Electrospinning of polymer nanofibers often begins with a single, straight, elongating, and electrified fluid jet that emanates from a droplet tip when the electric field at the surface is high enough. After some distance an electrically driven bending instability of the elongating jet occurs. For a polymer solution suitable for electrospinning, capillary instability does not cause the jet to become a spray of droplets. Under some conditions, a sequence of secondary jet branches emanates from the primary jet. This paper describes an experiment in which many closely spaced branches along the jet were observed during the electrospinning of a polycaprolactone solution. A theoretical description of the branching phenomenon is proposed. (c) 2005 American Institute of Physics.