And when for when, A.V.; into for in, A.V. Into the Name of the Lord Jesus (see Acts 8:16). So too Acts 10:48 of Cornelius and his company, "He commanded them to be baptized in the Name ( ἐν τῷ ὀνόματι) of Jesus Christ" (R.
V.). The formula of baptism, as commanded by the Lord Jesus himself, was, "In [or, 'into'] the Name ( αἰς τὸ ὔνομα) of the Father, and of the Son, and of the Holy Ghost" (Matthew 28:20). But the candidate always first made a profession of his faith in Jesus Christ, as in the A.
V. of Acts 8:37; and the effect of baptism was an incorporation into Christ so as to partake of his death unto sin and his life unto righteousness. It was, therefore, a true and compendious description of baptism, to speak of it as a baptism in (or into) the Name of Jesus Christ.
(See the Baptismal Service in the Book of Common Prayer.) There does not seem to be any difference of meaning between ἐν τῷ ὀνόματι and εἰς τὸ ὄνομα.