The four conductor cable is the correct cable for the S-31. It should terminate in a 4-pin plug, configured like the base of a standard 4-pin tube. If the S-31 is used as an extension speaker, it plugs into the 4-pin socket on the back of the source amplifier. You can see the 4-pin socket on the back of the amplifier in the photos. If the S-31 is used as a second channel for a stereo system, it plugs into the stereo preamp in the main unit.
I have a mono Model 295 Continental that was factory modified for stereo. It uses an S-31 as the second channel. My Continental has an AMP150 amplifier and an AMP177 preamp for the second channel. My S-31 has an AMP179 amplifier, which appears to be identical to the AMP150. The 4-pin plug connects to a socket on the AMP177.
The 4-pin plug on the back of the amp is shown on the schematic right at the input jack. Two of the four wires are the hot and return for the signal, the other two wires carry 6.3 volt heater power, I don't know what for, as they are not connected at the S-31 unit.