RCL Model 2804 "R-Vox" Dosimeter (late 1950s/early 1960s)
Unfortunately, I haven't been able to locate any references for this instrument. It appears to be an audible dose rate meter (sometimes referred to as a chirper). The higher the exposure rate, the higher the count rate.
The front of the case carries the instrument's name, the "R-VOX." This probably stands for "roentgen voice." In other words, it gives voice to the exposure rate. The back side of the case says: Model 2804, ORNL Q-1319, Radiation Counter Laboratories, Inc. This indicates that it was developed at Oak Ridge National Laboratory and commercialized by RCL.
The circular steel cap on the top of the case (toward the left in the photo) covers a battery compartment. The black knob on the top appears to be a sensitivity switch. It has three numbers on it, 0, 1 and 100, that might correspond to the exposure rate. Of course, it is also possible that this is a volume control.
Case: ca. 3" x 3 1/2" x 1 1/2"
The detector is almost certainly a Geiger Mueller tube. The R-Vox does incorporate one unusual feature: an internal mercury switch that would appear to turn the unit off when it is laid down. Or should it be "lay down?" I can never get those two straightened out and resorting to Bob Dylan's lyrics for guidance would probably be a mistake.