Inertial Navigation Systems (INS) use Inertial methods to provide a direct means of navigation. The old Carousels and Littons widely used in the 1970s, didn't only indicate present position, ground speed etc., they could be programmed with waypoints and coupled to the autopilot to fly great circle routes between the waypoints.
Inertial Reference Systems (IRS) use Inertial methods to compute aircraft motion and position data as outputs. The inertial reference data is then used by an independent navigation system, usually the FMS coupled to the autopilot, to compute and fly a flight path automatically. Since the FMS can provide automatic monitoring and updating of the inertially computed position (i.e. by autotuned DME/DME updates) an FMS/IRS navigation system is theoretically more accurate and requires less workload than a pure INS. For long over-water flights where there are no navaids, unless the aircraft is fitted with GPS, there is not much difference in accuracy until the aircraft comes within range of at least two DMEs.
Honeywell Ring Laser Gyros are widely recognised as being used in their IRS, fitted as standard in many 1980s aircraft. Less well known is that they were also used in a self contained navigation system (by definition an INS as it had no auto update) meant for regional aircraft and which could be coupled to the autopilot. We fitted the system into our B737s for charter flights to Christmas Island. (Lack of Navaids on the route) Since we didn't couple the system to the autopilot but used it only to provide a navigational display I suppose it was a case of using an INS as an IRS!
By the way, the triangular, ring lasers described are proprietary to Honeywell (Now General Electric) Other companies, notably Litton, use rectangular ring lasers.
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Through difficulties to the cinema