Variable Stars in M13. III. The Cepheid Variables and their Relation to Evolutionary Changes in Metal-poor BL Her Stars
New CCD photometry has been combined with published and unpublished earlier observations to study the three Cepheid variables in M13: V1, V2 and V6. The light curve characteristics in $B$, $V$ and $I_{\rm C}$ have been determined and the periods updated. A period change analysis shows all three stars have increasing periods but for V1 and V2 the rate of period increase does not appear to be constant over the 118 years of observation. The observed rates of period increase are in good agreement with the predictions of the Pisa theoretical models with helium abundance $Y = 0.25$. Theory suggests V1 and V6 have masses of $\sim0.57\,M_\odot$ and are in the redward-evolving final stage of the ``blue loop'' evolutionary phase that is produced when helium-shell ignition occurs. The larger period and period change rate for V2 indicate it has a mass of $\sim0.52\,M_\odot$. A study of eighteen metal-poor BL Her stars shows the observed period changes for such objects in general can be reasonably well explained using the predictions from horizontal branch evolutionary tracks. BL Her stars with periods less than $\sim$3 d and relatively large secular period change rates ($dP/dt\approx5-15$ d/Myr) are in the evolutionary stage before He-shell ignition; the remaining cases are stars that have already experienced He-shell ignition. Moreover, an analysis of crossing time through the instability strip indicates that it is likely that few, if any, BL Her stars have a He abundance as large as $Y = 0.33$.