  We use high-resolution capacitive dilatometry to study the low-temperature
linear magnetostriction of the bilayer ruthenate Sr$_3$Ru$_2$O$_7$ as a
function of magnetic field applied perpendicular to the ruthenium-oxide planes
($B\parallel c$). The relative length change $\Delta L(B)/L$ is detected either
parallel or perpendicular to the c-axis close to the metamagnetic region near
B=8 T. In both cases, clear peaks in the coefficient $\lambda(B)=d(\Delta
L/L)/dB$ at three subsequent metamagnetic transitions are observed. For $\Delta
L\perp c$, the third transition at 8.1 T bifurcates at temperatures below 0.5
K. This is ascribed to the effect of an in-plane uniaxial pressure of about 15
bar, unavoidable in the dilatometer, which breaks the original fourfold
in-plane symmetry.
