  We develop the theory of equiangular lines in Euclidean spaces. Our focus is
on the question of when a Seidel matrix having precisely three distinct
eigenvalues has a regular graph in its switching class. We make some progress
towards an answer to this question by finding some necessary conditions and
some sufficient conditions. Furthermore, we show that the cardinality of an
equiangular line system in $18$ dimensional Euclidean space is at most $60$.
