Velocity measurements on cracks in sapphire and glass have been made using Wallner lines. Contrary to the view that crack-branching in a given material occurs at a constant critical velocity, it is found that the branching velocity can vary with the stress on the crack. On the other hand, AfCb1/2 is constant. Here Af is the stress normal to the crack and Cb is the semi-length of a double-ended crack when it branches. A theory is given in terms of the (...) condition for inducing persistent growth of a new crack ahead of the running crack. (shrink)