An action for a string and a particle with two timelike dimensions is
proposed and analyzed. Because of the new gauge symmetries and associa
ted constraints, the motion of each system in the background of the ot
her is equivalent to the effective motion with a single timelike dimen
sion. The quantum constraints are consistent only in critical dimensio
ns. For the bosonic system in a flat spacetime the critical dimension
is 27 or 28, with signature (25,2) or (26,2), depending on whether the
particle is massive or massless, respectively. For the supersymmetric
case the critical dimensions are 11 or 12, with signature (9,2) or (1
0,2), under the same circumstances. Generalizations to multiparticles,
strings, and p-branes are outlined.