Among the many brunches of christianity there are those who baptize in the name of the Father, the Son and the Holy Spirit while others baptize in the name of Jesus only. This then becomes a point of doctrinal argument, even to the point of questioning the scripturalness of one's baptism based on the words that were said when the act of baptism took place.
Four elements are necessary for scriptural baptism. (1) a proper candidate, Acts 8:36-38. (2) A proper method (immersion), Matt. 3:16, Acts 8:38-39. (3) A proper reason, Rom 6:4-5, I Peter 3:21, Matt. 3:15. (4) a proper authority, Matt 28:18-20.
Jesus said "All power is given unto me in heaven and in earth.", Matt 28:18. "Power" means authority. Jesus has transferred that authority to his churches, Matt 28:19. He has given them the authority to make disciples, baptize them and teach them to observe his commandments. "In his name" means by his authority. The congregation is the proper authority for baptism.
Although Jesus said to baptize in the name of the Father, the Son and the Holy Ghost, we find at least three instances in the book of Acts that the churches baptized in the name of Jesus or in the name of the Lord. In Acts 8:14-16 we find "Now when the apostles which were at Jerusalem heard that Samaria had received the word of God, they sent unto them Peter and John: Who, when they were come down, prayed for them, that they might receive the Holy Ghost: (For as yet he was fallen upon none of them: only they were baptized in the name of the Lord Jesus.)" Also in Acts 10:48 the Scripture states, "And he commanded them to be baptized in the name of the Lord. Then prayed they him to tarry certain days." Again in Acts 19:5 we read, "When they heard this, they were baptized in the name of the Lord Jesus." Were the early churches, lead by the apostles unscriptural because they didn't say the right words? I believe they were not but rather baptized by the authority of Jesus or by the authority of the Lord.
Many practices have developed over the centuries. Sometimes these practices become customs and the customs become traditions. Later the traditions become law and people start to think that this is the way it has to be. The congregation, for example, must designate a person to actually perform the baptism. He baptizes by the authority of the church. This is usually the pastor but can be anyone the church designates. Since baptism is an immersion there must be sufficient water to accomplish it. This can be in a stream, river, lake, swimming pool or baptistery. It is logical to make clear to all those in attendance who is being baptized, why it is being done and under whose authority it is being performed. This promotes the testimony that baptism exhibits but there is no command in the Bible which states exactly what words should be said at the moment of baptism.
To say that the church building should be of a certain architecture or that the baptistery should be a specific size or that the one baptizing should hold up his right hand or should say a certain phrase is legalism. It is the type of activity practiced by the pharisees of whom Jesus said, "For they bind heavy burdens and grievous to be borne, and lay them on men's shoulders;...", Matt. 23:4. Jesus gave us a model prayer, Matt 6:9-13, but didn't expect us to repeat it word for word.
We should, as our baptist forefathers have done for centuries, faithfully follow the teachings of the Bible with reference to baptism but we must not entangle ourselves and divide ourselves over issues of custom and practice. I am in favor of the tradition that most of our churches have of saying something like, "I baptize you my dear brother in the name of the Father, and of the Son and of the Holy Spirit" but the presence or absence of such phrases is truly irrelevant.