What is the function of the dippers in a slip ring liquid starter?
Sounds like you're talking about a liquid rheostat, also called an electrolytic starter in the motor starter context. 

Ancient technology, once used for stage lighting dimmers and such like. The resistance varies with the electrode area immersed, either by moving the electrode(s) or the electrolyte level (eg. via displacement), as well as the resistivity of the liquid and the spacing of the electrodes. 

R = \$\rho L\over A\$ will apply if the configuration has a constant cross section, otherwise the calculations will be more complex. 

Here is a video showing the internals of such a device, and another showing a crude hand-operating exciting device. 

Frankly I'm not 100% sure which part is considered the dipper. I suspect it's the moving electrodes. That term does not seem to be used in US based patents.