  In his deep and prolific investigations of heat diffusion, Lam\'e was led to
the investigation of the eigenvalues and eigenfunctions of the Laplace operator
in an equilateral triangle. In particular he derived explicit results for the
Dirichlet and Neumann cases using an ingenious change of variables. The
relevant eigenfunctions are complicated infinite series in terms of his
variables. Here we first show that boundary value problems with simple boundary
conditions, such as the Dirichlet and the Neumann problems, can be solved in an
elementary manner. In particular for these problems, the unknown Neumann and
Dirichlet boundary values respectively, can be expressed in terms of a Fourier
series. Our analysis is based on the so called global relation, which is an
algebraic equation coupling the Dirichlet and the Neumann spectral values on
the perimeter of the triangle. As Lam\'e correctly pointed out, infinite series
are inadequate for expressing the solution of more complicated problems, such
as mixed boundary value problems. Here we show, utilizing further the global
relation, that such problems can be solved in terms of {\it generalized Fourier
integrals}.
