On what basis did the majority of court justices find school prayer unconstitutional

I'm thinking that we're talking here about the court case Engel v. Vitale, from 1962.  A New York state law required each school day to start with the Pledge of Allegiance and a nondenominational prayer.  Students were allowed to refrain from the prayer if they found it objectionable.  But the court decision found that the establishment of the prayer as a standard part of the school day in a public school was in violation of the First Amendment to the Constitution, which prohibits the government from making any law that establishes a religion.