Objection 1. It would seem that in God relation is not the same as person. For when things are identical, if one is multiplied the others are multiplied. But in one person there are several relations; as in the person of the Father there is paternity and common spiration. Again, one relation exists in two person, as common spiration in the Father and in the Son. Therefore relation is not the same as person.
Objection 2. Further, according to the Philosopher (Phys. iv, text. 24), nothing is contained by itself. But relation is in the person; nor can it be said that this occurs because they are identical, for otherwise relation would be also in the essence. Therefore relation, or property, is not the same as person in God.
Objection 3. Further, when several things are identical, what is predicated of one is predicated of the others. But all that is predicated of a Person is not predicated of His property. For we say that the Father begets; but not that the paternity is begetting. Therefore property is not the same as person in God.
On the contrary, in God "what is" and "whereby it is" are the same, according to Boethius (De Hebdom.). But the Father is Father by paternity. In the same way, the other properties are the same as the persons.
I answer that, Different opinions have been held on this point. Some have said that the properties are not the persons, nor in the persons; and these have thought thus owing to the mode of signification of the relations, which do not indeed signify existence "in" something, but rather existence "towards" something. Whence, they styled the relations "assistant," as above explained (28, 2). But since relation, considered as really existing in God, is the divine essence Itself, and the essence is the same as person, as appears from what was said above (Question 39, Article 1), relation must necessarily be the same as person.
Others, therefore, considering this identity, said that the properties were indeed the persons; but not "in" the persons; for, they said, there are no properties in God except in our way of speaking, as stated above (Question 32, Article 2). We must, however, say that there are properties in God; as we have shown (32, 2). These are designated by abstract terms, being forms, as it were, of the persons. So, since the nature of a form requires it to be "in" that of which it is the form, we must say that the properties are in the persons, and yet that they are the persons; as we say that the essence is in God, and yet is God.
Reply to Objection 1. Person and property are really the same, but differ in concept. Consequently, it does not follow that if one is multiplied, the other must also be multiplied. We must, however, consider that in God, by reason of the divine simplicity, a twofold real identity exists as regards what in creatures are distinct. For, since the divine simplicity excludes the composition of matter and form, it follows that in God the abstract is the same as the concrete, as "Godhead" and "God." And as the divine simplicity excludes the composition of subject and accident, it follows that whatever is attributed to God, is His essence Itself; and so, wisdom and power are the same in God, because they are both in the divine essence. According to this twofold identity, property in God is the same person. For personal properties are the same as the persons because the abstract and the concrete are the same in God; since they are the subsisting persons themselves, as paternity is the Father Himself, and filiation is the Son, and procession is the Holy Ghost. But the non-personal properties are the same as the persons according to the other reason of identity, whereby whatever is attributed to God is His own essence. Thus, common spiration is the same as the person of the Father, and the person of the Son; not that it is one self-subsisting person; but that as there is one essence in the two persons, so also there is one property in the two persons, as above explained (30, 2).
Reply to Objection 2. The properties are said to be in the essence, only by mode of identity; but in the persons they exist by mode of identity, not merely in reality, but also in the mode of signification; as the form exists in its subject. Thus the properties determine and distinguish the persons, but not the essence.
Reply to Objection 3. Notional participles and verbs signify the notional acts: and acts belong to a "suppositum." Now, properties are not designated as "supposita," but as forms of "supposita." And so their mode of signification is against notional participles and verbs being predicated of the properties.
Source: Summa Theologica (New Advent)