Here’s from the Catholic Encyclopedia. In summary:
- Solely in the Name of Christ is valid, but the full name is proper
- Using the Name of Jesus Christ invokes also the Father and the Spirit
- Biblical references to baptism in Jesus Christ are used to distinguish from the baptism of John, rather than to distinguish from the Trinity
- The Apostles would not introduce a novel form that goes against the teaching of Christ
- If Baptism should not include the Spirit in the form, then Paul’s surprise and re-baptism in Acts of those who had never heard of the Spirit makes no sense.
There has been a theological controversy over the question as to whether baptism in the name of Christ only was ever held valid. Certain texts in the New Testament have given rise to this difficulty. Thus St. Paul (Acts 19) commands some disciples at Ephesus to be baptized in Christ’s name: “They were baptized in the name of the Lord Jesus.” In Acts 10, we read that St. Peter ordered others to be baptized “in the name of the Lord Jesus Christ”. Those who were converted by Philip. (Acts 8) “were baptized in the name of Jesus Christ”, and above all we have the explicit command of the Prince of the Apostles: "Be baptized every one of you in the name of Jesus Christ, for the remission of your sins (Acts 2).
Owing to these texts some theologians have held that the Apostles baptized in the name of Christ only. St. Thomas, St. Bonaventure, and Albertus Magnus are invoked as authorities for this opinion, they declaring that the Apostles so acted by special dispensation. Other writers, as Peter Lombard and Hugh of St. Victor, hold also that such baptism would be valid, but say nothing of a dispensation for the Apostles. The most probable opinion, however, seems to be that the terms “in the name of Jesus”, “in the name of Christ”, either refer to baptism in the faith taught by Christ, or are employed to distinguish Christian baptism from that of John the Precursor. It seems altogether unlikely that immediately after Christ had solemnly promulgated the trinitarian formula of baptism, the Apostles themselves would have substituted another. In fact, the words of St. Paul (Acts 19) imply quite plainly that they did not. For, when some Christians at Ephesus declared that they had never heard of the Holy Ghost, the Apostle asks: “In whom then were you baptized?” This text certainly seems to declare that St. Paul took it for granted that the Ephesians must have heard the name of the Holy Ghost when the sacramental formula of baptism was pronounced over them.
The authority of Pope Stephen I has been alleged for the validity of baptism given in the name of Christ only. St. Cyprian says (Epistle 72) that this pontiff declared all baptism valid provided it was given in the name of Jesus Christ. It must be noted that the same explanation applies to Stephen’s words as to the Scriptural texts above given. Moreover, Firmilian, in his letter to St. Cyprian, implies that Pope Stephen required an explicit mention of the Trinity in baptism, for he quotes the pontiff as declaring that the sacramental grace is conferred because a person has been baptized “with the invocation of the names of the Trinity, Father and Son and Holy Ghost”.