Well, Melkite, I would like to post a synopsis, but the portion I was going to post doesn't deal with the entire doctrine, so it is likely that you will have many questions and objections from reading only that portion. Unless you read the whole thing, there will undoubtedly be difficulties. So I will recommend that you read as much as you like at your leisure from chapters 13-18 of this
book, particularly (at least) chapters 13, 14, and 15. I will post those three here, because they most pertain to this question, but I encourage you to read the others whenever you are able if you are interested. It is difficult to understand the manner in which all the various objections, problems, and difficulties are solved without reading the other portions.
Chapter 13: Augustine And Thomas
In his commentaries on the New Testament, St. Thomas carefully examined the principal texts regarding the Blessed Trinity, in the Synoptic Gospels, in the Gospel of St. John, and in the Epistles of St. Paul. He analyzes with special emphasis the formula of baptism, our Lord's discourse before His passion, and especially St. John's prologue. His guides throughout are the Fathers, Greek and Latin, who refuted Arianism and Sabellianism.
These scriptural studies led him to see clearly the part played by St. Augustine in penetrating into the meaning of our Lord's words on this supreme mystery. This debt of Thomas to Augustine must be our first study. We find here a very interesting and important chain of ideas. Unless we recall both the advantages and the difficulties presented by the Augustinian conception, we shall not be able to understand fully the teaching of St. Thomas.
Sabellius had denied real distinction of persons in the Trinity. Arius, on the other hand, had denied the divinity of the Son; Macedonius, that of the Holy Spirit. In refuting these opposite heresies, the Greek Fathers, resting on scriptural affirmation of three divine persons, had sought to show how this trinity of persons is to be harmonized with God's unity of nature. This harmony they found in the term "consubstantial," a term which by controversy grew more precise, and was definitively adopted by the Council of Nicaea. The Son, said the Greek Fathers, led particularly by St. Athanasius,  is consubstantial with the Father, because the Father who begets the Son communicates to that Son His own divine nature, not a mere participation in that nature. And since this Son is the Son of God, His redemptive merits have infinite value. And. the Holy Spirit, proceeding from the Father and the Son, is likewise God, consubstantial with the Father and the Son, without which consubstantiality He could not be the sanctifier of souls. .
Now these Greek Fathers thought of the divine processions rather as donations than as operations of the divine intelligence and the divine will. The Father, in begetting the Son, gives to that Son His own nature. And the Father and the Son give that divine nature to the Holy Spirit. The mode, they add, of this eternal generation and spiration is inscrutable. Further, following the order of the Apostles' Creed, they spoke of the Father as Creator, of the Son as Savior, of the Holy Spirit as Sanctifier. But their explanations left the road open to many questions.
Why are there two processions, and only two? How does the first procession differ from the second? Why is that first procession alone called generation? Why must there be one Son only? And why, in the Creed, is the Father alone called Creator, since creative power, being a characteristic of the divine nature, belongs also to the Son  and to the Holy Spirit? The Latin doctrine of appropriation is not found explicitly in the Greek Fathers.
St. Thomas, reading Augustine's work,  realized that this greatest of the Latin Fathers had taken a great step forward in the theology of the Trinity. St. Augustine's point of departure is the unity of God's nature, already demonstrated philosophically. Guided by revelation, he seeks the road leading from that unity of nature to the trinity of persons. This road, followed also by St. Thomas, is the inverse of that followed by the Greek Fathers.
In St. John's prologue, our Lord is called "the Word" and the "Only-begotten." These terms struck St. Augustine. Did they not offer an explanation of that generation which the Greek Fathers called inscrutable? The Son, proceeding from the Father, is called the Word. That divine Word is, not an exterior, but an interior word, a mental, intellectual word, spoken by the Father from all eternity. The Father begets the Son by an intellectual act, as our spirit conceives its own mental word.  But while our mental word is an accidental mode of our intellectual faculty, the divine word, like the divine thought, is substantial.  And while our spirit slowly and laboriously conceives its ideas, which are imperfect, limited, and necessarily manifold, to express the diverse aspects of reality, created and uncreated, the Father, on the contrary, conceives eternally one substantial Word, unique and adequate, true God of true God, perfect expression of all that God is and of all that God does and could do. Much light is thus thrown on the intimate mode of the Word's eternal generation. .
The saint also explains, in similar fashion, the eternal act of spiration.  The human soul, created to the image of God, is endowed with intelligence and with love. It not only understands the good, but also loves the good. These are its two highest faculties. If then the Only-begotten proceeds from the Father as the intellectual Word, we are led to think that the Holy Spirit proceeds from both by a procession of love, and that He is the terminus of this latter procession. Here, then, enter the divine relations.  The saint speaks thus: "It is demonstrated that not all predicates of God are substantial, but that some are relative, that is, as belonging to Him, not absolutely, but relatively to something other than Himself." The Father is Father by relation to the Son, the Son by relation to the Father, the Holy Spirit by relation to the Father and the Son.  This doctrine is the basis of Thomistic doctrine on the divine relations.
So far, then, we have the reason why there are two processions in God, and only two, and why the Holy Spirit proceeds, not only from the Father, but also from the Son, just as in us love proceeds from knowledge. St. Augustine, however, does not see why only the first procession is called generation, and why we are not to say that the Holy Spirit is begotten. On this point, and on many others, St. Augustine's doctrine awaits precision by St. Thomas.
A similar remark must be made on St. Augustine's doctrine concerning the question of appropriation. Starting from the philosophically demonstrated unity of God's nature, and not from the trinity of persons, he easily shows that not the Father alone is Creator, but also the Son and the Holy Spirit, since creative power is a characteristic of the divine nature, which is common to all three persons. This doctrine, through the course of centuries, becomes more precise by successive pronouncements of the Church.  St. Thomas is ever recurring to it. The three persons are one and the same principle of external operation. If then, in the Apostles' Creed, the Father is in particular called the Creator, He is so called by appropriation, by reason, that is, of the affinity between paternity and power. Similarly, the works of wisdom are appropriated to the Word, and those of sanctification to the Spirit of love. This theory of appropriation, initiated by St. Augustine,  finds final precision in St. Thomas,  and definitive formulation in the Council of Florence. .
Other difficulties still remain in St. Augustine's Trinitarian conception, difficulties which St. Thomas removes.  Here we note briefly the chief difficulties.
The generation of the Word is an intellective process. Now, since the intellective act is common to the three persons, it seems that generation, even to infinity, belongs to all three persons. St. Thomas answers. From the essential act of understanding, common to the three, we must distinguish the personal "act of speaking" (dictio): which is characteristic of the Father alone. .
A similar difficulty attends the second procession, which is the mode of love. Since all three persons love infinitely, each of them, it seems, should breathe forth another person, and so to infinity. But again, from that essential love which is common, we must distinguish, first, notional love, that is, active spiration, and secondly personal love, which is the Holy Spirit Himself. .
These distinctions are not to be found explicitly in St. Augustine. But in St. Thomas they appear as natural developments of St. Augustine's principles, in contrast to the conception prevalent in the Greek Fathers Let us note the chief advantages of this Augustino-Thomistic conception.
a) Starting from De Deo uno, it proceeds methodically, from what is better known to us to what is less knowable, the supernatural mystery of three divine persons.
b) It explains, by analogy with our own soul life, of mind and love, the number and characteristics of the divine processions, which the Greek Fathers declared to be inscrutable. Thus it gives the reason why there are two and only two processions, and why the Holy Spirit proceeds not only from the Father but also from the Son.
c) It shows more clearly why the three persons are but one single principle of operations ad extra, since divine activity derives from omnipotence, which is common to all three persons. Here lies also the reason why this mystery is naturally unknowable, since creative power is common to all three. .
These positive arguments of appropriateness show how far St. Augustine had progressed from the Greek conception, attained from a different viewpoint. The difficulties left unsurmounted by St. Augustine himself are due, not to deficient method, but to the sublimity of the mystery, whereas the difficulties in the Greek conception are due to imperfect method, which, instead of ascending from natural evidence to the mysterious, descends rather from the supernatural to the natural.
We will now examine the structure of De Trinitate as it appears in the Summa,  dwelling explicitly on the fundamental questions which virtually contain all the others. First, then, the divine processions.
Chapter 14: The Divine Processions
Following revelation, particularly as recorded in St. John's prologue, St. Thomas shows that there is in God an intellectual procession, "an intellectual emanation of the intelligible Word from the speaker of that Word." . This procession is not that of effect from cause (Arianism): nor that of one subjective mode from another (Modalism). This procession is immanent in God, but is a real procession, not merely made by our mind, a procession by which the Word has the same nature as has the Father. "That which proceeds intellectually (ad intra) has the very nature of its principle, and the more perfectly it proceeds therefrom the more perfectly it is united to its principle."  This is true even of our own created ideas, which become more perfect by being more perfectly united to our intellect. Thus the Word, conceived from eternity by the Father, has no other nature than that of the Father. And the Word is not like our word, accidental, but substantial, because God's act of knowledge is not an accident, but self-subsisting substance.
In Contra Gentes St. Thomas devotes long pages to this argument of appropriateness. The principle is thus formulated: "The higher the nature, the more intimately is its emanation united with it."  He illustrates by induction. Plant and animal beget exterior beings which resemble them, whereas human intelligence conceives a word interior to it. Yet this word is but a transient accident of our spirit, where thought follows after thought. In God, the act of understanding is substantial, and if, as revelation says, that act is expressed by Word, that Word must itself be substantial. It must be, not only the idea of God, but God Himself. .
Under this form St. Thomas keeps an ancient formula, often appealed to by the Augustinians, in particular by St. Bonaventure. It runs thus: Good is essentially self-diffusive.  The greater a good is, the more abundantly and intimately does it communicate itself.  The sun spreads light and heat. The plant, the animal, beget others of their kind. The sage communicates wisdom, the saint causes sanctity. Hence God, the infinite summit of all that is good, communicates Himself with infinite abundance and intimacy, not merely a participation in being, life, and intelligence, as when He creates stone, plant, animal, and man, not even a mere participation of His own nature, as when He creates sanctifying grace, but His own infinite and indivisible nature. This infinite self-communication in the procession of the Word reveals the intimacy and fullness of the scriptural sentence: "My Son art Thou, this day I beget Thee." .
Further,  this procession of the only-begotten  Son is rightly called generation. The living thing, born of a living thing, receives a nature like that of its begetter, its generator. In the Deity, the Son receives that same divine nature, not caused, but communicated. Common speech says that our intellect conceives a word. This act of conception is the initial formation of a living thing. But this conception of ours does not become generation, because our word is, not a substance, but an accident, so that, even when a man mentally conceives his own substantial self, that conception is still but an accidental similitude of himself, whereas the divine conception, the divine Word, is substantial, is not merely a similitude of God, but is God. Divine conception, then, is rightly called generation. Intellectual conception, purified from all imperfection, is an "intellectual generation," just as corporeal conception terminates in corporeal generation.
In this argument we have the highest application of the method of analogy. The Word of God, far from being a mere representative similitude of God the Father, is substantial like the Father, is living like the Father, is a person as is the Father, but a person distinct from the Father. .
There is in God a second procession, by the road of love, as love in us proceeds from the knowledge of good.  But this second procession is not a generation,  because love, in contrast with knowledge, does not make itself like its object, but rather goes out to its object. .
These two processions alone are found in God, as in us intelligence and love are the only two forms of our higher spiritual activity.  And in God, too, the second procession, spiration, presupposes the first, generation, since love derives from knowledge.
Further on St. Thomas  solves some difficulties inherent in St. Augustine's teaching on the divine processions. The three persons, he shows, have in common one and the same essential act of intellect, but it is the Father only who speaks the Word, a Word adequate and hence unique. To illustrate: Of three men faced with a difficult problem, one pronounces the adequate solution, while all three understand that solution perfectly. Similarly the three persons love by the same essential love, but only the Father and the Son breathe (by notional love) the Holy Spirit, who is personal love.  Thus love in God, whether essential or notional or personal, is always substantial.
Chapter 15: The Divine Relations
If there are real processions in God, then there must also be real relations. As in the order of nature, temporal generation founds two relations, of son to father and father to son, so likewise does the eternal generation of the Word found the two relations of paternity and filiation. And the procession of love also found two relations, active spiration and "passive" spiration. .
Are these relations really distinct from the divine essence? No.Since in God there is nothing accidental, these relations, considered subjectively in their inherence (esse in) are in the order of substance and are identified with God's substance, essence and existence. It follows then that the three persons have one and the same existence.  The existence of an accident is inexistence.  Now in God, this inexistence of the relations is substantial, hence identified with the divine existence, hence one and unique.
This position, so simple for St. Thomas, was denied by Suarez,  who starts from different principles on being, essence, existence, and relation. Suarez holds that even in the created order essence is not really distinct from existence, that relation, subjectively considered, in its inexistence, in its esse in, is identified with its objective essence, its esse ad. Hence the divine relations, he argues, cannot be real, unless each has its own existence. Thus he is led to deny that in God there is only one existence.  This is an important divergence, similar to that on the Incarnation, where the proposition of St. Thomas, that in Christ there is only one existence,  is also denied by Suarez.
Those divine relations which are in mutual opposition are by this very opposition really distinct one from the other.  The Father is not the Son, for nothing begets itself. And the Holy Spirit is not the Father nor the Son. Yet the Father is God, the Son is God, the Holy Spirit is God. Thus, by increasing precision, we reach the formula of the Council of Florence: In God everything is one, except where relations are opposite. .
Here enters the saint's response to an objection often heard. The objection runs thus: Things which are really identified with one and the same third thing are identified with one another. But the divine relations and the divine persons are really identified with the divine essence.  Hence the divine relations and the divine persons are identified with one another.
The solution runs thus: Things which are really identified with one and the same third thing are identified with one another; yes, unless their mutual opposition is greater than their sameness with this third thing. Otherwise I must say No. To illustrate. Look at the three angles of a triangle. Are they really distinct one from the other? Most certainly. Yet each of them is identified with one and the same surface.
Suarez,  having a different concept of relation, does not recognize the validity of this response. Instead of admitting with St. Thomas,  that the three divine persons by their common inexistence (esse in): have one and the same existence (unum esse): Suarez, on the contrary, admits three relative existences. Hence his difficulty in answering the objection just now cited. He solves it thus: The axiom that things identified with one third thing are identified with one another—this axiom, he says, is true in the created order only, but not universally, not when applied to God.
Thomists reply. This axiom derives without medium from the principle of contradiction or identity, and hence, analogically indeed, but truly, holds good also in God, for it is a law of being as such, a law of all reality, a law absolutely universal, outside of which lies complete absurdity.
Thus the doctrine of St. Thomas safeguards perfectly the pre-eminent simplicity of the Deity.  The three persons have but one existence. Hence the divine relations do not enter into composition with the divine essence, since the three persons, constituted by relations mutually opposed, are absolutely equal in perfection. .
A conclusion follows from the foregoing discussion. Real relations in God are four: paternity, filiation, active spiration, "passive" spiration. But the third of these four, active spiration, while it is opposed to passive spiration, is not opposed to, and hence not really distinct from, either paternity or filiation. .
This doctrine, perfectly self-coherent, shows the value of St. Augustine's conception, which is its foundation and guaranty.