrate and decay.
Thus it is entirely right to hold that the struggle of light against darkness is the struggle of the Church against the world; but we must add that even in this world the Church has One who works for her in secret, the hidden God who mysteriously enlightens every man, whose wisdom reaches from end to end of the universe, and who does not reap where He has not sown.
Other things being equal—that is to say, supposing an equal intensity of charity everywhere—membership of the Church by desire possesses a greater and greater degree of perfection as we pass from the non-baptized just to those of the traditionalist Protestant Churches, and then to those of the Graeco-Russian Churches. But by a very disconcerting paradox, the movement of conversion to the Church is not necessarily in direct, but rather in inverse, ratio to the religious perfection of these various groups. It may be that there is some mystery here like that of the Gentiles, whose conversion en masse is to precede the entry of Israel into the Church.
5. The Different A Attitudes That May Co-Exist With Membership By Desire
Turning now from groups of believers to individual persons, we note that membership by desire—that is to say the authentic movement of charity which effectively unites a soul to the Church—may co-exist with very diverse attitudes of mind, some of which may strike the faithful as rather strange. But it is not for the faithful, or for the theologians or even for the jurisdictional authority, to be the final judge of the salvation of each particular soul. That is for God alone. There are three typical attitudes, around which we may easily group the others.
First there is that of the catechumens. They have expressly asked for Baptism and the gates of the Church, which they know to be the Body of Christ, stand open before them. Their desire for her is fully conscious and explicit.
The second attitude is that of the unbaptized child who awakes at one and the same time to the life of reason and to the life of faith, and turns to his last end with a profound aspiration which will count as Baptism by desire and will bring him to the heart of the Kingdom of God. One grown to manhood in the forests, away from the company of men, and suddenly illumined by an inner inspiration showing him what to believe, would be in a similar position. In these two cases, and others like them, the desire that saves these men, though it springs from a faith vitalized by charity, is not always accompanied by explicit knowledge of Baptism or of the Church, nor even perhaps of the Incarnation and the Trinity: the explicit content of faith then amounting to two points which, in the supereminent mystery of their riches, contain all the articles of the creed: namely that "God is, and rewards those who seek after Him" (Heb. xi. 6).
The third attitude is that of men who are aware of the existence and activity of the Church, but who, far from seeming to move towards her, show themselves ill-disposed, perhaps oppose her with all their conscious powers, even persecute her; and yet do this because of insurmountable errors for which God does not hold them responsible, sincerely convinced as they are that they work for justice and truth. Their hostility to the Church can coexist with an authentic movement of faith working by charity, which attaches them closely to the very Church that they detest, but whose sons they already are. Newman had long given up "choosing his way" and was content to be led by the divine light; yet still the Church of Rome seemed to him to be allied with Antichrist. There are more things in a man's heart than are dreamt of in his philosophy; or even, often enough, in his theology.