The third thoracic ventral spinal rami exit the vertebral column through the intervertebral foramina between the pedicles of T3 and T4.  They emerge above the transverse processes of T4 and pass into the intercostal spaces between the pleura and internal intercostal membrane (fascia), where each ramus receives at least one grey (proximal) and white (distal) rami communicantes from adjacent sympathetic ganglia.

The rami then pass below the intercostal vessels in the neurovascular plane between the third internal and innermost intercostal muscles.  Before reaching the costal angle they give rise to collateral and lateral branches.  The collateral branches follow the inferior borders of the intercostal spaces and near the sternum they pass anterior to the internal thoracic vessels, pierce the internal intercostal muscles, external intercostal membrane and pectoralis major muscles to terminate as the third anterior cutaneous nerves of the thorax.  The lateral cutaneous branches pierce the intercostal muscles obliquely and then divide into anterior and posterior branches, which pierce serratus anterior to supply the overlying skin.

Supply
The third thoracic ventral rami supply the third intercostal muscles and adjacent skin over the front and sides of the chest.