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Without the definite article it depersonalizes the person of the Godhead. In Matthew 28:19 if it read in the name of the Father, of Son and of Holy Spirit, it would make them none persons, but since it reads in the Name of the Father and of the Son and of the Holy Spirit. We know that they are persons and singulkar becaise in the "name" is singular, in does not read in the "Names" which would mean that they are three seperate from one another manking them three God's and not one in three person.
I hope this helps with this question that you've asked.