What is the precise manner of expressing the relationship of the humanity (human nature) to the divine person of the Son?

When we say God is one essence with three persons, the Greeks would say that God is one essence with three substances (hypostasis).

In the Latin church, substantia was employed to translate hypostasis and together with essentia to translate ousia: “That which must be understood of persons, according to our usage, is to be understood of substances, according to the Greek usage: for they say three substances (hypostaseis), one essence, in the same way as we say three persons, one essence or substance (essentiam vel substantiam)” (Augustine, On the Trinity 7.4)

‘Substance’ means ‘that which stands under’ that is, in the sense of an underlying substratum. The idea is that God is one in the essential divine attributes but the three persons are spiritually 'differing personalities' of which their persons, or hypostatic character, is incommunicable to each other.

hypostasis - term employed in trinitarian and christological discussions. The Greek hypostasis can have a range of meaning. In keeping with its etymological signification, it means “to stand under,” hence its usage as equivalent to the Latin substantia (substance) and essentia (essence), or to the Greek ousia. It appears to mean “substance” or “essence” in its use in Heb. 1:3 (though not all translators agree). Likewise, the Nicene Creed of 325 uses it as synonymous with ousia when it condemns those who say that the Son is a different hypostasis or ousia from the Father. On the other hand, the word can also be used to designate an individual instance of an essence, that is, a person in a trinitarian or christological context. This range of meaning occasioned confusion in the earlier trinitarian debates, exacerbated when translating between Greek and Latin. Thanks in part to the work of the Cappadocian fathers in the fourth century, the term came to be restricted to the personal distinctions in the Godhead, rather than being used to designate the essence itself. Hence, the final Greek form of the trinitarian formula became “one ousia, three hypostases,” equivalent to the Latin-based formula “one substance/essence, three persons.” The expression hypostatic union is used in christological discussions. See also essentia and hypostatic union and monophysitism and subsistantia. (Shedd, W. G. T. (2003). Dogmatic theology (A. W. Gomes, Ed.) (3rd ed.) (956). Phillipsburg, NJ: P & R Pub.)

When applying the concept of essence and substance to the Incarnation, Christ’s person is the Unity or Essence and his two natures (divine and human) are substances of his person. Just as we could say ‘the nature of essence of God subsists in three persons’, so we could say:

’The humanity subsists in the Divine Person of the Son’, or ‘Christ subsists in both a human and divine nature’.

On the other hand, ‘The divine person of the Son subsists in the humanity’ can’t be true at all. This statement would make the humanity of Christ analogous to the essence of the Trinity and making the divine nature a distinct expression of humanity! This therefore has no theological meaning that makes sense.

Something to be careful about is that these terms are just used to express different distinctions of one person. In the Trinity it refers to the distinction of the Father, Son and Holy Spirit in One person. In the Incarnation the Divine and Human nature are distinguished in the Christ after the incarnation. However, we must remember that although the Divine nature is a sense a hypostasis of Christ, it is not in anyway really ‘standing under’ anything. It is only used in this way to say these natures stand under the Christ as a person logically. However, clearly it is the divine nature and not the human that is the root of Christ’s person, or ‘standing over’ it as it were before the incarnation. The second person of the Trinity is the root into which the human nature was grafted, not the other way around. Therefore the person of Christ is in itself kind of a hypostasis of God for he is still the Eternal Son, even though his divinity could be said to be a ‘hypostasis’ on account of his person taking upon humanity through the Incarnation. In this way both the Trinity and the Incantation use hypostasis or subsistence in a non-contradictory way.